PROTOTYPE
CONTEXTUALIZED
DAILY LESSON
PLANS IN
GRADE 11/12
STATISTICS AND
PROBABILITY
i
ii
LIST OF DEVELOPMENT TEAM MEMBERS
PROTOTYPE AND CONTEXTUALIZED DAILY LESSON PLANS IN GRADE
11/12 (STATISTICS AND PROBABILITY)
WRITERS
MILADEN DESPABILADERAS
ERLYN M. LACSA
ANALYN M. VELOSO
ADELFA C. DITAN
RICHELLE D. DIONEDA
MA. JECCA L. AZAS
RUEL G. FRAGO
MA. CIELO BERMUNDO
MICHAEL DOMANAIS
JASMIN A. JAO
PINKY D. DESTACAMENTO
MARKSON B. MEJIA
AMORY R. BORINGOT
KATHLEEN DUCAY
ARWIN D. BONTIGAO
NELVIN EBIO
CATHERINE GERONIMO
DEMONSTRATION TACHERS
TITO GUATNO
ERLYN M. LACSA
RUEL G. FRAGO
AMORY R. BORINGOT
MARILYN HULAR
RICHELLE DIONEDA
KATHLEEN DUCAY
PINKY O. DESTACAMENTO
JASMIN A. JAO
ANNALYN M. VELOSO
MARKSON B. MEJIA
JECCA AZAS
MILADEN DESPABILDERAS
ARWIN BONTIGO
EDITORS
MA. THERESA DUAZO
ROWENA H. BORJA
ELENA D. HUBILLA
Education Program Supervisor-1
Mathematics
MONSERAT D. GUEMO, Ph. D.
CID Chief Supervisor
MARIVIC P. DIAZ, Ed. D.
OIC, Assistant Schools Division Superintendent
Dr. NYMPHA D. GUEMO
Schools Division Superintendent
iii
iv
School
Teacher
Grade Level
Learning Area
Time & Date
Quarter
11
Statistics and
Probability
3rd
I.OBJECTIVES
A. Content
Standards
B. Performance
Objective
C. Learning
Competencies/
Objectives
( Write the LC
code for each)
II.CONTENT
III.LEARNING
RESOURCES
A. References
1.Teachers Guide
pages
2.Learners
Material Pages
3. Textbook
Pages
B. Other Learning
Resources
IV. PROCEDURE
A. Reviewing past
lesson or
Presenting the
new lesson
The learner demonstrates understanding of key concepts of
random variables and probability distributions.
The learner is able to apply an appropriate random variable for
a given real-life problem (such as in decision making and
games of chance).
 The learner illustrates a random variable (discrete
and continuous). (M11/ 12SP- IIIa-1)
 The learner distinguishes between a discrete and
continuous random variable. (M11/ 12SP – IIIa-2)
EXPLORING RANDOM VARIABLES
Statistics and probability
Rene R. Belecina, Elisa S. Bacacay, Efren B. Mateo,pp.2 –8
Worksheets
COUNTABLE or MEASURABLE?
Identify whether the given situation is countable or measurable.
The students will raise their right hand if the situation is
countable, left if it is measurable.
1. Number of students inside the classroom
2. Amount of salt needed to cook chicken tinola
3. Number of likes your recent post received
4. Capacity of an auditorium
5. Length of the chalkboard
1
B. Establishing a
purpose of the
new lesson
The teacher should have summarized the learners’ answers in
the previous activity as follows:
COUNTABLE
Number of students inside
the classroom
Number of likes your recent
post received
MEASURABLE
Amount of salt needed to
cook tinola
Capacity of an auditorium
Length of chalkboard
C. Presenting
Examples/
instances of the
new lesson
D. Discussing new
concepts and
practicing new
skills no.1.
Let the learners identify the key words identifying countable
and measurable variables.
1. The teacher will discuss what variable is.
A variable is a characteristic that is observable or
measurable in every unit of the universe. Variables can be
broadly classified as either qualitative or quantitative. And
quantitative can be classified into discrete and
continuous.
2. The students will be asked to determine the variables in the
activity they performed.
3. The teacher will explain quantitative and qualitative
variables, as well as discrete and continuous variables.
4. Let the students classify discrete and continuous variables
from the given situations in the activity.
CREATE YOUR GROUP PROFILE
To create a group profile in statistics class, the members
of each team will fill up the following data:
 NAME OF THE STUDENT
 GENDER
 AGE
 NUMBER OF SIBLINGS
 DAILY ALLOWANCE
 RELIGION
 HEIGHT IN CM
 WEIGHT IN KG
 FINAL GRADE IN GENERAL MATH SUBJECT
After gathering the data, each team will make a creative group
profile on a cartolina.
RUBRICS:
ORGANIZATION OF THE DATA – 15
CREATIVITY – 10
TOTAL: 25
The students will make a summary on the classifications of the
data gathered in their group profile through a table.
QUANTITATIVE
QUALITATIVE
Age
Name of student
Number of siblings
Gender
Daily allowance
Religion
Height in cm
Weight in kg
Final grade in General Math
2
Guide Questions:
1. When do you say that the variable is qualitative?
2. When do you say that the variable is quantitative?
3. Among the quantitative variables, which are discrete? Why?
4. Among the quantitative variables, which are continuous?
Why?
E. Discussing new
concepts and
practicing new
skills no.2
The students will classify the listed quantitative variables in the
activity CREATE YOUR GROUP PROFILE as discrete or
continuous by putting the data in the correct column.
DISCRETE
CONTINUOUS
Height in cm
Weight in kg
Final grade in Gen. Math
Age
Number of siblings
Daily allowance
The teacher will discuss what discrete and continuous
variables are.
A random variable is a discrete random variable if
its set of possible outcomes is countable.
A random variable is a continuous random variable
if it takes on values of a continuous scale. Often,
continuous random variables represent measured
data, such as heights, weights and temperatures.
F. Developing
Mastery
(Leads to
Formative
Assessment 3.)
WHAT AM I?
The students will classify the listed quantitative variables below
as discrete or continuous by putting the data in the correct
column.
1.the number of patients attributed to dengue
2. the average amount of electricity consumed per household
per month
3. the number of patient arrivals per hour at a hospital
4. the number of voters who reported for registration
5. the amount of sugar in a cup of coffee
DISCRETE
The number of patients
attributed to dengue
the number of patient arrivals
per hour at a hospital
the number of voters who
reported for registration
G. Finding
practical
application of
concepts and
skills in daily living
CONTINUOUS
The average amount of
electricity consumed per
household per month
the amount of sugar in a cup
of coffee
Make a survey regarding the use of cellphone of 5 of your
classmates using the following variables. For each of them,
classify the qualitative and the quantitative. Distinguish a
quantitative variable as to discrete or continuous.
1. Number of family members with cellphone
2. Type of ownership
3. Length (in minutes) of longest call made on each cellphone
3
4. Amount paid for cellphone load per month
H. Making
Generalization
and abstraction
about the lesson
1. How do you classify quantitative and qualitative variable?
2. How do you distinguish discrete and continuous variable?
I. Evaluating
learning
Classify whether the variable is qualitative or quantitative. If
quantitative, distinguish if discrete or continuous.
ADVANCED LEARNERS
1. the number of dropouts in
a school for a period of 10
years
2. the number of defective
computers produced by a
manufacturer per year
3. the number of points
scored in a basketball game
4. the heights of a varsity
players in a school in meters
5. the length of time spent in
playing video games in
minutes
J. Additional
activities for
application and
remediation
AVERAGE LEARNERS
1. gender of athletes for
Palarong Bikol
2. speed of a car
3. number of school days per
semester
4. the number of accidents
per year in an intersection
5. the number of deaths
attributed to lung cancer
Reflection:
In life, what are countable treasures? What are measurable
treasures? If you are to choose, which do you prefer to keep,
countable or measurable treasures? Why?
V- REMARKS
VI-REFLECTION
VII-OTHERS
A. No. of learners
who earned 80%
in the evaluation
B. No. of learners
who require
additional
activities for
remediation who
scored below
80%
C. Did the
remedial
lessons work?
No. of learners
who have caught
up with the
lesson
D. No. of learners
who continue to
4
require
remediation
E. Which of my
teaching
strategies worked
well? Why did
these work?
F. What
difficulties
did I encounter
which my
principal or
supervisor can
help me solve?
G. What
innovation
or localized
materials did I
use/discover
which I wish to
share with other
teachers?
5
School
Teacher
Grade Level
Learning Area
Time & Date
Quarter
11
Statistics and
Probability
3rd
I.OBJECTIVES
A. Content
Standards
B. Performance
Objective
C. Learning
Competencies/
Objectives
( Write the LC
code for each)
II.CONTENT
III.LEARNING
RESOURCES
A. References
1.Teachers
Guide pages
2.Learners
Material Pages
3. Textbook
Pages
B. Other Learning
Resources
IV.PROCEDURE
A. Reviewing
past lesson or
Presenting the
new lesson
The learner demonstrates understanding of key concepts of
random variables and probability distributions.
The learner is able to apply an appropriate random variable for
a given real-life problem (such as in decision making and games
of chance).
 The learner finds the possible values of a random
variable.
( M11/12SP – IIIa-3)
EXPLORING RANDOM VARIABLES
Statistics and probability
Rene R. Belecina, Elisa S. Baccay, Efren B. Mateo, pp.2 8
Each team will perform an experiment using coins and dice to
answer the following questions. An answer board is provided for
each team. Every correct answer is equivalent to 5 points. The
three teams with the highest score will be declared winners.
1. In how many ways can a coin fall?2
2. In how many ways can a die fall?6
3. In how many ways can two coins fall?4
4. In how many ways can two dice fall?36
B. Establishing a
purpose of the
new lesson
GIVE ME MY SAMPLE SPACE
Each team will complete the table by identifying the sample
space for the given event.
EVENT
1. Tossing a coin
2. Rolling a die
SAMPLE SPACE
H,T
1,2,3,4,5,6
(H,T), (H,H), (T,H), (T,T)
(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
3. Tossing two coins
4. Rolling two dice
6
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)
Guide Questions:
1. How many outcomes did you have in tossing a coin?
2. How many outcomes did you have in rolling a die?
3. How many outcomes did you have in tossing two coins?
4. How many outcomes did you have in rolling two dice?
5. How did you represent the outcomes of each event?
This activity leads you to the understanding of Sample Space
and Finding the Value of the Random Variable.
C. Presenting
Examples/
instances of the
new lesson
Two dice are rolled. Let X be the random variable representing
the 6 spots/dots that occur. Find the value of random variable
X.
SAMPLE SPACE
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
7
VALUE OF THERANDOM
VARIABLE X
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
1
1
1
6,4
6,5
6,6
1
1
2
The value of random variable X are 0,1 and 2
D. Discussing
new concepts
and practicing
new skills no.1
The teacher will discuss how to find the Value of Random
Variable.
E. Discussing
new concepts
and practicing
new skills no.2
Suppose an experiment is conducted to determine the distance
that a certain type of car will travel using 10 L of gasoline over a
prescribed test course. If distance is a random variable, can you
determine the value of random variable? Why? Why not?
Lead the students to understanding on continuous random
variable.
F. Developing
Mastery (Leads
to Formative
Assessment 3.)
Three coins are tossed. Let Z be the random variable
representing the number of heads that occur. Find the values
of the random variable Z.
SAMPLE SPACE
H,H,H
H,H,T
H,T,H
H,T,T
T,H,H
T,H,T
T,T,H
T,T,T
VALUE OF THERANDOM
VARIABLE Z
3
2
2
1
2
1
1
0
The value of the random variable Z are 0,1,2 and 3
G. Finding
practical
application of
concepts and
skills in daily
living
Suppose a cellphone buyer wants to buy four units of
cellphones. Randomly, how would he know that the cellphone
he chose is defective or not?
Let D represent the defective cellphone and N represents the
non-defective cellphone. If we let X be the random variable
representing the number of defective cellphones, show the
values of the random variable x. Complete the table below to
show the values of the random variable.
SAMPLE
SPACE
D,D,D,D
D,D,D,N
D,D,N,D
D,D,N,N
D,N,D,D
D,N,D,N
VALUE OF THE RANDOM VARIABLE X
(number of defective cellphones)
4
3
3
2
3
2
8
D,N,N,D
D,N,N,N
N,D,D,D
N,D,D,N
N,D,N,D
N,D,N,N
N,N,D,D
N,N,D,N
N,N,N,D
N,N,N,N
2
1
3
2
2
1
2
1
1
0
The value of the random variable X are 0,1,2,3and 4.
H.Making
Generalization
and abstraction
about the lesson
I. Evaluating
learning
How do you find the values of a random variable?
Find the possible values of the random variable.
ADVANCED LEARNERS
From a box containing 4 black balls and 2 green balls, 3 balls
are drawn in succession. Each ball is placed back in the box
before the next draw is made. Let G be a random variable
representing the number of green balls that occur. Find the
values of the random variable G.
POSSIBLE
VALUE OF RANDOM VARIABLE G
OUTCOMES
AVERAGE LEARNERS
A shipment of five computers contains two that are slightly
defective. If a retailer receives three of these computers at
random, list the elements of the possible outcomes using D
for defective and N for non- defective computers. To each
sample point assign a value x of random variable x of the
random variable X representing the number of computers
purchased by the retailer which are slightly defective. Find the
values of the random variable X.
POSSIBLE
OUTCOMES
VALUE OF RANDOM VARIABLE G
J. Additional
activities for
application and
remediation
9
V- REMARKS
VI-REFLECTION
VII-OTHERS
A. No. of
learners
who earned 80%
in the evaluation
B. No. of
learners
who require
additional
activities for
remediation who
scored below
80%
C. Did the
remedial
lessons work?
No. of learners
who have caught
up with the
lesson
D. No. of
learners
who continue to
require
remediation
E. Which of my
teaching
strategies worked
well? Why did
these work?
F. What
difficulties
did I encounter
which my
principal or
supervisor can
help me solve?
G. What
innovation
or localized
materials did I
use/discover
which I wish to
share with other
teachers?
10
School
Teacher
Grade Level
Learning Area
Time & Date
Quarter
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II.
III.
CONTENT
LEARNING
RESOURCES
A. References
1. Teacher’s guide
pages
2. Learner’s material
pages
3. Textbook Pages
4. Additional
materials from
learning resource
(LR) portal:
B. Other Learning
Resources
IV.
PROCEDURES
A. Reviewing previous
lesson or presenting new
lesson
11
Statistics and
Probability
3rd
The learner demonstrates understanding of key
concepts of random variables and probability
distributions.
The learner is able to apply an appropriate random
variable for a given real-life problem (such as in
decision making and games of chance).
The learner is able to:
 Illustrate a probability distribution for a discrete
random variable and its properties. M11/12SPIIIa-4
 Construct the probability mass function of a
discrete random variable and its corresponding
histogram. M11/12SP-IIIa-5
 Compute probabilities corresponding to a given
random variable. M11/12SP-IIIa-6
CONSTRUCTING PROBABILITY DISTRIBUTION
117-127
NONE
NONE
Statistics and Probability by Rene R. Belecina, Elisa S.
Bacacay, and Efren B. Mateo
Ask the learners to provide information on how many
siblings they have by asking them to raise their hands
as the teacher calls the no. of siblings they have by
starting with 0,1,2,..
No. of
Frequency
Relative
Siblings
Frequency
0
1
2
3
4
5
6
7
8
11
9
10
Total:
Draw a histogram to represent relative frequency.
Emphasize that the values on the y- axis represent
these relative frequencies (in percent). Have them add
the areas, and show that the sum is 100%. Ask them if
this is a coincidence or this is expected?
B. Establishing a purpose
for the lesson
C. Presenting
examples/instances of the
new lesson
 Present the objectives of the lesson:
 Illustrates a probability distribution for a discrete
random variable and its properties.
 Constructs the probability mass function of a
discrete random variable and its corresponding
histogram.
 Computes probabilities corresponding to a given
random variable.
 Introduce the properties of the probability.
Properties of Probability Distributions of Discrete
Random Variable
1. The probability of each value of the random
variable must be between or equal to 0 and 1. In
symbol, we write it as 0 ≤ P(x) ≤ 1.
2. The sum of the probabilities of all values of the
random variable must be equal to 1. In symbol,
we write it as ∑ P(x) = 1.
 Present the example.
Suppose three coins are tossed. Let Y be the
random variable representing the number of tails
that occur. Find the probability of each values of the
random variable Y.
Solution:
STEPS
1. Determine the
sample space.
Let H
represent head
and T
represent Tail.
2. Count the
number of tails
in each
outcome in the
sample space
and assign a
number to this
outcome.
12
SOLUTION
The sample space for this experiment
is:
S=
{TTT, TTH, THT, HTT, HHT, HTH, THH, HHH)}
Possible
Outcomes
TTT
TTH
THT
HTT
HHT
Value of the
Random Variable
Y (Number of tails)
3
2
2
2
1
3. Write the
possible values
of the random
variable Y
representing
number of tails.
Assign
probability
Values P(Y) to
each value of
the random
variable.
HTH
THH
HHH
Number
of tails
(Y)
0
1
2
3
1
1
0
Probability
P(Y)
1/8
3/8
3/8
1/8
The probability Distribution or the Probability Mass
function of Discrete Random Variable Y
Number of tails
Y
Probability P(Y)
D. Discussing new
concepts and practicing
new skills #1

0
1/8
1
3/8
2
3/8
3
1/8
The students will look for a partner and distribute a
worksheet for each pair.
Let T be a random variable giving the number of heads
in three tosses of a coin. List the elements of the
sample space S for the three tosses of the coin and
find the probability of each of the values of the random
variable T. (10 mins)
13
STEPS
1. Determine the
sample space.
2. Count the number
of heads in each
outcome in the
sample space
and assign this
number to this
outcome.
SOLUTION
S=
{TTT, TTH, THT, HTT, HHT, HTH, THH, HHH)}
Possible
Value of the
Outcome
Random
s
Variable
T(Number of
heads)
TTT
TTH
THT
HTT
HHT
HTH
THH
HHH
0
1
1
1
2
2
2
3
3. Write the number
of possible values
and assign
probability values
to each random
variable.

Number Probability of
of
P(T)
heads
(T)
0
1/8
1
3/8
2
3/8
3
1/8
Make a probability Mass Function of Discrete
Random Variable T.
Number of
Heads (T)

0
Probability
1/8
P(T)
Construct a Histogram.
14
1
2
3
3/8
3/8
1/8
 Call two volunteer pairs to share their output to the
class.
E. Discussing new
concepts and practicing
new skills #2
F. Developing mastery

The students will form 5 groups and the teacher
will provide the worksheet to each group. The
group activity is good for 15 mins.
Two balls are drawn in succession without replacement
from an urn containing 5 red balls and 6 blue balls. Let
Z be the random variable representing the number of
blue balls. Construct the probability distribution of the
random variable Z.
STEPS
1. Determine the
sample space.
2. Count the number of
blue balls in each
outcome in the
sample space and
assign this number to
this outcome.
SOLUTION
S= { RR, RB, BR, BB }
Possibl
e
Outco
mes
RR
RB
BR
BB
3. Write the number of
possible values and
assign probability
values to each
random variable.

Probabili
ty P(Z)
¼
½
¼
Make a probability Mass Function of Discrete
Random Variable Z.
Number of blue balls (Z)
Probability P(Z)

Number
of Blue
Balls (Z)
0
1
2
Value of
the
Random
Variable Z
(Number
of Blue
Balls)
0
1
1
2
Construct a Histogram.
15
0
1/4
1
1/2
2
1/4
G. Finding practical
applications of concepts,
and skills in daily living
 Let them post their output on the board and each
group will critic the output of the other group.
Practical Application
 The daily demand for copies of a newspaper at a
variety store has the probability distribution as
follows:
Number of copies X
Probability P(X)
0
0.06
1
0.14
2
0.16
3
0.14
4
0.12
5
0.10
6
0.08
7
0.07
8
0.06
9
0.04
10
0.03
What is the probability that three or more copies will be
demanded in a particular day?
0.64 or 64 %
What is the probability that the demand will be at least
two but not more than six?
0.6 or 60%
H. Making generalization
I. Evaluate learning


How do you construct probability distribution?
How do you make the histogram for a probability
distribution? Give the steps in constructing the
histogram for a probability distribution.
The following data show the probabilities for the
number of Banana Chipssold in SHS Canteen:
Number of Banana
Chips
0
1
2
3
4
5
6
7
8
9
10
a. Find P(X≤ 2) =0.5
b. Find P(X≥ 7)=0.13
c. Find P(1≤ X ≤ 5)= 0.81
16
Probability P(x)
0.100
0.150
0.250
0.140
0.090
0.080
0.060
0.050
0.040
0.025
0.015
d. Construct a histogram
V. REMARKS
VI. REFLECTION
A. No. of learners
who earned 80%
in the evaluation
B. No. of learners
who require
additional activities
for remediation
who scored below
80%
C. Did the remedial
lessons work?
No. of learners
who have caught
up with the lesson
D. No. of learners
who continue to
require
remediation
E. Which of my
teaching strategies
worked well? Why
did this works?
F. What difficulties
did I encounter
which my principal
or supervisor can
help me solve?
G. What innovation or
localized materials
did I use/discover
which I wish to
share with other
teachers?
17
School
Grade Level
Learning
Area
Quarter
Teacher
Time & Date
I – OBJECTIVES
A. Content Standards
B. Performance
Standard
C. Learning
Competencies/
Objectives3
II – CONTENT
III – LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learners’ Materials
3. Textbook pages
4. Additional Materials
from Learning
Resource (LR)
portal
5. Other Learning
Resources
IV – PROCEDURE
A. Reviewing past
lesson or
presenting the new
lesson
Eleven
Statistics and
Probability
Third Quarter
The learner demonstrates understanding of
key concepts of random variables and
probability distributions.
The learner is able to apply an appropriate
random variable for a given real-life problem
(such as in decision making and game of
chance).
 The learner illustrates and calculates the
variance of a discrete random variable.
M11/12SP-IIIb-1, M11/12SP-IIIb-2
 The learner interprets the variance of a
discrete random variable. M11/12SPIIIb-3
 The learner solves problems involving
the variance of a discrete random
variable. M11/12SP-IIIb-4
Computing the Variance of a Discrete Random
Variable
31 - 40
Statistics and Probability (Rex Book Store)
by: Rene R. Belecina
1. Start the lesson by “The Longer the
Better Game”
Mechanics:
a. Group the class into 4 or 5 groups
b. Give each group a printed pictures of
bananas
c. Each group will measure the sizes of the
bananas
d. They will compute the mean, variance
and standard deviation of the data
gathered
18
e. The group that finishes first will be the
winner.
Ask: How do you get the mean, the
variance and the standard deviation?
 (Recall that the average of a given set of
data is a measure of central tendency.
Inform them that the expected value –
being an average – measures the center
of the distribution of the possible values
of X.)
 The variance and standard deviation
describe the amount of spread,
dispersion, or variability of the items in a
distribution. So using the standard
deviation we have a “standard way of
knowing what is normal, what is extra
large or extra small.
B. Establishing a
purpose for the
new lesson
Ask:
How do you describe the spread or
dispersion in a probability distribution?
Our lesson for today will teach us how to
compute the variance and standard
deviation of a discrete probability
distribution.
C. Presenting
Present a contextualized problem to the class.
examples/instances
 The number of pentel pens sold per day
of the new lesson
at the canteen, along with its
probabilities, is shown in the table
posted on the board. Compute the
variance and the standard deviation of
the probability distribution by following
the given steps.
Number of Pentel
Probability
Pens Sold (X)
P(X)
1
0
10
2
1
10
3
2
10
2
3
10
2
4
10
19
D. Discussing new
concepts and
practicing new skill
#1
(After filling in the table, add another column on
the right and let the class subtract the mean
from the value of the random variable X)
No. of
Pentel
Pens
Sold
(X)
Probability
𝑋
∙ 𝑃(𝑋)
𝑋− 𝜇
0
0 – 2.2 =
- 2.2
P(X)
1
10
2
10
3
10
2
10
2
10
0
1
2
3
4
2
10
6
10
6
10
8
10
1 – 2.2 =
1.2
2 – 2.2 =
-0.2
3 – 2.2 =
0.8
4 – 2.2 =
1.8
(Let them square the result an write it on the
column added in the right side)
No. of
Pentel
Pens
Sold
(X)
P(X)
𝑋 ∙ 𝑃(𝑋)
𝑋
− 𝜇
0
- 2.2
4.84
1.2
1.44
-0.2
0.04
0.8
0.64
1.8
3.24
1
10
2
10
3
10
2
10
2
10
0
1
2
3
4
2
10
6
10
6
10
8
10
(𝑋
− 𝜇) 2
(Let the students multiply the result in the 5th
column by the corresponding probability P(X))
No. of
Pentel
Pens
Sold
(X)
P(X)
0
1
10
20
𝑋
∙ 𝑃(𝑋)
0
𝑋
− 𝜇
- 2.2
(𝑋
− 𝜇)2
(𝑋
− 𝜇)2
∙ 𝑃(𝑋)
4.84
0.484
1
2
3
4
2
10
3
10
2
10
2
10
2
10
6
10
6
10
8
10
1.2
1.44
0.288
-0.2
0.04
0.012
0.8
0.64
0.128
1.8
3.24
0.648
(Tell the students to get the sum in the 6the
column)
𝜎 2 = ∑(𝑥 − 𝜇 )2 ∙ 𝑃(𝑋) = 1.56
 This is now the variance of the
probability distribution
Ask:
-
How do we get the standard deviation?
 To get the standard deviation, simply
get the square of the variance.
E. Discussing new
concepts and
practicing new skill
#2
(Present the alternative Procedure in Finding
the Variance and Standard Deviation of a
Probability Distribution found in page 35 of the
textbook.)
F. Developing
Mastery
Ask:
What does the variance tell us?
How about the standard deviation?
G. Finding practical
applications of concepts
and skills in daily living
Present a sample word problem to the class.
(The teacher will decide if the activity will be
done by pair or by group)

When three coins are tossed, the
probability distribution for the random
variable X representing the number of
heads that occur is given below.
Compute the variance and the standard
deviation of the probability distribution.
21
Number of
Heads (X)
Probability P(X)
0
1
2
3
H. Making
Generalization
1
8
3
8
3
8
1
8
(Checking of output)
Ask:
-What does the variance of the probability
distribution tell us?
-How do you interpret the variance of a
probability distribution?
-How do you get the variance of discrete
random variable?
-How do you get the standard deviation of
discrete random variable?
(Solicit ideas/answers from the class and post
it on the board)
(Present the formula for the variance of the
discrete random variable)
Formula for the Variance and Standard
Deviation of a Discrete Probability Distribution
The variance of a discrete random variable with
a discrete probability distribution is given by the
formula:
𝜎 2 = ∑(𝑥 − 𝜇 )2 ∙ 𝑃 (𝑋)
The standard deviation of a discrete random
variable with a discrete probability distribution
is given by the formula:
𝜎 2 = √(𝑋 − 𝜇 )2 ∙ 𝑃(𝑋)
where:
X
= value of the random variable
P(X) = probability of the random variable X
𝜇
= mean of the probability distribution
22
I. Evaluate learning
(The teacher will distribute an activity sheet for
the evaluation)
Solve.
Find the variance and standard deviation of the
probability distribution of the random variable
X, which can take only the values 1, 2 and 3,
10
given the P(1) = 33,
1
12
P(2) = 3, and P(3) = 33.
J. Additional Activities
V. REMARKS
VI. OTHERS
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson worked? No.
of learners who
have caught up
with the lesson
D. No. of learners who
continue to require
remediation
E. Which of the
teaching strategies
worked well? Why
did these work?
F. What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
G. What innovation or
localized materials
did I used/discover
which I wish to
share with other
teachers?
23
School
Grade Level
Learning
Area
Quarter
Teacher
Time & Date
Eleven
Statistics and
Probability
Third Quarter
I. OBJECTIVES
A. Content Standards
B. Performance
Standards
C. Learning
Competencies /
Objectives
(Write the LC code for
each)
II. CONTENT
III.
LEARNINGRESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s Materials
pages
3. Textbook pages
4. Additional Materials
for Learning
B. Other Learning
Resources
The learner demonstrates understanding of key
concepts of normal probability distribution.
The learner is able to accurately formulate and solve
real-life problems in different disciplines involving
normal distribution.
The learner illustrates a normal random variable and its
characteristics.( M11/12SP-IIIc-1)
The learner construct normal curve (M11/12SP-IIIc-2)
Specific Objectives:
1. Illustrate a normal random variable
2. Determine / enumerate the characteristics of a
normal random variable / probability distribution.
3. Cite real- life examples involving normal distribution.
4. Sketch / Construct a normal curve which represents
a normal distribution.
Normal Probability Distribution and its
Characteristics


Next Century Mathematics (Statistics and
Probability) Senior High School by Jesus P.
Mercado pages 308-321
Statistics and Probability by Rene R. Belecina et.
Al. First Edition
IV. PROCEDURES
A. Reviewing
I. Ask the leader of the day to do the routinely activities:
a. lead the prayer
previous lesson
or presenting the
b. do the head counting
new lesson
c. recapitulation of the previous lesson
(the teacher thank the leader of the day’s effort)
II. Ask the students to do the activity “Let’s be United”
(refer to sheet no. 1)
24
Group Activity
Students are to form a figure out of the pieces of the
puzzle
Group1: Graph skewed to the right
Group 2:Graph skewed to the left
Group 3:Graph of a normal distribution
Group4: Sketch of a negatively skewed
Group 5: Sketch of a positively skewed
Group 6: Sketch of a normal curve
III. Class discussion:
The teacher facilitates the discussion on the different
aspects or characteristics of each graph/ sketch/ figure
through the following questions:
1. What have you formed? Say something about
the figure.
2. Is there similar graphs? In what sense?
3. If we are to group the graphs / figures you
formed, which should be together?
4. How do these grouped figures differ from the
other groups?
(for a bigger class the teacher can select
representatives to do the activity especially those who
were identified as good performers in class, then the
rest of the class observes)
B. Establishing a
purpose for the
lesson
The teacher would say: “Today we are guided by
the following objectives … “
A. The teacher presents a power point (any visual) of
the objectives of the lesson.
B. (The presentation of today’s rule during class
discussion is encouraged if any)
C. Presenting
examples/
instances of the
new lesson
A. Based on the observations from the previous
activity the teacher discusses the difference among
the positively and negatively skewed and the graph
of normal distribution.
(Mention that there are so many continuous random
variables, such as IQ scores, heights of people, or
weights have histograms that have bell-shaped
distributions.)
B. Show them the picture to let them see the real- life
application of the normal curve.
25
1. What have you noticed with the picture shown?
2. If we are to locate the middle part, what can you say
on the left or right part of the figure?
3. Is the given figure best describes a normal probability
distribution? Why?
C. Discussing
1. Let the students watch the video on normal
new concepts
distribution and its properties
and practicing
(the students has to take down notes on the
new skills #1
properties)
2. Discussion of the properties (this can be done
through the video or after watching the video)
3. (include) Draw a picture of the normal (bellshaped) curve
Emphasize the following statements about the normal
curve:
• The total area under the normal curve is equal to 1.
• The probability that a normal random variable X
equals any particular value a, P(X=a) is zero (0) (since
it is a continuous random variable).
• Since the normal curve is symmetric about the mean,
the area under the curve to the right of m equals the
area under the curve to the left of m which equals ½,
i.e. the mean m is the median.
 Emphasize also to learners that every normal
curve (regardless of its mean or standard
deviation) conforms to the following "empirical
rule" (also called the 68-95-99.7 rule):
• About 68% of the area under the curve falls within 1
standard deviation of the mean.
• About 95% of the area under the curve falls within 2
standard deviations of the mean.
• Nearly the entire distribution (About 99.7% of the
area under the curve) falls within 3 standard
deviations of the mean.
4. Explain that the graph of the normal distribution
depends on two factors: the mean m and the
standard deviation σ.
5.
D. Discussing
Sketching Normal Curve
new concepts The teacher shows the normal curve to the class and
and practicing the process on how to sketch the curve. (the teacher
new skills #2
26
should give emphasis on the properties of the normal
curve)
E. Developing
mastery
(Leads to
Formative
Assessment 3)
A. Present the properties through PPT then ask
the students to perform the activity with the
group of 10 students (any desired group size of
the teacher):
Group 1: Sketch a normal curve then label the
parts of the curve showing the properties of the
curve.(puzzle like or anything related to arts)
Group 2: Create a convo (conversation about
the properties of the normal distribution )
Group 3: Make a Jingle of about the properties
of the normal distribution.
*The teacher can add other skills/ talents of the
students ass observed by the teacher.
F. Finding
practical
Let students cite some example in real- life where
applications
of concepts
they can see the normal curve or distribution.
and skills in
daily living
G. Making
generalization
Let them answer the question;
s and
abstractions
“What are the properties of the normal distribution?”
about the
lesson
H. Evaluating
Distribute sheet 1 to the students.
Learning
V. REMARKS
VI. REFLECTION
A. No. of learners who
earned 80% in the
evaluation.
B. No. of learners who
require additional activities
for remediation who
scored below 80%.
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson.
27
D. No. of learners who
continue to require
remediation
E. Which of my teaching
strategies worked well?
Why did these work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did I
use/discover which I
wish to share with other
teachers?
Evaluation
Test I. TRUE OR FALSE:
1. The standard normal distribution is also called normal curve.
2. The area under a normal curve is 100.
3. The mean of a standard normal curve is 3.
4. The curve of a normal distribution extends indefinitely at the tails.
5. The shape of the normal probability distribution is symmetric about the mean.
Test II
Give and label each normal curve below with the correct characteristics / properties.
28
School
Grade Level
Learning
Area
Quarter
Teacher
Time & Date
Eleven
Statistics and
Probability
Third Quarter
I. OBJECTIVES
A. Content Standards
B. Performance
Standards
C. Learning
Competencies /
Objectives
(Write the LC code for
each)
II. CONTENT
III.
LEARNINGRESOURCES
A. References
Teacher’s Guide
pages
2. Learner’s Materials
pages
3. Textbook pages
4. Additional Materials for
Learning
B. Other Learning
Resources
The learner demonstrates understanding of key
concepts of normal probability distribution.
The learner is able to accurately formulate and solve
real-life problems in different disciplines involving
normal distribution.
The learner identifies regions under the normal
curve corresponding to different standard
normal values. (M11/12SP-IIIc-3)
Specific Objectives: The learner will be able to:
Read and utilize the z- table correctly.
Draw a sketch of a normal curve
Identify regions under the normal curve
corresponding to different standard normal
values.
Regions under the Normal Curve
Next Century Mathematics (Statistics and
Probability) Senior High School by Jesus P.
Mercado pages 308-321
Statistics and Probability by Rene R. Belecina
et. Al. First Edition
https://int.search.myway.com/search/AJimage.j
html?&n=7858e4f9&p2=%5E0D%5Exdm495%
5ETTAB03%5Eph&pg=AJimage&pn=4&ptb=2E
73C0C4-6C67-4509-B7E8BCE1002E511C&qs=&searchfor=normal+distri
bution+curve&si=61279913649036127991365303&ss=sub&st=tab&tpr=jrel2&trs=
wtt&ots=1570210308912&imgs=1p&filter=on&i
mgDetail=true
IV. PROCEDURES
Reviewing previous I. Ask the leader of the day to do the routinely activities:
lesson or
lead the prayer
presenting the new
do the head counting
lesson
recapitulation of the previous lesson
29
(the teacher thank the leader of the day’s effort)
II. Ask the students to recall the definition of the
standard normal curve
(A standard normal curve is a normal probability
distribution that has a mean (μ) equals 0 and a
standard deviation (σ) equals 1)
*the teacher may present the concept on the
board or any visual aid he/ she may have then
show them a picture or an example of a normal
curve with the properties of a normal
distribution / curve.
Establishing a
purpose for the
lesson
Presenting
examples/
instances of the
new lesson
I.
Discussing
new concepts
and practicing
new skills #1
A. The teacher would say: “Today we are
guided by the following objectives … “
The teacher presents a power point (any visual)
of the objectives of the lesson.
B. (The presentation of today’s rule during class
discussion is encouraged if any)
A. Let them recall the role of the standard deviation in
the normal curve.
(the distance or units at the bottom part of the
curve is the standard deviation σ)
B. The teacher presents the normal curve
divided in desired portions.
*through this, the teacher can give preview of
the lesson about the regions under the normal
curve.
The teacher present the video on “Normal
Distribution Table - Z-table Introduction”
Ask the students to perform the activity. (to
check their skill on the utilization of the z- table)
By triad: Give the corresponding area between z= 0
and each of the following: (Check the answer after 5
minutes)
P(z=0.23)
answer : 0.0910
P(z=1.09)
answer: 0.3621
P(z=2.01)
answer: 0.4778
P(z=-0.98)
answer: 0.3365
P(z=-0.03)
answer: 0. 0120
The teacher shall do the corrections/ fixing of
the mistakes committed by the students in the
utilization of the z- table. The teacher has to
inform the students of the other possible z-
30
table. (the teacher can present the different ztable trough a power point presentation)
Discussing new
concepts and
practicing new skills
#2
A. The teacher shall discuss the proportions of
areas under the normal curve through a video
presentation on “ Normal Distribution Explained Simply (part 1)” and “b Normal
Distribution - Explained Simply (part 2)”
B. GROUP ACTIVITY
In a group of 5 students, the teacher asks the
students to give their ideas of the following
statement ask them to support their answers
illustrating each situation in a normal curve:
(answer can be written in a manila paper or
through a PPT. The students have to identify
the correct statement base on the equivalent
proportions of areas under the normal curve).
1. Z – score -2 and 2 covers 95.44%
2. The area from z- score 1 to 2 is 15%.
3. The total area between z= -1 and z= +1 is
0.6826
Answers:
1. The statement is correct or true by adding
the values from the table P(Z=2)=0.4772 or
47.72% and P(Z=-2)=0.4772 or 47.72% the
sum is 0.9544 or 95.44%.
2. The area covered from z= 0 to z=+1 is
0.3413 or 34.13% and from z=0 to z= +2 is
0.4772 0r 47.72% then the difference of the
values is 0.1359 or 13.59% not 15%. Therefore,
the statement is incorrect or false.
3. The sum of the values from z= 0 to z= -1
which is 0.3413 and from z= 0 to z=+1 which is
0.3413, is 0.6826. So, the statement is true or
correct.
31
*the teacher should give the correct illustration
of each statement as she/ he checks and
explains the answers (the teacher can use a ppt
or an IM for normal curve)
**ICT INTEGRATION
If computers are available, show learners that
we could alternatively use Excel to obtain (a)
and (b). Merely enter the command =
NORMSINV(0.5832)
and generate the value of z as 0.210086 for
(a). While for (b), we enter the command
= NORMSINV(1-0.8508)
and thus find z as –1.03987.
* the teacher can explore some z- scores for further
drills on the ICT integration.
A. Developing
mastery
(Leads to Formative
Assessment 3)
B. Finding
practical
applications of
concepts and
skills in daily
living
C. Making
generalizations
and
abstractions
about the
lesson
D. Evaluating
Learning
V.
Let the students perform the activity on the areas
under the normal curve. (see attached sheet 1) ;
(different colors can be use if desired)
Ask the students to give their real- life examples of
having regions or areas or a figure parallel or related
to the lesson. (example: covered area in cleaning the
floor/ applying floor wax in an specific area / region of
the floor) * Creativity and imaginative skill of the
teacher is highly encouraged.
Present the normal curve with the common / usual
proportions under the normal curve. Let the students
give at least 1 visible proportions of the areas under
the normal curve.
Let the students perform attached sheet 2
REMARKS
VI. REFLECTION
A. No. of learners who
earned 80% in the
evaluation.
B. No. of learners who
require additional activities
for remediation who scored
below 80%.
32
C. Did the remedial lessons
work? No. of learners
who have caught up with
the lesson.
D. No. of learners who
continue to require
remediation
E. Which of my teaching
strategies worked well?
Why did these work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did I
use/discover which I wish
to share with other
teachers?
33
Sheet 1
Shade the normal curves with its given corresponding z- score
then identify the proportions of areas under the normal curve.
Greater than z= 1.065
Between z= -2. 08 and
z= 0.78.
Between z= 0. 12 and
z= 1.96
P(-1.53 < 0.45)
From mean (μ) to z= 2.05
Between z=-1.12 and z= 1.12
34
Sheet 2
EVALUATION
A. Determine the proportions of the areas under the following normal
curves.
1.
2.
-2
1
-1
3.
1
2
4.
-2
-0.64
-2
-1
B. Illustrate and give the proportions of the regions under the normal
curve with the following z- scores.
5. 𝑧 = 0.38 ; 𝑓𝑟𝑜𝑚 𝑧 = 0
6. 𝑧 = −1.29 ; 𝑓𝑟𝑜𝑚 𝑧 = 0
7. 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑧 = 2.01 𝑎𝑛𝑑 𝑧 = 2.93
−0.67
8. 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑧 =
9. 𝑏𝑒𝑙𝑜𝑤 𝑧 = 1.37
10. 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑜𝑓 𝑧 = 2.03
35
2 2.5 2
School
Grade Level
Learning
Area
Quarter
Teacher
Time & Date
Eleven
Statistics and
Probability
Third Quarter
I. OBJECTIVES
A. Content Standards
B. Performance
Standards
C. Learning
Competencies /
Objectives
(Write the LC code for
each)
II. CONTENT
III.
LEARNINGRESOURCES
A. References
Teacher’s Guide
pages
2. Learner’s Materials
pages
3. Textbook pages
The learner demonstrates understanding of key
concepts of normal probability distribution.
The learner is able to accurately formulate and solve
real-life problems in different disciplines involving
normal distribution.
The learner converts a normal random variable
to a standard normal variable and vice versa.
(M11/12SP-IIIc-4)
Specific Objectives: The learner will be able to:
1. Find the z- value that corresponds to a score X
2. Utilize/ use z- table independently and correctly.
3. Convert a normal random variable to a standard
normal variable and vice versa
4. Sketch the normal curve with convert a normal
random variable to a standard normal variable
and vice versa
Conversion of a normal random variable to a
standard normal variable and vice versa
Next Century Mathematics (Statistics and Probability)
Senior High School by Jesus P. Mercado pages 308321
Statistics and Probability by Rene R. Belecina et. Al.
First Edition
4. Additional Materials
for Learning
B. Other Learning
Resources
IV. PROCEDURES
Reviewing
I. Ask the leader of the day to do the routinely activities:
previous lesson or
lead the prayer
presenting the new
do the head counting
lesson
recapitulation of the previous lesson
(the teacher thank the leader of the day’s effort)
36
II. Ask the students to recall the properties of a normal
curve/ distribution through the activity:
A. Sketch the normal curve with the following
properties
1. μ= 0, σ=1
2. μ= 35, σ=3
3. μ=98, σ= 2.5
4. μ=105, σ= 4
5. μ= 100, σ= 20
*the teacher shall focus on the baseline of the
normal curve and the standard deviation. The
distance of each unit should be reviewed.
Establishing a
purpose for the
lesson
A. The teacher presents the objectives of the
lesson through a power point presentation.
B. The teacher presents a normal curve with the
converted raw scores. Let the students
determine the μ and σ.
Presenting
examples/
instances of the
new lesson
A. Let them recall the role of the standard deviation in
the normal curve.
(the distance or units at the bottom part of the curve is
the standard deviation σ)
J. Discussing
The teacher presents the video on the derivation of
new concepts
the formula
and practicing
new skills #1
The areas under the normal curve are given in terms
of z- values or scores. Either the z- score locates X
within a sample or within a population.
The formula for calculating z is :
For population data
For sample data
𝑋−𝜇
𝑧=
𝜎
Where :
X- given measurement
μ- population mean
σ- population standard deviation
37
𝑧=
𝑋 − 𝑋̅
𝑠
s- sample standard deviation
X- sample mean
Discussing new
concepts and
practicing new
skills #2
*raw scores may be composed of large values, but
large values cannot be accommodated at the base
line of the normal curve. So, they need to be
converted into scores for convenience without
sacrificing the meaning associated to the raw score.
A. Group Activity: Solve for the equivalent z- score
of the problem assigned to your group, then sketch
the normal curve showing the calculated z- score that
corresponds to the raw score X.
Group 1- Given the mean μ= 60 and the standard
deviation σ= 5 of a population, find the z- value that
corresponds to score X= 54.
Group 2- Given the mean μ= 78 and the standard
deviation σ= 13 of a population, find the z- value that
corresponds to score X= 88.
Group 3- Given the mean μ= 45 and the standard
deviation σ= 3 of a population, find the z- value that
corresponds to score X= 40.
Group 4- Given the mean μ= 128 and the standard
deviation σ= 2.6 of a population, find the z- value that
corresponds to score X= 131.
Group 5- Given the mean μ= 155 and the standard
deviation σ= 6.5 of a population, find the z- value that
corresponds to score X= 147.5.
B. Let them present their output through a
manipulative normal curve made – up of
cardboard.
*The teacher can make her/ his own rubrics
according to the ability of the students.
*the teacher should give the correct illustration of
each statement as she/ he checks and explains
the answers (the teacher can use a ppt or an IM
for normal curve)
E. Developing
mastery
(Leads to
Formative
Assessment 3)
A. By triad. : Give the missing value;
1.
2.
3.
4.
5.
B.
X= 23, μ=32, σ=8, z=?
μ=231, σ= 120, X= 250, z=?
μ=127, σ= 5, X= 98, z=?
μ=450, σ= 15, z=-1.5, X=?
σ= 5, X= 98, z=2.21, μ=?
Checking of the answer may be done through a
quick check where the teacher will give the
answers or if the students seem to be slow in
understanding the concept, the solution of each
problem shall be presented.
38
F. Finding
Ask the students to give their real- life examples of
practical
having small or large things which need to be converted
applications
just to fit in an actual scenario
of concepts
* Creativity and imaginative skill of the teacher is highly
and skills in
encouraged.
daily living
G. Making
Ask the students to give the summary of the
generalization
lesson.
s and
abstractions The teacher shall present the formulae to the students
about the
through a PPT.
lesson
The formula for calculating z is :
For population data
For sample data
𝑋−𝜇
𝑧=
𝜎
𝑧=
Where :
X- given measurement
μ- population mean
σ- population standard deviation
s- sample standard deviation
X- sample mean
H. Evaluating
Learning
Let the students perform attached sheet 1
V. REMARKS
VI. REFLECTION
A. No. of learners who
earned 80% in the
evaluation.
B. No. of learners who
require additional activities
for remediation who
scored below 80%.
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation
E. Which of my teaching
strategies worked well?
Why did these work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
39
𝑋 − 𝑋̅
𝑠
G. What innovation or
localized materials did I
use/discover which I
wish to share with other
teachers?
EVALUATION
Shade the normal curves with its corresponding z- score after
converting the raw score to its standard normal variable.
X= 69
μ=75
σ= 14
z=?
X= 219
μ=200
σ=21
z=?
X= 950
μ=1000
σ=25
z=?
X= 12
μ=20
σ=6.5
z=?
Solve for the missing value.
X= 250
σ=15.5
z=1.65
μ=?
40
X= 100
z=-0.98
μ= 112
σ=?
School
Grade Level
Learning
Area
Quarter
Teacher
Time & Date
Eleven
Statistics and
Probability
Third Quarter
I. OBJECTIVES
The learner demonstrates understanding of key
concepts of normal probability distribution.
The learner is able to accurately formulate and solve
B. Performance Standards real-life problems in different disciplines involving
normal distribution.
The learner computes probabilities and
percentiles using the standard normal table.
(M11/12SP-IIIc-d1)
C. Learning
Specific Objectives: The learner will be able to:
Competencies /
5. Recall the concept on the reading of
Objectives
probabilities on the z- table.
(Write the LC code for
6. Find the z- scores when probabilities are
each)
given.
7. Computes the probabilities and percentiles
using the standard normal table.
A. Content Standards
II. CONTENT
Locating Percentiles Under the Normal Curve
III.
LEARNINGRESOURCES
A. References
Teacher’s Guide
pages
2. Learner’s Materials
pages
3. Textbook pages
4. Additional Materials for
Learning
B. Other Learning
Resources
Next Century Mathematics (Statistics and
Probability) Senior High School by Jesus P.
Mercado pages 308-321
Statistics and Probability by Rene R. Belecina
et. Al. First Edition
a.
https://www.google.com/search?q=percentile
&oq=percentile&aqs=chrome..69i57j0l5.2828j
0j9&sourceid=chrome&ie=UTF-8
b.
IV. PROCEDURES
A. Reviewing previous
lesson or presenting the
new lesson
I. Ask the leader of the day to do the routinely
activities:
lead the prayer
do the head counting
recapitulation of the previous lesson
41
(the teacher thank the leader of the day’s effort)
II. Ask the students to recall the process on how to
read values from the z- table by asking the students to
give the equivalent probability of the following; (this
can be a quiz bee with the help of the power point)
1. z= 0.12
2. z=-2.13
3. z=1.28
4. z=2.48
5. z=-0.87
6. z=-1.24
7. z= -2.09
8. z= 2.01
9. z= 1.72
10. z= 0.04
B. Establishing a purpose
for the lesson
C. Presenting examples/
instances of the new lesson
D. Discussing new concepts
and practicing new skills #1
C. The teacher presents the objectives of the
lesson through a power point presentation.
D. The teacher ask: “Which of the following are
familiar to you?”
a. First Honor
b. Top five
c. Eliminated candidates are the below 10%
d. Scholars are the top two
e. Remediation session is for students at the
bottom 5.
*the teacher shall ask the students to give the
meaning of each situation above.
A. (optional) the teacher can make a huge
normal curve and ask the students to stand on
the position of the following:(this can be done
by group)
1. Above z= 2.00
2. Below z = 0.08
3. More than z= 1.54
4. Less than or equal to z=-1.34
5. To the right of z= 0.49
The teacher presents and asks the opinion of
the class about the picture.
42
*the idea of the percentile shall be given
emphasis and be defined.
Percentile - each of the 100 equal groups into
which a population can be divided according to
the distribution of values of a particular
variable.
A percentile is a measure used in statistics
indicating the value below which a given
percentage of observations in a group of
observations fall
E. Discussing new concepts
C. The teacher shall present the following
and practicing new skills #2
considerations or important things to
remember when we are given probabilities
and we know their corresponding z- scores.
1. A probability value corresponds to an area
under the normal curve.
2. In the Table of Areas Under the Normal Curve,
the numbers in the extreme left and across the
top are z- scores, which are the distances
along the horizontal scale. The numbers in the
body of the table are areas or probabilities.
3. The z- scores to the left of the mean are
negative values.
D. Group Activity:
Ask the students to sketch the following:
Group 1: P25
Group 2: P65
Group 3: P88
Group 4: P90
Group 5: P98
E. Let them give the meaning of the assigned
percentile to their group.
F. Ask them to present the illustrations(for the
wrong sketch the teacher should check or
correct the illustration)
G. Discussion of how to determine the z- score of
every percentile.
The 95th percentile is z= 1.645
 .95/2 = 0.45 where there is no exact
0.45 in the table so therefore we get
the nearest values z=1.65 (0.4505)
and the z= 1.64 (0.4495) by
interpolation the value now is z=
1.645.
H. Ask the students to give the z – score of their
assigned percentile as stated above (B)
F. Developing mastery
(Leads to Formative
Assessment 3)
A. Let them perform the activity by pair:
1. Find the upper 10% of the normal curve. Illustrate
the normal curve.
43
2. The results of a nationwide aptitude test in
Mathematics are normally distributed with m=80
and s= 15. What is the percentile rank of a score
84?
I. Check their answer and resolve the
misconceptions committed by the students.
G. Finding practical
applications of concepts and
skills in daily living
H. Making generalizations
and abstractions about the
lesson
I. Evaluating Learning
Ask them to give their own example of the percentile
rank (students can mention their rank after taking the
quiz or any test they had)
Is a normal curve useful in visualizing the positions of
the scores or the rank? Why do you think so? Write
your thoughts in a piece of paper.
Let the students perform attached sheet 1
V. REMARKS
VI. REFLECTION
A. No. of learners who
earned 80% in the
evaluation.
B. No. of learners who
require additional activities
for remediation who scored
below 80%.
C. Did the remedial lessons
work? No. of learners who
have caught up with the
lesson.
D. No. of learners who
continue to require
remediation
E. Which of my teaching
strategies worked well?
Why did these work?
F. What difficulties did I
encounter which my
principal or supervisor can
help me solve?
G. What innovation or
localized materials did I
use/discover which I wish
to share with other
teachers?
44
EVALUATION
1. Sketch the 85th percentile.
2. Present the procedure in calculating the P99 of the normal curve then draw.
3. What is the percentile rank of a score of 56 from the normally distributed NAT
results with mean of 75 and σ= 20. Draw.
45
School
Grade Level
Learning
Area
Quarter
Teacher
Time & Date
Eleven
Statistics and
Probability
Third Quarter
I.
OBJECTIVES
A. Content
The learner demonstrates understanding of key concepts of
Standard
sampling ad sampling distributions of the sample mean.
B. Performance
Standard
The learner is able to apply suitable sampling and sampling
distributions of the sample mean to solve real-life problems
in different disciplines.
C. Learning
Competency/Obj
ectives
The learner illustrates random sampling.
M11/12SP-IIId-2.
II.
CONTENT
Random Sampling
III.
LEARNING RESOURCE
References
1. Jose Dilao S., Orines F and Bernabe J. (2009).
Advanced Algebra, Trigonometry and Statistics. SD
Publications, Inc. pp 234-236.
2. Ocampo J. & Marquez W. (2016). Senior High
Conceptual Math and Beyond Statistics and Probability.
Brilliant Creations Publishing, Inc. pp.86-93.
Other Learning
https://www.youtube.com/watch?v=xh4zxC1OpiA
Resource
IV.
PROCEDURES
A. Reviewing
Recall from our study of probability that the number of
previous lessons combinations of n objects taken r at a time is obtained by
or presenting the using the formula.
new lesson
𝑛!
C (n, r) = (𝑛−𝑟)!𝑟! 𝑤ℎ𝑒𝑟𝑒 𝑛 ≥ 𝑟
Evaluate the following:
1.C (5, 3)
2. C (10, 4)
3. C (9, 6)
4. C (8, 2)
5. C (7, 6)
B. Establishing a
purpose for the
lesson
The students will explain their solutions.
To prepare the students in the lesson, activities are as
follows:
A.A sample of investment experts was asked to give their
opinion as to where they would invest their money. The
following are their responses.
Stocks
Real estate
Real estate
Precious metals Art
Precious metals
Real state
Precious metals Commodities
Art
Precious metals Foreign money
46
Precious metals
Stocks
Stocks
Real estate
Commodities
Stocks
Real estate
Commodities
Commodities
Foreign money
Stocks
Stocks
Real estate
Stocks
Real estate
Stocks
Real estate
Precious metals
Real estate
Real estate
Foreign money
Construct a table to show the frequency distribution of the
given responses.
Types of Investment
C. Presenting
Examples/Instan
ces of the
Lesson
Frequency
Norma wants to know the common number of children her
classmates’ families have. Which of the following samples is
a good representation of the class? Why?
1.A sample consisting of Norma’s friends
2.A sample consisting of students belonging to rich families.
3. A sample consisting of students whose names were drawn
from a box all the names of students in Norma’s class.
Wrong conclusion may be inferred from samples given in
numbers 1 and 2. This sample will not represent the correct
number of children the families of Norma’s classmates have.
The sample in a number 3 in the best representation of the
class.
This is idea of representativeness leads to the importance of
random sampling, a method of drawing out a sample from a
population without a definite plan, purpose, or pattern.
D. Discussing New
concepts and
Practicing New
Skills # 1
E. Developing
Mastery
Let students analyze the video in the
https://www.youtube.com/watch?v=xh4zxC1OpiA
link-
After watching the video presentation, the students will
define random sampling and state its uses.
Group activity for 10 minutes. The students are task
to:
1. Create problem that involves random sampling.
2. Construct a table that show frequency distribution of
the samples.
3. What learning discovered in doing such activity?
Would you be able to use this in your life? How and
why?
47
The rubrics will be used in scoring the performance of the
group.
Categor
4
3
2
1
ies
Excellen Satisfact Developi
Beginning
t
ory
ng
Mathe
Demons Demons Demonstr Shows lack
matical trates a trates a ates
of
Concep thoroug satisfact incomplet understandin
t
h
ory
e
g and have
underst underst understa severe
anding
anding
nding
misconceptio
of the
of the
and has
ns.
topic
uses it
some
and
to
misconce
uses it
simplify
ptions.
accurate the
ly to
problem
solve
.
the
problem
Accura All
The
Generally Errors in
cy of
computa computa , most of computations
comput tion are tion are the
are severe.
ation.
correct
correct. computati
an are
ons are
logically
not
present
correct.
ed
Organiz Highly
Satisfact Somewh Illogical and
ation of organize orily
at
obscure. No
the
d, flows organize cluttered. logical
report
smoothl d.
Flow is
connections
y, and
Sentenc not
of ideas.
observe e flow is consisten Difficult to
s logical generall tly
determine
connecti y
smooth,
the meaning.
ons of
smooth
appears
points.
and
disjointed
logical.
.
Particip
ation of
the
membe
rs
All
member
s take
part in
the
activity,
support
and
encoura
ge
others in
the
group
member
48
Almost
90-99%
take
part in
the
activity,
support
and
encoura
ge
others in
the
group
member
Almost
80-89%
take part
in the
activity,
support
and
encourag
e others
in the
group
members
do not
find fault
Almost 7079% take
part in the
activity,
support and
encourage
others in the
group. There
isa conflict
relationship
in doing the
activity.
F. Making
generalization
and abstraction
about the lesson
G. Evaluating
Learning
s do not s do not in one
find fault find fault another,
in one
in one
open to
another, another, comment
open to open to s and
commen commen criticism.
ts and
ts and
criticism criticism
.
.
What is random sampling?
Random sampling is a method by which every
element of a population has a chance of being
included in a sample. That is, the elements that
compose the sample are taken without purpose. The
more elements in the sample, the better the chances
of getting a true picture of the whole population.
Determine whether the following is a random sample or not.
Explain your answer.
1.To select the students to attend the summer workshop in
Sorsogon, the teacher told her class to count off, and then
selected those even-numbered students for the workshop.
2. To study the average number of years a family has stayed
in Barangay Guinlajon, the barangay captain chose to
interview the families around his residence.
3. To find the average number of dengue victims in hospitals
per day, a researcher made a list of all hospitals in Sorsogon
Province, and then selected every fifth in the list.
4. A survey of the prevailing cost of rice was undertaken in
the seven key cities of the country.
5. To select students for MTAP competition, the school
math coordinator decided to screen competitive students
from junior high school.
V. REMARKS
VI. REFLECTION
A. No. of learners
who earned 80% in
the evaluation.
B. No. of learners
who require
additional activities
for remediation who
scored below 80%.
C. Did the remedial
lessons work? No.
of learners who
have caught up with
the lesson.
D. No. of learners
who continue to
require remediation
49
E. Which of my
teaching strategies
worked well? Why
did these work?
F. What difficulties
did I encounter
which my principal
or supervisor can
help me solve?
G. What innovation
or localized
materials did I
use/discover which I
wish to share with
other teachers?
50
School
Grade Level
Learning
Area
Quarter
Teacher
Time & Date
Eleven
Statistics and
Probability
Third Quarter
I.
OBJECTIVES
A. Content
The learner demonstrates understanding of key concepts of
Standard
sampling ad sampling distributions of the sample mean.
B. Performance
Standard
The learner is able to apply suitable sampling and sampling
distributions of the sample mean to solve real-life problems in
different disciplines.
C. Learning
Competency/
Objectives
The learner distinguishes between parameter and statistic.
M11/12SP-IIId-3.
II. CONTENT
Parameter and Statistic
III. LEARNING RESOURCE
References
3. Ocampo J. & Marquez W. (2016). Senior High Conceptual
Math and Beyond Statistics and Probability. Brilliant
Creations Publishing, Inc. pp.86-93.
2. Supplementary Statistics Topics. Retrieved from
https://www2.southeastern.edu/Academics/Faculty/dgurne
y/Math241/StatTopics.html
5. Surbhi (2017). Difference Between Statistic and Parameter
Retrieved
from
https://keydifferences.com/differencebetween-statistic-and-parameter.html
Other Learning
https://www.youtube.com/watch?v=M-L8C2aOf7E
Resource
IV.
PROCEDURES
A. Reviewing
Jumble the letters that corresponds to the given definition.
previous
1. AATD- facts and statistics collected together for reference or
lessons or
analysis.
presenting the 2. NIOTALUPOP- an aggregate observation of subjects
new lesson
grouped together by a common feature
3. ELPMSA- a small part or quantity intended to show what the
whole is like.
4.UAIESMMRZ- give a brief statement of the main points of
(something).
5. PRMTRSAAEE- a numerical or other measurable factor
forming one of a set that defines a system or sets the conditions
of its operation.
B. Establishing a
purpose for
the lesson
Let students analyze the given definition and comparison
chart of statistic and parameter
In statistics vocabulary, we often deal with the terms parameter
and statistic, which play a vital role in the determination of the
sample size. Parameter implies a summary description of the
characteristics of the target population. On the other extreme,
51
the statistic is a summary value of a small group of population
i.e. sample.
-Definition of Statistic
A statistic is defined as a numerical value, which is obtained
from a sample of data. It is a descriptive statistical measure and
function of sample observation. A sample is described as a
fraction of the population, which represents the entire
population in all its characteristics. The common use of statistic
is to estimate a particular population parameter.
From the given population, it is possible to draw multiple
samples, and the result (statistic) obtained from different
samples will vary, which depends on the samples.
-Definition of Parameter
A fixed characteristic of population based on all the elements of
the population is termed as the parameter. Here population
refers to an aggregate of all units under consideration, which
share common characteristics. It is a numerical value that
remains unchanged, as every member of the population is
surveyed to know the parameter. It indicates true value, which
is obtained after the census is conducted
C. Presenting
Examples/Inst
ances of the
Lesson
The students will distinguish the parameter and statistic in the
given statements.
1.A researcher wants to know the average weight of females
aged 22 years or older in Sorsogon. The researcher obtains
the average weight of 54 kg, from a random sample of 40
females.
-Solution: In the given situation, the statistics are the average
weight of 54 kg, calculated from a simple random sample of
40 females, in Sorsogon while the parameter is the mean
weight of all females aged 22 years or older.
2.A researcher wants to estimate the average amount of water
consumed by male teenagers in a day. From a simple random
sample of 55 male teens the researcher obtains an average of
1.5 litres of water.
52
-Solution: In this question, the parameter is the average
amount of water consumed by all male teenagers, in a day
whereas the statistic is the average 1.5 litres of water
consumed in a day by male teens, obtained from a simple
random sample of 55 male teens
D. Discussing
New concepts
and Practicing
New Skills # 1
E. Developing
Mastery
Let the students analyze the video in
https://www.youtube.com/watch?v=M-L8C2aOf7E
the
link-
After watching the video presentation, the students will reflect
to the difference between parameter and statistic and connect
it to real life.
Group activity for 10 minutes. The students are task to:
4. Create statements that involves parameter and
statistic.
5. What learning discovery did you found useful in your
daily life activities?
The rubrics will be used in scoring the performance of the
group.
Categories
4
Excellent
3
Satisfactory
2
Developing
1
Beginning
Mathem
atical
Concept
Demonstr
ates a
thorough
understa
nding of
the topic
and uses
it
accuratel
y to solve
the
problem
All
computati
on are
correct
an are
logically
presente
d
Highly
organize
d, flows
smoothly,
and
observes
logical
connectio
ns of
points.
Demonstr
ates a
satisfacto
ry
understa
nding of
the uses
it to
simplify
the
problem.
Demonstra
tes
incomplete
understan
ding and
has some
misconcep
tions.
Shows
lack of
understan
ding and
have
severe
misconcep
tions.
The
computati
on are
correct.
Generally,
most of
the
computatio
ns are not
correct.
Errors in
computatio
ns are
severe.
Satisfact
orily
organize
d.
Sentence
flow is
generally
smooth
and
logical.
Somewhat
cluttered.
Flow is not
consistentl
y smooth,
appears
disjointed.
Illogical
and
obscure.
No logical
connection
s of ideas.
Difficult to
determine
the
meaning.
Accurac
y of
computa
tion.
Organiza
tion of
the
report
53
Participa
tion of
the
member
s
All
members
take part
in the
activity,
support
and
encourag
e others
in the
group
members
do not
find fault
in one
another,
open to
comment
s and
criticism.
Almost
90-99%
take part
in the
activity,
support
and
encourag
e others
in the
group
members
do not
find fault
in one
another,
open to
comment
s and
criticism.
Almost 8089% take
part in the
activity,
support
and
encourage
others in
the group
members
do not find
fault in one
another,
open to
comments
and
criticism.
Almost 7079% take
part in the
activity,
support
and
encourage
others in
the group.
There isa
conflict
relationshi
p in doing
the
activity.
F. Making
generalization
and
abstraction
about the
lesson
Differentiate parameter to statistic.
G. Evaluating
Learning
Problems (1) through (6) below each present a statistical
study*. For each study, identify both the parameter and the
statistic in the study.
-Parameters are numbers that summarize data for an entire
population. Statistics are numbers that summarize data from a
sample, i.e. some subset of the entire population
1) A researcher wants to estimate the average height of women
aged 20 years or older. From a simple random sample of 45
women, the researcher obtains a sample mean height of 63.9
inches.
2) A nutritionist wants to estimate the mean amount of sodium
consumed by children under the age of 10. From a random
sample of 75 children under the age of 10, the nutritionist
obtains a sample mean of 2993 milligrams of sodium
consumed.
3) Nexium is a drug that can be used to reduce the acid
produced by the body and heal damage to the esophagus. A
researcher wants to estimate the proportion of patients taking
Nexium that are healed within 8 weeks. A random sample of
224 patients suffering from acid reflux disease is obtained, and
213 of those patients were healed after 8 weeks.
4) A researcher wants to estimate the average farm size in
Kansas. From a simple random sample of 40 farms, the
researcher obtains a sample mean farm size of 731 acres.
54
5) An energy official wants to estimate the average oil output
per well in the United States. From a random sample of 50 wells
throughout the United States, the official obtains a sample
mean of 10.7 barrels per day.
6) An education official wants to estimate the proportion of
adults aged 18 or older who had read at least one book during
the previous year. A random sample of 1006 adults aged 18 or
older is obtained, and 835 of those adults had read at least one
book during the previous year.
J. Additional
activities for
application or
remediation
V. REMARKS
VI. REFLECTION
A. No. of
learners who
earned 80% in
the evaluation.
B. No. of
learners who
require
additional
activities for
remediation
who scored
below 80%.
C. Did the
remedial
lessons work?
No. of learners
who have
caught up with
the lesson.
D. No. of
learners who
continue to
require
remediation
E. Which of my
teaching
strategies
worked well?
Why did these
work?
F. What
difficulties did I
encounter
which my
principal or
55
supervisor can
help me solve?
G. What
innovation or
localized
materials did I
use/discover
which I wish to
share with
other
teachers?
56
School
Grade Level
Learning
Area
Quarter
Teacher
Time & Date
Eleven
Statistics and
Probability
Third Quarter
I.
OBJECTIVES
A. Content
The learner demonstrates understanding of key concepts of
Standard
sampling ad sampling distributions of the sample mean.
B. Performance
Standard
The learner is able to apply suitable sampling and sampling
distributions of the sample mean to solve real-life problems
in different disciplines.
C. Learning
Competency/O
bjectives
M11/12SP-IIId-4.
The
learner
identifies
distributions of statistics (sample mean)
II.
sampling
CONTENT
Identifying Sampling Distributions of statistics (sample
mean)
III.
LEARNING RESOURCE
References
Ocampo J. & Marquez W. (2016). Senior High
Conceptual Math and Beyond Statistics and Probability.
Brilliant Creations Publishing, Inc. pp.86-93.
Other Learning
https://www.youtube.com/watch?v=xh4zxC1OpiA
Resource
IV.
PROCEDURES
A. Reviewing
Find the mean of the following sets of data.
previous
Set of data
Mean
lessons or
1.18, 19, 20, 21, 22,21, 20, 19, 17, 17, 16, 16, 16
presenting the
2.5,3,6,9, 7,2,10,8
new lesson
3.18,16,19,22,20, 15,23,21,21
4.76,69,63,82,29,83,64,71,76
5.36,37,37,38,23,30,35
B. Establishing a
purpose for the
lesson
C. Presenting
Examples/Insta
nces of the
Lesson
Suppose we have a population of size N with a mean 𝜇, and
we draw or select all possible samples of size n from this
population. Naturally, we expect to get different values of the
means for each sample. The sample means may be less
than, greater than, or equal to the population mean 𝜇.
The sample means obtained will from a frequency and the
corresponding probability distribution can be constructed.
This distribution is called the sampling distribution of the
sample means.
How do we construct the sampling distribution of the sample
means? Study the given example.
A population consists of five values (Php2, Php 3, Php 4, Php
5, Php6). A sample of size 2 is to be taken from this
population.
a. How many samples are possible? List them
and compute the mean of each sample.
57
D. Discussing
New concepts
and Practicing
New Skills # 1
E. Developing
Mastery
b. Construct the histogram of the sampling
distribution of the sample means.
The following table gives the monthly salaries
Officer
Salary
A
8
B
12
C
16
D
20
E
24
F
28
1.How many samples are possible? List them and compute
the mean of each sample?
2. Construct the sampling distribution of the sample means.
3. Construct the histogram of the sampling distribution of the
sample means.
Group activity for 10 minutes. The students are task
to:
1. Create problem that involves sampling distributions
of statistics (sample mean).
2. Construct sampling distribution and histogram of the
sample means
3. What learning discovered in doing such activity?
Would you be able to use this in your life? How and
why?
The rubrics will be used in scoring the performance of the
group.
Categories
4
Excellent
3
Satisfactory
2
Developing
1
Beginning
Mathe
matical
Concep
t
Demons
trates a
thoroug
h
underst
anding
of the
topic
and
uses it
accurate
ly to
solve
the
problem
All
computa
tion are
correct
an are
logically
present
ed
Demons
trates a
satisfact
ory
underst
anding
of the
uses it
to
simplify
the
problem
.
Demonstr
ates
incomplet
e
understa
nding
and has
some
misconce
ptions.
Shows lack
of
understandin
g and have
severe
misconceptio
ns.
The
computa
tion are
correct.
Generally Errors in
, most of computations
the
are severe.
computati
ons are
not
correct.
Accura
cy of
comput
ation.
58
Organiz
ation of
the
report
Highly
organize
d, flows
smoothl
y, and
observe
s logical
connecti
ons of
points.
Satisfact
orily
organize
d.
Sentenc
e flow is
generall
y
smooth
and
logical.
Somewh
at
cluttered.
Flow is
not
consisten
tly
smooth,
appears
disjointed
.
Illogical and
obscure. No
logical
connections
of ideas.
Difficult to
determine
the meaning.
Particip
ation of
the
membe
rs
F. Making
generalization
and abstraction
about the
lesson
G. Evaluating
Learning
All
Almost
Almost
Almost 70member 90-99% 80-89%
79% take
s take
take
take part part in the
part in
part in
in the
activity,
the
the
activity,
support and
activity,
activity,
support
encourage
support support and
others in the
and
and
encourag group. There
encoura encoura e others
isa conflict
ge
ge
in the
relationship
others in others in group
in doing the
the
the
members activity.
group
group
do not
member member find fault
s do not s do not in one
find fault find fault another,
in one
in one
open to
another, another, comment
open to open to s and
commen commen criticism.
ts and
ts and
criticism criticism
.
.
What is sampling distribution of sample means?
-It is the frequency distribution of the sample means taken
from a population.
A. Determine the number of different samples of the given
size n that can be drawn from the given population of size
N.
N
N
Number of Possible Samples
7
3
15
5
50
4
10
3
25
4
B. Random samples of size n=2 are drawn from a finite
population consisting of numbers 5, 6,7,8,and 9.
a. How many possible samples are there?
59
b .List all the possible samples and the corresponding
mean for each sample.
c. Construct the sampling distribution of the sample means.
d. Construct the histogram for the sampling distribution of
the sample means. Describe the shape of the histogram.
V. REMARKS
VI. REFLECTION
A. No. of learners
who earned 80% in
the evaluation.
B. No. of learners
who require
additional activities
for remediation who
scored below 80%.
C. Did the remedial
lessons work? No.
of learners who
have caught up with
the lesson.
D. No. of learners
who continue to
require remediation
E. Which of my
teaching strategies
worked well? Why
did these work?
F. What difficulties
did I encounter
which my principal
or supervisor can
help me solve?
G. What innovation
or localized
materials did I
use/discover which I
wish to share with
other teachers?
60
School
Grade Level
Learning
Area
Quarter
Teacher
Time & Date
I. OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competency/Objectives
Write the LC code for
each
I.CONTENT
II.LEARNING
RESOURCES
A. Reference
1.Teacher’s Guide pages
2.Learner’s Material
pages
3.Textbook pages
4.Additional Materials
from Learning
Resource(LR) Portal
B. Other Learning
Resources
IV. PROCEDURES
A. Reviewing previous
lesson or
presenting the new
lesson
Eleven
Statistics and
Probability
Third Quarter
The learner demonstrates understanding of key
concepts of sampling and sampling distributions of the
sample mean.
The learner is able to apply suitable sampling and
sampling distributions of the sample mean to solve
real-life problems in different disciplines.
The learners shall be able to finds the mean, variance
and the standard deviation of the sampling distribution
of the sample mean.
M11/12SP-IIId-5
Sampling and Sampling Distributions
K-12 Curriculum Guide
Statistics and Probability by Belencina, Baccay &
Mateo
pp. 110-119
Calculator, manila paper, pentel pen, projector and
laptop
Tell the class that the
sampling distribution of
the sample means is
actually the probability
distribution of the
sample mean
Start the lesson with a
review on how to
construct the sampling
distribution of the
sample mean.
Consider a population
consisting of 1,2,3,4
and 5. Suppose
sample size 2 are
drawn from this
population
Construct the sampling
distribution of the
sample mean.
61
Step 1:Determine the
number of possible
samples that can be
drawn from the
population using the
combination formula
5!
5𝐶2 = (5−2)!2!
5𝑥4𝑥3!
=
3!2!
= 20/2
5𝐶2 = 10
𝑛!
n𝐶𝑟 = (𝑛−𝑟)!𝑟!
Step 2:List all possible
samples and compute
the mean of each sample
Or through the use of scientific
Calculator
Keystroke:
5
Sample
Mean
Step 3: Construct a
frequency distribution of
the sample means
obtained in step 2
Sam Frequ
ple ency
Mea
n
𝑥̅
Proba
bility
P(x)
Tot
al
62
n𝐶𝑟
Sample
1,2
1,3
1,4
1,5
2,3
2,4
2,5
3,4
3,5
4,5
2
= Display
Mean
1.50
2.00
2.50
3.00
2.50
3.00
3.50
3.50
4.00
4.50
Sample Frequency Probability
Mean
P(x)
𝑥̅
1.50
1
1/10
2.00
1
1/10
2.50
2
2/10 or
1/5
2
2/10 or
1/5
3.50
2
2/10 or
1/5
4.00
1
1/10
4.50
1
1/10
Total
10
1.00
B. Establishing a
purpose for the lesson
Tell the class that on this lesson we shall continue to compute the
mean and variance of the sampling distribution of the sample mean
C. Presenting
examples/instances of
the
new lesson
Consider a population
consisting of 1,2,3,4 and
5. Suppose sample size
2 are drawn from this
population
𝜇=
Σ𝑥
𝑛
1+2+3+4+5
5
𝜇=
𝜇=3.00
Compute the population
mean from the given
example
D. Discussing new
concepts and
practicing new
skills # 1
Challenge the students
to compare this mean to
the mean of the sampling
distribution of the sample
mean after the next
activity had been done.
Discuss the steps on
how to find the mean and
variance of the given
sampling distribution
(PPT)
ICT Integration
Activity 2
Consider a population
consisting of 1,2,3,4 and
5. Suppose sample size
2 are drawn from this
population
Find the mean and
variance of the sampling
distribution of the sample
mean?
Remind the students to
follow the s
1. Construct the
sampling distribution of
the sample mean.
2. Compute the mean of
the sampling distribution
of the sample mean (𝜇𝑥̅
63
Answers:
𝑋̅
P(𝑋̅)
1.50
2.00
2.50
3.00
3.50
4.00
4.50
Total
1/10
1/10
1/5
1/5
1/5
1/10
1/10
1.00
𝑋̅ ∙ P(𝑋̅)
0.15
0.20
0.50
0.60
0.70
0.40
0.45
̅
̅ )=
𝚺𝑿 ∙ P(𝑿
3.00
̅)
̅ ∙ P(𝑿
𝜇𝑥̅ = 𝚺𝑿
𝜇𝑥̅ = 3.00 mean of the
sampling distribution of the
means
by multiplying the sample
mean by the
corresponding probability
and add the results.
3. Compute the variance
(𝜎 2 𝑥̅ ) of the sampling
distribution of the sample
Mean using the
formula
𝜎 2 𝑥̅ = Σ P(𝑋̅) ∙ (𝑋̅
2
– 𝜇)
4. Compute the standard
deviation by finding the
square root of the
variance
𝜎𝑥̅ = √Σ P(𝑋̅) ∙ (𝑋̅ – 𝜇)2
Complete the table below
𝑋̅ P 𝑋̅ (𝑋̅ P(𝑋̅)
(
–
–
∙ (𝑋̅
𝑋̅ 𝜇 𝜇) – 𝜇)2
2
)
T
o
t
a
l
Σ
P
(
𝑋̅
)
=
Σ
P(𝑋̅)
∙ (𝑋̅
– 𝜇)2
=
What is now the mean
and the variance of the
given sampling
distribution?
𝑋̅
1.
5
0
2.
0
0
2.
5
0
3.
0
0
3.
5
0
4.
0
0
4.
5
0
T
ot
al
𝑋̅
–
𝜇
(𝑋̅ –
𝜇)2
P(𝑋̅) ∙ (𝑋̅ –
𝜇)2
2.2
5
0.225
1.0
0
0.100
0.2
5
0.050
1/
5
1.
50
1.
00
0.
50
0.
00
0.0
0
0.000
1/
5
0.
50
0.2
5
0.050
1/
1
0
1/
1
0
1.
0
0
1.
00
1.0
0
0.100
1.
50
2.2
5
0.225
P
(
𝑋̅
)
1/
1
0
1/
1
0
1/
5
ΣP(𝑋̅)∙(𝑋̅–
𝜇)2 =0.750
𝜎 2 𝑥̅ = Σ P(𝑋̅ ) ∙ (𝑋̅ – 𝜇)2
𝜎 2 𝑥̅ = 0.750 Variance of the
sampling distribution of the
sample mean
64
E. Discussing new
concepts and
practicing new
skill #2
From the activity ask the
students to compute the
standard deviation by
finding the square root of
the variance.
F. Developing
mastery leads
to Formative
Assessment
After the discussion, divide the class into 4 groups and
distribute worksheets and materials.
Group 1. Construct Me
Group 2- Meant to be
Group 3.Difference and its Square
Group 4:Your Square Root, My Standard
G. Making
generalization
and abstraction
about the
lesson
Give the summary through question and answer.
1. What are the steps in computing the mean, variance
and standard Deviation of the sampling distribution of the
sample mean?
2. How do you compare mean of the sample means and
the mean of population?
H. Evaluating
Learning
J. Additional
activities for
application or
remediation
Evaluate the students base on the results of their output.
𝜎𝑥̅ = √Σ P(𝑋̅) ∙ (𝑋̅ – 𝜇)2
= √0.750
𝜎𝑥̅ = 0.87
So, the standard deviation of
the sampling distribution of the
sample mean is.87
From a group of eight students in your class. Determine
the general weighted average of the members of the group
and list all possible samples of size 2 and their
corresponding mean. Construct the sampling distribution
and solve the mean , variance and standard deviation of
the sampling distribution of the sample mean.
V. REMARKS
VI. REFLECTION
A. No of learners who
earned 80% in the
evaluation
B. No of learners who
require additional
activities for
remediation who
scored
below 80%
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson
D. No of learners who
continue to require
remediation
65
E. Which of my
teaching strategies
worked well? Why did
these work?
F. What difficulties did I
encounter which
my principal or
supervisor can help me
solve?
G. What innovation or
localized materials did I
use/discover which I
wish to share with
other teachers?
66
Worksheets
Group1. Construct Me
Given the population 1,3,4,6 and 8.
Suppose the sample size of 3 are drawn
from this population.
Construct the sampling distribution of the
Sample Mean
Step1. List all possible samples of size 3
and their corresponding mean
Sample
Mean
2.67
3.33
3.67
4.00
4.33
5.00
5.67
6.00
Step 2 Construct the sampling distribution
of the sample means
Sample
Mean
𝑥̅
Frequency
Probability
P(𝑥̅ )
Total
67
Sample
1,3,4
1,3,6
1,3,8
1,4,6
1,4,8
1,6,8
3,4,6
3,4,8
3,6,8
4,6,8
Mean
2.67
3.33
4.00
3.67
4.33
5.00
4.33
5.00
5.67
6.00
Sample
Mean
𝑥̅
2.67
3.33
Frequency
Probability
P(𝑥̅ )
1
1
1/10
1/10
3.67
1
1/10
4.00
1
1/10
4.33
5.00
2
2
2/10 or 1/5
2/10 or 1/5
5.67
1
1/10
6.00
Total
1
10
1/10
1.00
Group 2: Meant to Be.
Solve the mean of the sampling distribution
of the mean.
Sample
Mean
𝑥̅
2.67
Probability
P(𝑥̅ )
3.67
1/10
4.00
1/10
4.33
2/10 or 1/5
5.00
2/10 or 1/5
5.67
1/10
6.00
1/10
Total
1.00
𝑥̅ ∙ P(𝑥̅ )
1/10
What is now the mean of the sampling
3.33 of the sample
1/10
distribution
mean?
Sample
Probability
Mean
P(𝑥̅ )
𝑥̅ ∙ P(𝑥̅ )
𝑥̅
̅)
̅ ∙ P(𝑿
𝜇𝑥̅ 2.67
= 𝚺𝑿
1/10
0.267
𝜇𝑥̅ 3.33
= 4.40 mean1/10
of the sampling
0.333
distribution of the means
3.67
1/10
0.367
4.00
1/10
0.400
4.33
2/10 or 1/5
0.866
5.00
2/10 or 1/5
1.00
5.67
1/10
0.567
6.00
1/10
0.600
total
1.00
Σ𝑥̅ ∙ P(𝑥̅ ) =4.40
Group 3: Difference and its square
If the mean 𝜇 of the population is 5.
Step 1. Subtract the population mean (𝜇)
from each sample (𝑥̅ ).
Sampl Probabilit
𝑥̅ - 𝜇
e
y
Mean
P(𝑥̅ )
𝑥̅
2.67
1/10
3.33
1/10
3.67
1/10
4.00
1/10
4.33
2/10 or
1/5
5.00
2/10 or
1/5
5.67
1/10
6.00
1/10
2.67
1/10
total
1.00
68
Sampl
e
Mean
𝑥̅
2.67
3.33
3.67
4.00
4.33
5.00
5.67
6.00
total
Probability
P(𝑥̅ )
𝑥̅ - 𝜇
1/10
1/10
1/10
1/10
2/10 or 1/5
2/10 or 1/5
1/10
1/10
1.00
-1.73
-1.07
-0.73
-0.40
-0.07
0.60
1.27
1.60
Step 2:Square the difference 𝑥̅ - 𝜇
Sampl Probabilit
y
e
𝑥̅ - 𝜇
(𝑥̅ − 𝜇 )2
P(𝑥̅ )
Mean
𝑥̅
2.67
3.33
3.67
4.00
4.33
1/10
1/10
1/10
1/10
2/10 or
1/5
2/10 or
1/5
1/10
1/10
1.00
5.00
5.67
6.00
total
Sample
Mean
𝑥̅
Probability
P(𝑥̅ )
𝑥̅ - 𝜇
(𝑥̅ − 𝜇 )2
2.67
3.33
1/10
1/10
-1.73
-1.07
3.67
4.00
4.33
1/10
1/10
2/10 or
1/5
2/10 or
1/5
1/10
1/10
1.00
-0.73
-0.40
-0.07
2.993
1.145
0.533
0.160
0.005
0.60
0.360
1.27
1.60
1.613
2.560
-1.73
-1.07
-0.73
-0.40
-0.07
5.00
0.60
5.67
6.00
total
1.27
1.60
Group 4: Your Square Root, My Standard
Compute the Variance and Standard Deviation of the sampling distribution of the
Means if the mean 𝜇 of the population is 5.
Step 1: Multiply (𝑥̅ − 𝜇)2 by its corresponding Probability P(𝑥̅ ) and add the results
Sample
Mean
𝑥̅
2.67
3.33
3.67
4.00
4.33
5.00
5.67
6.00
total
Probabili
ty
P(𝑥̅ )
1/10
1/10
1/10
1/10
2/10 or
1/5
2/10 or
1/5
1/10
1/10
1.00
𝑥̅ - 𝜇
P(𝑥̅ )∙ (𝑥̅ −
𝜇)2
Samp
le
Mean
𝑥̅
Probab
ility
2.67
3.33
3.67
4.00
4.33
1/10
1/10
1/10
1/10
2/10
or 1/5
2/10
or 1/5
1/10
1/10
1.00
-1.73
-1.07
-0.73
-0.40
-0.07
(𝑥̅
− 𝜇)2
2.993
1.145
0.533
0.160
0.005
0.60
0.360
5.00
1.27
1.60
1.613
2.560
5.67
6.00
total
P(𝑥̅ )
𝑥̅ - 𝜇
(𝑥̅ − 𝜇)2
-1.73
-1.07
-0.73
-0.40
-0.07
2.993
1.145
0.533
0.160
0.005
0.300
0.115
0.053
0.016
0.001
0.60
0.360
0.072
1.27
1.60
1.613
2.560
0.161
0.256
P(𝑥̅ )∙ (𝑥̅ −
𝜇)2
Σ P(𝑥̅ )∙ (𝑥̅ −
𝜇)2 =0.974
What is now the variance of the sampling
distribution of the sample mean?
𝜎 2 𝑥̅ = Σ P(𝑋̅ ) ∙ (𝑋̅ – 𝜇)2
𝜎 2 𝑥̅ = 0.974 Variance of the sampling distribution
of the sample mean
̅ ) ∙ (𝑋
̅ – 𝜇)2
𝜎𝑥̅ = √Σ P(𝑋
= √0.974
𝜎𝑥̅ = 0.990 standard deviation of the sampling
distribution of the sample mean
69
Daily Lesson Plan in Statistics and Probability
Grade 11/12
Quarter 3 Week 5
I.
OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competency/Obj
ectives
The learner demonstrates understanding of key concepts
of sampling ad sampling distributions of the sample mean.
The learner is able to apply suitable sampling and sampling
distributions of the sample mean to solve real-life problems
in different disciplines.
M11/12SP-III-e-2. The learner illustrates the Central Limit
Theorem.
II.
CONTENT
Central Limit Theorem
III.
LEARNING RESOURCE
References
1. Commission on Higher Education & Philippine Normal
University (2016). Teaching Guide for Senior High
School: Statistics and Probability. pp.242-261
2. Woodward, E. (2019). Ed's Intro to Prob and Stats.
Retrieved
from
https://legacy.cnx.org/content/col12133/1.1/ pp. 364.
3. Holmes, A., Illowsky, B., & Dean, S. (2019). Retrieved
from
https://opentxtbc.ca/introbusinessstatopenstax/chapter
/usng-the-central-limit-theorem.
Other Learning
Resource
1. Calculator, manila paper, permanent markers,
ruler/meter stick, Diagram of the different shapes of
distributions
retrieved
from
http://mathcenter.oxford.emory.edu/site/math117/shap
eCenterAndSpread/
IV.
PROCEDURES
A. Reviewing
previous
1. Describe the shape of the following distribution.
lessons or
presenting the
new lesson
Image downloaded from
http://mathcenter.oxford.emory.edu/site/math117/shapeCenterAndSpread/
70
Options
A. Symmetric, unimodal, bell-shaped
B. Uniform
C. Skewed right
D. Skewed left
E. Symmetric, bimodal
F. Non-symmetric, bimodal
B. Establishing a
purpose for the
lesson
What will be the effect of increasing the sample size on the
shape of the sampling distribution of the sample mean
given that the samples are selected at random?
C. Presenting
Examples/Instan
ces of the
Lesson
The learners will be asked to write their hypothesis on their
notebook. The teacher will inform the learners that in order
to test their hypotheses, they will be asked to perform an
activity. At this point, the learners will be divided into six
groups. Each group will be given a copy of the worksheet
to be used and other materials needed to accomplish the
task.
The class will be divided into 4 groups. Provide each group
with the materials needed in accomplishing their tasks such
as dice, Hand-outs, permanent markers, calculator, manila
paper and coloring materials. (See attached Hand-outs.)
Tasks:
Group 1: Construct a probability distribution of the random
variable X defined by the outcomes of rolling a die. Draw its
corresponding histogram. What is the shape of the
distribution?
Group 2. Ask one member of the group to roll 2 dice.
Consider this as trial 1. Compute the sample mean of the
faces showing. For example, 1 and 2, thus the mean is
1+2
= 1.50. Record the outcomes and the mean of the
2
samples on the hand-out provided to your group (Hand-out
1.A) The same person will continue rolling the dice until 20
trials. After completing all the required trials, construct a
probability distribution of the sample means and construct
its corresponding histogram. Describe the shape of the
distribution.
Group 3. Ask one member of the group to roll 5 dice.
Consider this as trial 1. Compute the sample mean of the
faces showing. For example, 1, 3,4,5 and 2, thus the mean
1+3+4+5+2
is
= 3.00. Record the outcomes and the mean of
5
the samples on the hand-out provided to your group (Handout 1.B) The same person will continue rolling the dice until
20 trials. After completing all the required trials, construct a
probability distribution of the sample means and construct
its corresponding histogram. Describe the shape of the
distribution.
71
Group 4. Ask one member of the group to roll 10 dice.
Consider this as trial 1. Compute the sample mean of the
faces showing. For example, 1,1,1,4,4,5,6,2,3 and 2, thus
1+1+1+4+4+5+6+2+3+2
the mean is
= 2.90. Record the
10
outcomes and the mean of the samples on the hand-out
provided to your group (Hand-out 1.C) The same person
will continue rolling the dice until 20 trials. After completing
all the required trials, construct a probability distribution of
the sample means and construct its corresponding
histogram. Describe the shape of the distribution.
D. Discussing New
concepts and
Practicing New
Skills # 1
The learners will be given at most 2 minutes to present their
group outputs. The teacher then checks the histogram
constructed by each group. Once all of the groups’ outputs
are checked, ask the learners to compare the histograms of
the sampling distributions of the sample mean when n=2,
n=5 and n=10 and compare this to the original population
distribution constructed by Group 1. What happens to the
shape of the sampling of the sample means when the
sample size increases?
E. Developing
Mastery
Show the following sets of diagrams to the learners. Let
them answer the guide questions afterwards.
(A)
(B)
(C)
Images retrieved from https://opentxtbc.ca/introbusinessstatopenstax/chapter/usng-thecentral-limit-theorem
Guide Questions
1. What is the shape of the population distribution in Set
A? in Set B? in Set C?
72
2. What happens to the shape of the sampling of the
sample means when the sample size increases?
3. Complete the statement below about central limit
theorem.
The central limit theorem for sample means says that
as the sample size_________, the sampling distribution
of the sample mean grows closer to a ________,
regardless of the shape of the original population
distribution. (increases, normal distribution)
F. Making
generalization
and abstraction
about the lesson
G. Evaluating
Learning
The central limit theorem for sample means says that
as the sample size increases, the sampling distribution
of the sample mean grows closer to a normal
distribution, regardless of the shape of the original
population distribution.
When the variable has a distribution that is not a Normal
distribution, the sample means are not normally
distributed unless the sample size is large enough.
(Generally, a good rule of thumb is to use a sample size
of at least 30, to ensure a sampling distribution that will
be approximately normal. Unless of course the original
population is known to be normal, in which case the
sampling distribution of the sample mean will be
guaranteed to normal.)
Choose only 1 of the suggested tasks. (See attached
worksheets 1-A to 1-D)
73
WORKSHEET 1-A
I. Write O if the statement is TRUE and X if otherwise.
1. The Central Limit Theorem tells us that as sample sizes get larger, the sampling
distribution of the sample means will become normally distributed, even if the
data within each sample are not normally distributed. (TRUE)
2. The shape of the sampling distribution of the means becomes left skewed if
random samples of size n becomes larger. (FALSE)
3. The central limit theorem states that as the sample size increases, the shape
of the distribution of the sample values look more and more normal. (FALSE)
II. Read and analyze the situations below. Write a short explanation for your answer.
4. A certain study involving senior high school students’ number of hours spent in
social media in a day shows a strongly skewed distribution with a mean of 5.2
hours and a standard deviation of 2.4 hours. What is the shape of the sampling
distribution of the sample means of 55 randomly selected senior high school
students if 55 is considered to be a large sample? Justify your answer.
74
WORKSHEET 1-B. What’s Your Muddiest Point?
I’m most confused about _____________________________________________
___________________________________________________________________
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75
WORKSHEET 1-C. 3-2-1
Accomplish the table below by writing 3 things that you have learners today, 2 things
that you found interesting and 1 question that you still have in mind.
Things I learned today
3
Things I found interesting
2
Questions I still have
1
76
WORKSHEET 1-D. T-L-R
Accomplish the table below by writing your initial hypothesis in the first column. In the
second column, write all the things that you have learned throughout the session and
in the third column, write a short reflection about your learnings. Is your hypothesis
correct? Can you cite real life situations or phenomena wherein the concept of central
limit theorem can be applied?
What I think
What I learned?
Reflection
(Write your initial
(Write the things that
hypothesis before the
you learned today.)
conduct of the activity)
77
HAND-OUT 1.A
SAMPLING DISTRIBUTION OF SAMPLE MEANS (n=2)
Name: _______________________ _______________________
_______________________ _______________________
_______________________ _______________________
_______________________ _______________________
Tabulation of Results.
Trials
Samples (X)
Example
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1,2
Sample Means
(round off to the
nearest hundredths)
̅)
(𝑿
1.50
Complete the probability distribution of the sample means below. You may add columns if
needed.
̅)
(𝑿
̅)
P (𝑿
Draw the histogram of the sampling distribution of the sample means (n=2).
Describe the shape of the distribution.
78
HAND-OUT 1.B
SAMPLING DISTRIBUTION OF SAMPLE MEANS (n=5)
Name: _______________________ _______________________
_______________________ _______________________
_______________________ _______________________
_______________________ _______________________
Tabulation of Results.
Trials
Samples (X)
Example
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1, 3,4,5,2
Sample Means
(round off to the
nearest hundredths)
̅)
(𝑿
3.00
Complete the probability distribution of the sample means below. You may add columns if
needed.
̅)
(𝑿
̅)
P (𝑿
Draw the histogram of the sampling distribution of the sample means (n=2).
Describe the shape of the distribution.
79
HAND-OUT 1.C
SAMPLING DISTRIBUTION OF SAMPLE MEANS (n=10)
Name: _______________________ _______________________
_______________________ _______________________
_______________________ _______________________
_______________________ _______________________
Tabulation of Results.
Trials
Samples (X)
Example
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1,1,1,4,4,5,6,2,3,2,
Sample Means
(round off to the
nearest hundredths)
̅)
(𝑿
2.90
Complete the probability distribution of the sample means below. You may add columns if
needed.
̅)
(𝑿
̅)
P (𝑿
Draw the histogram of the sampling distribution of the sample means (n=2).
Describe the shape of the distribution.
80
Daily Lesson Plan in Statistics and Probability
Grade 11/12
Quarter 3 Week 5
I.
OBJECTIVES
A. Content Standard
The learner demonstrates understanding of key
concepts of sampling ad sampling distributions of the
sample mean.
B. Performance
Standard
The learner is able to apply suitable sampling and
sampling distributions of the sample mean to solve reallife problems in different disciplines.
C. Learning
Competency/Objec
tives
M11/12SP-III-e-3. The learner defines sampling
distribution involving sampling distribution of the sample
mean using the Central Limit Theorem.
II.
CONTENT
Sampling distribution involving sampling
distribution of the sample mean using the Central
Limit Theorem
III.
LEARNING RESOURCE
References
6. Commission on Higher Education & Philippine
Normal University (2016). Teaching Guide for
Senior High School: Statistics and Probability.
pp.242-261
7. Woodward, E. (2019). Ed's Intro to Prob and Stats.
Retrieved from
https://legacy.cnx.org/content/col12133/1.1/ pp.
364.
8. Holmes, A., Illowsky, B., & Dean, S. (2019).
Retrieved from
https://opentxtbc.ca/introbusinessstatopenstax/chap
ter/usng-the-central-limit-theorem.
Other Learning
Resource
IV.
PROCEDURES
A. Reviewing
previous lessons
or presenting the
new lesson
B. Establishing a
purpose for the
lesson
Calculator, manila
ruler/meter stick
paper,
permanent
markers,
2. Recall of the essential formulas for calculating the
mean and variance of the sampling distribution of the
sample means.
Mean of the sampling distribution
µ𝑥̅ = 𝑋̅ • 𝑃(𝑋̅)
Variance of the sampling distribution
𝜎 2 𝑥̅ = ∑ 𝑃(𝑋̅) − (𝑋̅ − µ)2
What will be the effect of increasing the sample size on
the (a) mean and (b) spread of the sampling distribution
of the sample mean given that the samples are selected
at random?
81
The learners will be asked to write their hypothesis on
their notebook. The teacher will inform the learners that
in order to test their hypotheses, they will be asked to
perform an activity. At this point, the learners will be
divided into six groups. Each group will be given a copy
of the worksheet to be used and other materials needed
to accomplish the task.
C. Presenting
Examples/Instance
s of the Lesson
Utilizing the same groupings and accomplished handouts in the previous lesson, the learners will be asked to
compute the mean and the standard deviation of the
population distribution and of the sampling distribution of
the sample means.
Group 1. Compute the mean and standard deviation of
the probability distribution of the random variable X
defined by the outcomes of rolling a die.
Group 2. Compute the mean and the standard deviation
of the sampling distribution of the sample means (n=2).
Group 3. Compute the mean and the standard deviation
of the sampling distribution of the sample means (n=5).
Group 4. Compute the mean and the standard deviation
of the sampling distribution of the sample means (n=10).
D. Discussing New
concepts and
Practicing New
Skills # 1
Sampling Distribution
of the Sample Means
n=2
n=5
Population
n=
10
N=
6
Mean
Standard
Deviation
Guide Questions:
1. What happens to the mean/Expected value (EV) of
the sampling distribution of the sample means when
the sample size increases?
2. What happens to the standard deviation/ standard
error (SE) of the sampling distribution of the sample
means when the sample size increases?
E. Developing
Mastery
F. Making
generalization and
abstraction about
the lesson
The mean of the sampling distribution of the
sample mean will always be the same as the
mean of the original population regardless of the
sample size. 𝝁𝑿̅ = μx
̅ , 𝝈𝑿̅ = 𝝈 , is the
The standard deviation of 𝑿
√𝒏
standard error of the mean (SEM) for samples
with replacement.
82
If X is a random variable with mean μx and
standard deviation σx and either X is normally
̅ ∼ N(μx, 𝛔𝑿 )
distributed or n ≥ 30, then 𝑿
√𝒏
G. Evaluating
Learning
Choose only one activity from the following activities:
Activity 1 (Individual Task)
Determine the mean 𝜇𝑋̅ , variance 𝜎𝑋2̅ and standard
deviation 𝜎𝑋̅ for each item.
1. A random sample of size 49 is taken with
replacement from a population with µ = 82.4 and σ =
60.
2. A random sample of size 36 is taken with
replacement from a population with µ = 48 and σ =
6.5.
3. A random sample of size 49 is taken with
replacement from a population with µ = 28.6 and σ =
12
4. A random sample of size 36 is taken with
replacement from a population with µ = 120 and σ =
20.
5. A random sample of size 100 is taken with
replacement from a population with µ = 28.6 and σ =
25.
Activity 2. TIC-TAC-TOE (Pair Activity)
(see attached guide)
Activity 3. What’s Your Muddiest Point?
I’m most confused about
____________________________________________
____________________________________________
____________________________________________
Activity 4. 3-2-1
Accomplish the table below by writing 3 things that you
have learners today, 2 things that you found interesting
and 1 question that you still have in mind.
Things I learned today
3
2
Things I found interesting
Questions I still have
1
83
Activity 2. TIC-TAC-TOE (Pair Activity)
This activity will be accomplished by pair. Player 1 will use X mark while Player 2 will
use O as his mark. To place a mark, the player must correctly solve the given problem.
Players takes turn. The player who succeeds in placing three of their marks in a
horizontal, vertical or diagonal row wins the game
.
A random sample of size
49
is
taken
with
replacement
from
a
population with µ = 82.4
and σ = 60. Find 𝜇𝑋̅ .
A random sample of size
36
is
taken
with
replacement
from
a
population with µ = 48 and
σ = 6.5. Find 𝜎𝑋̅ .
A random sample of size
49
is
taken
with
replacement
from
a
population with µ = 28.6
and σ = 12. Find 𝜎𝑋2̅ .
A random sample of size
36
is
taken
with
replacement
from
a
population with µ = 120
and σ = 20. Find 𝜎𝑋2̅ .
A random sample of size
100
is
taken
with
replacement
from
a
population with µ = 28.6
and σ = 25. Find 𝜇𝑋̅ .
A random sample of size
100
is
taken
with
replacement
from
a
population with µ = 28.6
and σ = 25. Find 𝜎𝑋̅ .
A random sample of size
100
is
taken
with
replacement
from
a
population with µ = 28.6
and σ = 25. Find 𝜎𝑋̅ .
A random sample of size
100
is
taken
with
replacement
from
a
population with µ = 28.6
and σ = 25. Find 𝜎𝑋2̅ .
A random sample of size
100
is
taken
with
replacement
from
a
population with µ = 28.6
and σ = 25. Find 𝜇𝑋̅ .
84
Daily Lesson Plan in Statistics and Probability
Grade 11/12
Quarter 3 Week 5-6
I.
OBJECTIVES
A. Content Standard
The learner demonstrates understanding of key concepts
of sampling ad sampling distributions of the sample mean.
B. Performance
Standard
The learner is able to apply suitable sampling and
sampling distributions of the sample mean to solve reallife problems in different disciplines.
C. Learning
Competency/Obje
ctives
M11/12SP-III-e-f-1. The learner solves problems involving
sampling distributions of the sample mean.
II.
CONTENT
Solving Word Problems on Sampling Distribution of
Sample Mean
III.
LEARNING RESOURCE
Teacher’s Guide
pp. 261-264, pp. 191-192
Other Learning
Cumulative Distribution Function (CDF) of the Standard
Resource
Normal Curve TG pp. 191-192
Central Limit Theorem –Worksheet (Mean) Retrieved from
https://lhsblogs.typepad.com/files/central-limit-theoremworksheet-mean.pdf
IV.
PROCEDURES
A. Reviewing
1. Find the area under the normal curve given the
previous lessons
following conditions.
or presenting the
a. To the left of z = 1 Answer: 0.8849
new lesson
b. To the right of z = 1.75
Answer: 0.0401
a. Between z = 0.5 and z = 2.5
Answer: 0.3023
85
b. Between z = -2.5 and z = -1.2
Answer: 0.1089
2. Find the z values for each of the following:
a. µ = 50, σ = 4 and X = 45
Answer: z = -1.25
b. µ = 30, σ = 10 and X = 20
Answer: z = 1.00
c. µ = 50, σ = 25 and 𝑋̅ = 45, n = 100.
Answer: z = -2.00
d. µ = 30, σ = 10 and 𝑋̅ = 25, n = 25.
Answer: z = -2.50
3. Establishing a
purpose for the
lesson
Inform the learners that the aim of the lesson is to solve
word problems on sampling distribution of the sample
mean.
How do we solve word problems on sampling distribution
of the sample mean?
4. Presenting
Examples/Instanc
es of the Lesson
Present the following problems to the learners.
Fresh Cola uses a filling machine to fill plastic
bottles with soda. The contents of every bottle vary
according to a normal distribution with µ = 253 ml
and σ= 3 ml.
(a) What is the probability that an individual bottle
contains less than 250 ml?
(b) If 10 bottles are randomly selected, what is the
probability that the mean of the samples will be
less than 250 mL?
5. Discussing New
concepts and
Practicing New
Skills # 1
Guide the students in solving the word problem above.
Below are the solutions to the problem.
Problem: Fresh Cola uses a filling machine to fill plastic
bottles with soda. The contents of every bottle vary
86
according to a normal distribution with µ = 253 ml and σ=
3 ml.
a. What is the probability that an individual bottle contains
less than 250 ml?
Steps
Solution
1. Identify
the
µ = 253 ml
given/fact σ= 3 ml
s in the
X = 250
problem.
2. Identify
what is
P (X < 250)
asked for.
3. Identify
the
formula
to be
used?
The problem deals with an individual
data obtained from the population, so
we will use the formula Z=
𝜎
to
standardize 250.
Z=
Z=
𝑋−µ
𝜎
250−253
−3
4. Solve the
problem.
𝑋−µ
3
Z=
3
Z = -1
We shall find P (X < 250) by getting the
area under the normal curve.
P(X < 250) = P (z < -1)
= 0.5000 – 0.3413
= 0.1587
5. State the
final
answer.
So, the probability that a randomly
selected bottle will contain less than
250 ml is 0.1587 or 15.87%.
87
b. If 10 bottles are randomly selected, what is the
probability that the mean of the samples will be less
than 250 mL?
Steps
1. Identify
the
given/f
acts in
the
proble
m.
2. Identify
what is
asked
for.
3. Identify
the
formula
to be
used?
Solution
µ = 253 ml
σ= 3 ml
𝑋̅ = 250
n= 10
P (X < 250)
Here, we are dealing with data about the
sample means. So, we will use the
formula Z=
𝑋̅ −µ
𝜎
√𝑛
to standardize 250.
Z=
Z=
𝑋̅−µ
𝜎
√𝑛
250−253
3
√10
Z= -3.16
̅ < -3.16) by getting the
We shall find P (𝑋
area under the normal curve.
4. Solve
the
proble
m.
̅ < -3.16) = P (z < -3.16)
P (𝑋
= 1.0000– 0.9992
= 0.0008
5. State
the
final
answer
.
So, the probability that 10 randomly
selected bottles will have a mean less
than 250 ml is 0.0008 or 0.08%.
88
6. Developing
Mastery
For Slow learners, use Worksheet 1-A and 1-B.
For Average and Advanced learners, use Worksheet 1-C
and 1-D.
7. Making
generalization
and abstraction
about the lesson
Z=
𝜎
( used to gain information about an individual data
value when the variable is normally distributed.)
Z=
8. Evaluating
Learning
𝑋−µ
𝑋̅−µ
𝜎
√𝑛
( used to gain information when applying the
central limit theorem about a sample mean when the
variable is normally distributed or when the sample size is
30 or more.)
For Slow Learners, distribute Worksheet 2-A or 2-B to the
learners together with a z-table.
For Average and Advanced Learners, distribute both
Worksheet 2-A and 2-B to the learners together with the
z-Table.
89
WORKSHEET 1- A
Duck Eggs
Problem: The weights of the eggs produced by a certain breed of ducks are normally
distributed with mean 70 grams and standard deviation of 10 grams. What is the
probability that one duck egg selected at random weigh more than 75 grams?
Steps
1. Identify the given/facts in the
problem.
2. Identify what is asked for.
3. Identify the formula to be
used?
Solution
µ=
σ=
X=
P (X > 75)
The problem deals with an individual data
obtained from the population, so we will use the
formula Z=
𝑋−µ
𝜎
to standardize 75.
4. Solve the problem.
Z=
𝑋−µ
𝜎
Substitute the values of X, µ and σ.
Z=
Z=
Z=
We shall find P (X > 75) by getting the area
under the normal curve.
P(X > 75) = P (z > ____ )
= ________
5. State the final answer.
So, the probability that a randomly selected duck
egg will weigh greater than 75 grams is
_______.
90
WORKSHEET 1- B
Duck Eggs
Problem: The weights of the eggs produced by a certain breed of ducks are normally
distributed with mean 70 grams and standard deviation of 10 grams. What is the
probability that the average weight of the 12 duck eggs selected at random will be
more than 75 grams?
Steps
1. Identify the given/facts in the
problem.
2. Identify what is asked for.
Solution
µ=
σ= 10
𝑋̅ =
n= 12
P (X < 250)
3. Identify the formula to be
used?
Here, we are dealing with data about the
sample means. So, we will use the formula Z=
𝑋̅ −µ
𝜎
√𝑛
to standardize 75.
4. Solve the problem.
Z=
Z=
𝑋̅−µ
𝜎
√𝑛
10
√12
Substitute the values of 𝑋̅and µ.
Z=
̅ > 75) by getting the area
We shall find P (𝑋
under the normal curve.
̅ > 75) = P (z > _______ )
P (𝑋
= _____________
5. State the final answer.
So, the probability that 12 randomly selected
duck eggs will have a mean greater than 75
grams is ___________.
91
WORKSHEET 1- C
Duck Eggs
Problem: The weights of the eggs produced by a certain breed of ducks are normally
distributed with mean 70 grams and standard deviation of 10 grams. What is the
probability that one duck egg selected at random weigh more than 75 grams?
Steps
1. Identify the
problem.
Solution
given/facts
in
the
2. Identify what is asked for.
3. Identify the formula to be used?
4. Solve the problem.
5. State the final answer.
92
WORKSHEET 1- D
Duck Eggs
Problem: The weights of the eggs produced by a certain breed of ducks are normally
distributed with mean 70 grams and standard deviation of 10 grams. What is the
probability that the average weight of the 12 duck eggs selected at random will be
more than 75 grams?
Steps
1. Identify the
problem.
Solution
given/facts
in
the
2. Identify what is asked for.
3. Identify the formula to be used?
4. Solve the problem.
5. State the final answer.
93
Worksheet 2-A
Pregnancy
The length of human pregnancies from conception to birth varies according to a
distribution that is approximately normal with mean 264 days and standard deviation
16 days. Consider 15 pregnant women from a rural area. Assume they are equivalent
to a random sample from all women.
(a) What's the probability that a single pregnant woman is pregnant for less than 250
days?
Steps
Solution
1. Identify the given/facts in the
problem.
2. Identify what is asked for.
3. Identify the formula to be used?
4. Solve the problem.
5. State the final answer.
(b) What is the probability that the sample mean length of pregnancy lasts for less than
250 days?
94
Steps
1. Identify the
problem.
Solution
given/facts
in
the
2. Identify what is asked for.
3. Identify the formula to be used?
4. Solve the problem.
5. State the final answer.
95
Worksheet 2-B.
Life Expectancy
A certain study on the life expectancy of people in a Country A revealed that the mean
age at death was 80 years and the standard deviation was 10 years.
(a) What is the probability that an individual selected at random will be less than 50
years old?
Steps
Solution
1. Identify the given/facts in the
problem.
2. Identify what is asked for.
3. Identify the formula to be used?
4. Solve the problem.
5. State the final answer.
96
(b) If a sample of 100 people from this country is selected, find the probability that the
mean life expectancy will be less than 50 years.
Steps
Solution
1. Identify the given/facts in the
problem.
2. Identify what is asked for.
3. Identify the formula to be used?
4. Solve the problem.
5. State the final answer.
97
Daily Lesson Plan in Statistics and Probability
Grade 11/12
Quarter 3 Week 6
I.
OBJECTIVES
A. Content
Standard
The learner demonstrates understanding of key concepts of
estimation of population mean an population proportion.
B. Performance
Standard
The learner is able to estimate the population mean and
population proportion to make sound inferences in real-life
problems in different disciplines.
C. Learning
Competency/Obj
ectives
M11/12SP-III-f-2. The learner illustrates point and interval
estimations.
M11/12SP-III-f-3. The learner distinguishes between point
and interval estimations.
II.
CONTENT
Point and Interval Estimations
III.
LEARNING RESOURCE
Teacher’s Guide
pp. 316-319
Other Learning
Resource
IV.
PROCEDURES
A. Reviewing
previous
lessons or
presenting the
new lesson
B. Establishing a
purpose for the
lesson
Meta-cards of different colors, permanent marker, masking
tape
Frayer Model diagram
School Division, S. (2018, October 29). Frayer Model. OER
Commons.
Retrieved
July
24,
2019,
from
https://resourcebank.ca/authoring/1499-frayer-model.
A quick review of the concept of parameter and statistic will
be made before discussing the concept of Point and Interval
estimate.
Task 1. Parameter & Statistic
Below are measures that describe some characteristics of
a given sample or population. Raise your right hand if the
identified measure is a statistic. If the given measure is a
parameter, raise your left hand.
1. Sample mean (𝑥̅ )
2. Population mean (µ)
3. Sample variance (𝑠2 )
4. Population variance (σ2 )
5. Sample standard deviation (s)
6. Population standard deviation (σ)
7. Sample proportion (𝑝̂ )
8. Population proportion (p)
At this point, reiterate to the learners that in reality, we do
not have the whole population to work on. Hence, a
representative obtained through random sampling is
important in making inference about the population and or
98
its parameters. In making inferences about the population,
learners can either provide a value or values for the
parameter or evaluates a statement about a parameter.
Inform the learners that the focus of the lesson is on the two
ways of estimation (point and interval estimation) and
differentiate one from another.
C. Presenting
Examples/Instan
ces of the
Lesson
Task 2. Guessing Game
(This is a modified activity of the activity found in TG p. 317)
Distribute three meta-cards to the learners. The learners will
be asked answer the statements/phrases below by writing
their responses on specified meta-cards.
Red Meta1. His/her “best” guess of your age by
card
giving a single number
Yellow meta- 2. His/her “best guess of the range
card
wherein your age will likely fall.
Green metacard
3. His/her confidence from 0% (not
confident) to 100% (very confident)
in his/her educated guess of the
range of values un number 2.
Afterwards, ask the learners to post their responses on the
specific posting area for each color of the meta-card.
Randomly pick three students to briefly explain his/her basis
for his/her written response.
Note: Aside from teacher’s age, other variables such as
price of rice, daily allowance, etc. can also be used.
D. Discussing New
concepts and
Practicing New
Skills # 1
1. Discuss the responses on the activity to the learners,
emphasizing the following points:
a. There are no right or wrong numerical value for the
given answers. However, there might be
misconceptions or misunderstanding of the
concepts when they provided answers.
b. On the red meta-cards, the learners should have
written one logical number between 21 and 65
(inclusive). Their guess of your age should be
between 21 and 65 for it to be logical since one
usually starts working at the age of 21 and retires at
the age of 65 (compulsory retirement age).
c. On the yellow meta-card, the learners should have
written a logical range of values or set of values with
upper and lower limits. The lower limit should be at
least 21 while the upper limit should be at most 65.
d. On the green meta-card, the learners should have
written a percentage within 0% to 100%.
99
Inform the learners of your true age. Take note also of
how many learners gave the correct point estimate and
confidence interval estimate that included your age.
2. Inform the learners that the numbers that they wrote on
the red meta-cards can be considered point estimate.
The range of values or set of values that they wrote on
the yellow meta-cards can be considered as interval
estimate. The percentage that they wrote on the green
meta-cards can be considered as confidence
coefficient. When interval estimate is combined with
confidence coefficient, it is now referred to as
confidence interval estimate.
Task 3. Let’s define the Estimates
Note: For Advanced learners, let them accomplish the
Frayer model below by groups. For average and slow
learners, you may want to present them with an
accomplished Frayer model for Point estimate and
Interval estimate.
For group 1 and 2
Definition
A point estimate is a
numerical value and it
identifies a location or
position in the distribution
of possible values.
Characteristics
- Single value as
estimate
- There are two
possibilities
(right or wrong)
- No confidence
coefficient
involved
Point
estimate
Examples
The mean volume of 11
bottles of cola is 12 oz.
On the average, there are
89 customers in the
restaurant.
100
Non-Examples
The mean volume of 11
bottles of cola ranges from
12-13 oz.
On the average, there are
89-100 customers in the
restaurant.
For group 3 and 4
Definition
An interval estimate
is a range of values
where most likely
the true value will
fall.
Characteristics
- More than one
possible values
as estimates
- Involves
confidence
coefficient
Interval
estimate
Examples
Non-Examples
The mean volume of
11 bottles of cola
ranges from 12-13
oz.
On the average, there
are 89-100 customers
in the restaurant.
On the average, there are 89
customers in the restaurant.
The mean volume of 11
bottles of cola is 12 oz.
Note: The Frayer Model is a graphical organizer used for
word analysis and vocabulary building. This four-square
model prompts students to think about and describe the
meaning of a word or concept by (a) Defining the term, (b)
Describing its essential characteristics, (c) Providing
examples of the idea, and (d) Offering non-examples of the
idea. (Sun West School Division, 2018)
E. Discussing New Task 4. Compare and Contrast
concepts
and Using the Venn Diagram below, compare and contrast
Practicing New “Point Estimation” and “Interval Estimation”.
Skills # 2
Note: For Advanced learners, let them accomplish the Venn
diagram below by group but for average and slow learners,
you may want to give them choices.
Point
Estimation
101
Interval
Estimation
F. Developing
Mastery
Task 5. Is it true?
Get a piece of paper. Your teacher will a statement for you.
If the statement is correct, write TRUE. Otherwise, write
FALSE.
1. The sample statistic s (sample standard deviation) is a
point estimator.
2. The population mean (µ) is a point estimator.
3. The sample mean (𝑥̅ ) is a point estimate of the
population mean (µ)
4. The sample proportion (𝑝̂ ) is an interval estimate of the
population proportion p.
5. In the statement, “About 80%-85% of the youths in
Sorsogon City voted during the SK election.”, the phrase
“about 80%-85% denotes an interval estimate.
6. Point and Interval estimates of a population parameters
are from sample statistic.
G. Finding
Application
of Can you cite other real-life phenomenon involves point
Concepts
in estimate? How about real-life phenomenon involves
Daily
Living interval estimate?
H. Making
generalization
and abstraction
about the lesson
I.
Evaluating
Learning
A point estimate is a numerical value and it identifies a
location or position in the distribution of possible values.
A point estimate of a population parameter is a single
value of a sample statistic.
An interval estimate is a range of values where most
likely the true value will fall.
Directions. The statements below contain statistic which
can be classified as either point estimate or an interval
estimate. Read each item carefully and write PE if the
underlined measure is a point estimate and IE if interval
estimate.
1. The average weight of 12 Grade 11 students is
50.5 kilograms.
2. The average wait time in Restaurant A is 10
minutes.
3. 49 Grade 11 students were asked about their
age and recorded a mean age of 17 years old.
4. The average weight of 100 randomly selected
mangoes in the crate ranges from 10 grams to
20 grams.
102
5. The average time spent for social media of 20
randomly selected Grade 11 students is 4
hours in a day.
6. The average weight of newborn babies in
Hospital A is around 2.6 kg to 3.6 kg.
7. 10 out of 50 randomly selected professionals
are in favor of the implementation of national
ID.
8. The mean weight of 40 randomly selected
Grade 7 students is around 40 to 45
kilograms.
9. The age of beginning Grade 1 pupil is 6-7
years old.
Note: The evaluation can also be implemented through a
pair-work in in a form of a game called Tic-Tac-Toe. See
attached sample Tic-Tac-Toe Board
103
Tic-Tac-Toe
This activity will be accomplished by pair. Player 1 will use X mark while Player 2 will
use O as his mark. To place a mark, the player must correctly identify whether the
highlighted estimator is a point estimate or an interval estimate. Players takes turn.
The player who succeeds in placing three of their marks in a horizontal, vertical or
diagonal row wins the game.
The average weight of 12
Grade 11 students is 50.5
kilograms.
The average weight of
100 randomly selected
mangoes in the crate
ranges from 10 grams to
20 grams.
10 out of 50 randomly
selected professionals
are in favor of the
implementation of
national ID.
The average wait time in
Restaurant A is 10
minutes.
49 Grade 11 students
were asked about their
age and recorded a mean
age of 17 years old.
The average time spent
for social media of 20
randomly selected Grade
11 students is 4 hours in
a day.
The average weight of
newborn babies in
Hospital A is around 2.6
kg to 3.6 kg.
The mean weight of 40
randomly selected Grade
7 students is around 40
to 45 kilograms.
104
The age of beginning
Grade 1 pupil is 6-7
years old.
Daily Lesson Plan in Statistics and Probability
Grade 11
I.
OBJECTIVES
A. Content Standard
B. Performance Standard
C. Learning
Competency/Objectives
Write the LC code for
each
II.
CONTENT
III.
LEARNING
RESOURCES
I. Reference
Teacher’s Guide pages
Learner’s Material pages
Textbook pages
Additional Materials from
Learning
Resource(LR) Portal
J. Other Learning
Resources
IV.
PROCEDURES
A. Reviewing previous
lesson or presenting the
new lesson
The learner demonstrates understanding of key concepts of
estimation of population mean and of population proportion.
The learner is able to estimate the population mean and
population proportion to make sound inferences in real-life
problems in different disciplines.
The learner identifies point estimator for the population mean
M11/12SP-IIIf-4
Estimation of Parameters
Statistic and Probability by Belencina, Baccay & Mateo pp.179-183
Senior High Conceptual Math and Beyond Statistics and
Probability pp. 146-151
1.
2.
3.
4.
1 fair coins, Manila paper, pentel pen, projector, laptop
Preliminary Activity: Word Scramble(Power Point Presentation))
Unscramble the list of letters found at the screen, to spell out
words
.
answer
U
N
Population
T
Point
Estimator
T
O
T
O
P
P
I
O
M
I
T
S
A
R
S
Sample
Mean
N
Proportion
B. Establishing a purpose
for the lesson
C. Presenting
examples/instances of
the new lesson
Tell the class about the objectives of the lesson on how proportions about
populations are expressed and computed.
Present examples to the class( PPT)
Great Escape
On a typical morning Anthony ask his TV viewers if they believe
or do not believe the “great escape” story of a group of soldiers
105
from a perceived adversary. At the end of the show, he reported
that 68% of the respondents believe the story.
Guided Question:
What does the report meant to you?
What do you understand by the expression 68%?
D. Discussing new concepts
and practicing new skills
#1
Discuss further the example presented and answer the guided
question.
1) 68% is also called proportion
2)Percentages are preferred when reporting frequencies of
subsets of
population.
3)On computing percentage are first converted to proportions in
decimal
form.
4) Proportion also represent probabilities. So, the probability that
all TV viewers favor the great escape story id 0.68.
5) And that who do not favor is 1 – 0.68 = 0.32 or 32%
E. Discussing new concepts
and practicing new skill
#2
Tell the class that the percentage expression is called proportion.
Have them define the proportion.
Proportion – is a fraction expression with the number of
favorable responses on the numerator and the total number of
respondents on the denominator
Continue the discussion on how to solve the sample proportion.
Use the symbol for the discussion of the probability
n = number of observations in simple random sample
𝑝 = population proportion
𝑝̂ = sample proportion (read “p hat”)
Where: 𝑝̂ =
𝑥 (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠)
𝑛(𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠)
F. Developing mastery
leads to Formative
Assessment
Activity 3: Differentiated Activities
Divide the class into three groups and distribute the manila
paper, pentel pen and worksheet .
Remind the students to read carefully the instruction. Ask the
group to select their leader and present the output to the class.
Group 1: NEW KIND OF SNACKS
Group 2: My Head and My Tail
Group 3: Job Satisfaction
G. Finding practical
application of concept
and skill in daily living.
Ask the students how useful are proportions in the reports of
survey results?
Possible answer:
1. To determine the easiest and convenient way the population
proportion.
e. A sort of summarizing
H. Making generalization
and abstraction about the
lesson
Give the
summary of the
lesson through
106
Answers:
question and
answer
1. What is the
mean of the
sampling
distribution of
𝑝̂ ?
2. What is the
point estimator
of the population
proportion p?
3. What is the
formula use to
solve the point
estimator of p?
I.
Evaluating Learning
1. p
2. 𝑝̂ where
3. 𝑝̂ =
𝑥 (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠)
𝑛(𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠)
and 𝑞̂ = 1- 𝑝̂
Evaluate the results by counting the correct answer from the
Formative Test
J. Additional activities for
application or
remediation
V.
REMARKS
VI.
REFLECTION
A. No of learners who earned
80% in the
evaluation
B. No of learners who require
additional
activities for remediation who
scored
below 80%
C. Did the remedial lessons
work? No. of
learners who have caught up
with the
lesson
D. No of learners who continue
to require remediation
E. Which of my teaching
strategies worked well? Why did
these work?
F. What difficulties did I
encounter which
my principal or supervisor can
help me
solve?
G. What innovation or localized
materials did I use/discover
which I wish to share with other
teachers?
107
Group 1
Worksheet 1
NEW KIND OF SNACKS
Activity 1: New kind of snacks
A random selection of school children were asked whether they Like (1), Do not Like
(0) whether they like or not, a new kind of snacks served by the school cafeteria.
1
0
1
1
1
1
1
0
0
1
0
1
0
0
0
Procedure:
1. Tally the responses on the table
Response
Tally
Like(code 1)
Do not Like (code 0)
Cannot decide (code 2)
1
1
1
1
1
1
1
1
1
0
1
0
1
1
0
Frequency
Total
3. Write the tally marks as whole numbers.
4. Complete the entries in the table
5. Answer the guided questions:
a. What is the proportion of respondents who like the new snacks?
b. What is the proportion of respondent who do not like the new snack?
c. What is the prop
Answers:
Response
Like(code 1)
Do not Like (code 0)
Tally
IIII – IIII- IIII-IIII
IIII - IIII
Total
Frequency
20
10
30
5. Answer the guided questions:
a. What is the proportion of respondents who like the new snacks?
𝑝̂ =
=
𝑥
𝑛
20
30
𝑝̂ = 0.67 or 67%
b. What is the proportion of respondent who do not like the new snack?
𝑞̂ = 1- 𝑝̂
= 1- 0.67
𝑞̂ = 0.33 or 33%
108
Group 2
Worksheet 2
Getting the Head
Do the following task
1. Toss a fair of coin 10 times.
2. Record the result in the table. Use 0 for heads and 1 for tails.
1st Trial
Head
Tails
3. Counts the number of heads occurring and denote this as x
4.Compute 𝑝̂ =
𝑥
𝑛
for the first trial. This is called 𝑝̂1
5. Repeat step 1 to 3 times
2nd Trial
Head
Tails
3rd Trial
Head
Tails
6. Compute the mean or average of the 𝑝̂ values
𝑝̂=
𝑝̂1 + 𝑝̂2 + 𝑝̂3
3
What is now, the point estimator of the population proportion p?
7. Compute the proportion of getting Tail using the formula 𝑞̂ = 1- 𝑝̂
Group 3
Worksheets 3
Nora conducted a survey for a fast food restaurant owner who wanted to know the
level of acceptability of a new food combo among customer.
A tally of the frequencies yielded the following results.
Frequency (f)
Very Acceptable (VA)
182
Acceptable (A)
74
Guided Question:
1. What is the total number of respondents?(n)
109
2. What percentage of the respondents find the new food combo very acceptable?( 𝑝̂
)
3. What percentage of the respondents fond the new food combo acceptable?( 𝑞̂
Answers:
1. 256
2. 182
3. 29%
110
School
Teacher
Grade Level
Learning Area
Time & Date
Quarter
I. Objectives
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II. Topic
A. References
B. Teacher’s guide
pages
C. Learner’s material
pages
D. Textbook pages
E. Additional
materials from
learning resource
portal
F. Other learning
resources
III. Procedure
11
Statistics and
Probability
3rd
The learner demonstrates understanding of key
concepts of estimation of population mean and
population proportion.
The learner is able to estimate the population mean and
population proportion to make sound inferences in reallife problems in different disciplines.
M11/12SP-IIIF-4 and M11/12SP-IIIF-5
1. Explain the how to determine the point estimate
of the population mean.
2. Identify point estimator for the population mean.
3. Compute for the point estimate of the population
mean.
Point Estimate
Mercado, Jesus P., et al., Next Century Mathematics
Statistics and Probability, 2016
Belecina, Rene R., et al., Statistics and Probability, 2016
pen, board, calculators, visual aids
A. Reviewing
 Daily routine (opening prayer, checking of attendance)
previous lesson
 What do you mean by parameter?
and presenting the  In what situation in the field of business, agriculture,
new lesson
education, technology and health you use or take
sample to describe the characteristics of the
population? Is it valid or acceptable? Why?
 Present a picture of group of people or crowd.
According to reports, there are 500,000 people
who attended the PenafraciaTranslacion.
Source: Nino N. Luces/ Manila Bulletin File Photo
111
What you do think is the basis of the said report? Did
the reporter count the actual number of participants?
B. Establishing the
purpose of the
lesson
C. Presenting a
examples/
instances of a new
lesson
 Present the learning objectives. (Write or post the
learning objectives on the board)
How can you estimate the lifespan or life expectancy of
Filipinos?
Provide a set of data showing the lifespan of 30
Filipinos:
82
65
72
78
67
78
75
39
82
67
75
78
78
50
78
48
67
58
34
58
78
91
75
40
88
68
70
78
93
66
 Divide the class into four(4) groups and let the muse
their calculator to determine the average lifespan of:
 Set A: Any five (5) randomly selected data
 Set B: First ten(10) data (first row)
 Set C: First twenty (20) data (first two rows)
 Set D: All the given (30) data
 Let each group representative present their answer on
the board.
 Ask the class to compare the different results.
Guide questions:
 What can you say about the computed mean
from Set A, B and C?
 Which of answer is closer to the population
mean (Set D)?
 What if we will take 25 randomly selected
sample scores, how would you describe its
mean in relation to the entire population mean?
 Relate students’ answer(s) to Central Limit Theorem.
(If random samples of size n are drawn from a
population, then as n becomes larger, the sample
means approaches the normal distribution, regardless
of the shape of the population distribution.)
D. Discussing new
concepts and
practicing new
skills #1
 Emphasize to the class that the previous activity is a
process of point estimation
 State and discuss the meaning of:
- Estimation is the process of finding parameter
value.
112
-

E. Discussing new
concepts and
practicing new
skills #2
Estimate is a value or range of values that
approximate a parameter based on sample
statistics computed from sample data.
Point Estimate is a specific numerical value of
the population parameter.
Based from the previous example:
a) What population parameter we used to estimate
the population?
b) Why do we use mean as point estimator instead
of median or mode? Emphasize the advantages
of using mean and the limitations of median and
mode as point estimator. Identify the sample
mean, median and mode of the given data and
compare it to the population mean.
c) What are the other properties of good estimator?
 Considering the same data above (the life expectancy
of 30 Filipino samples), do you think the average or
mean of the means from each column of data will be
the same as your answer in set D? Why?
82
65
72
78
67
78
75
39
82
67
75
78
78
50
78
48
67
58
34
58
78
91
75
40
88
68
70
78
93
66
Can you consider the mean of the means as the
point estimate of the population parameter?
Emphasize to the students the difference between
the mean of the means of: (a) samples and (b)
population.
F. Developing
Mastery
 Activity: “Do you know me?”
1. Ask each student to estimate Grade 11 students’
profile by determining the averages of each of the
following: number of siblings, weight, height,
average daily allowance and sleeping time. Let
them write the answers in their notebook.
2. By group, let them gather basic information about
their classmates’ profile:
a. Number of siblings in the family (Group 1)
b. Weight
(Group 2)
c. Height
(Group 3)
d. Average daily allowance
(Group 4)
e. Sleeping time
(Group 5)
 Let each group representative present their findings.
 Ask:
Are the estimated averages/ means the same as the
actual class means?
What makes your estimate afar or closer to the
actual mean?
113
G. Finding Practical
Applications of
Concepts and
skills in daily life
H. Making
Generalization and
abstraction about
the lesson
Think –Pair - Share
 What is the importance of point estimate?
 Think of instances or real life situations where
estimation is used in the field of technology, science
and economics.
 By dyad: share your insights or answers to your
partner.
 Ask at least 10 student–representatives to present
their answers to the class.




I.
Evaluating
Activities
What are the significant learning you gained from our
lesson today?
What is the point estimator of the population mean?
How do you calculate the point estimate of the
population mean?
Why is it important to determine the point estimate of
the population mean?
Compute the point estimate of the population mean for
the given data.
Math Quiz Scores of 30 students randomly
selected from Grade 11 and 12 classes
23
33
44
41
34
J. Additional
Activities for
Application or
Remediation
V-REMARKS
39
45
38
31
43
48
38
38
44
46
38
41
40
40
26
33
27
42
37
45
46
45
27
28
38
Gather data on the cellular phone load expense of 50
students in our school and find its mean.
What are the implications of the data/ information you
gathered?
VI-REFLECTION
A. No. Of learners who
earned 80% on the
formative
assessment
B. No.of learners who
require additional
activites for
remediation.
C. Did the remedial
lesson work? No. of
learners who have
caught up with the
lesson.
114
D. No. of learners who
continue to require
remediation
E. Which of my
teaching strategies
worked well? Why
did these work?
F. What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
G. What innovation or
localized materials
did I use/ discover
which I wish to
share with other
teachers?
115
School
Teacher
Time and Date
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
I. OBJECTIVES
A. CONTENT
STANDARD
B. PERFORMANCE
STANDARD
C. LEARNING
COMPETENCIES /
OBJECTIVES
(Write the LC Code
for each)
II. CONTENT
III. LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s Material
pages
3. Textbook pages
4. Additional Materials
from Learning
Resource (LR)
portal
B. Other Learning
Resources
IV. PROCEDURE
A. Presenting the new
lesson.
B. Establishing a
purpose for the
lesson.
The learner demonstrates understanding of key
concepts of estimation of population mean and
population proportion.
The learner is able to estimate the population mean
and population proportion to make sound inferences
in real-life problems in different disciplines.
Identifies the appropriate form of the confidence
interval estimator for the population mean when: (a)
the population variance is known, (b) the population
variance is unknown, and (c) the Central Limit
Theorem is to be used. (M11/12SP-IIIg-1)
Interval Estimation for a Mean
Statistics and Probability for Senior High School
Authors: Christian Paul O. Chan Shio & Maria Angeli
T. Reyes
Pages 194 - 200
ADVANCED LEARNERS AVERAGE LEARNERS
In the previous lesson, you have learned how to
construct a point estimator of a population parameter.
However, a much better method for estimating a
parameter would be to incorporate a “margin of error”
to construct an interval that contains the true
parameter value. This method is called interval
estimation.
In interval estimation, two numbers are calculated
based on sample data, forming an interval where the
parameter’s value is expected to lie. In this case, the
formula is called interval estimator, while the range of
values obtained is called interval estimate or a
confidence interval.
The confidence coefficient, denoted by 1 – ά, is
the probability that a confidence interval will contain
the estimated parameter.
116
Confidence Interval for the Population Mean
(large sample or normal population, σ is known)
A (1 – ά) 100% confidence interval for μ is given by
𝜎
𝜎
(𝑥̅ − 𝑧ά ∙
,
𝑥̅ + 𝑧ά ∙
)
2 √𝑛
2 √𝑛
where 𝑥̅ = sample mean;
𝑎
𝑧ά = z-value that leaves an area of to the
2
2
right;
𝜎 = population standard deviation; and
𝑛 = sample size
Confidence Interval for the Population Mean
(large sample, σ is unknown)
A (1 – ά) 100% confidence interval for μ is given by
𝑠
𝑠
(𝑥̅ − 𝑧ά ∙
,
𝑥̅ + 𝑧ά ∙
)
√𝑛
√𝑛
2
2
where 𝑥̅ = sample mean;
𝑎
𝑧ά = z-value that leaves an area of to the
2
2
right;
𝑠 = sample standard deviation; and
𝑛 = sample size
C. Presenting examples
/ instance of the new
lesson.
Example 1: Find and interpret a 95% confidence
interval for the population mean given that 𝑛 = 36, 𝑥̅ =
13.1, 𝑎𝑛𝑑 𝜎 = 3.42.
Example 1: Find and interpret a 95% confidence
interval for the population mean given that 𝑛 = 64, 𝑥̅ =
15.4, 𝑎𝑛𝑑 𝑠 = 2.27.
D. Discussing new
concepts and
practicing new skills.
Solution for example 1:
For a 95% confidence interval, 1 – ά = 0.95, so ά =
0.05. Using the z – table, we have 𝑧ά = 𝑧0.05 = 𝑧0.025 =
2
2
1.96. Substituting these values into the formula for the
confidence interval for a population mean when σ is
known, we have
𝜎
𝜎
(𝑥̅ − 𝑧ά ∙
,
𝑥̅ + 𝑧ά ∙
)
2 √𝑛
2 √𝑛
3.42
3.42
= (13.1 − 1.96 ∙
,
13.1 + 1.96 ∙
)
√36
√36
= (13.1 − 1.12, 13.1 + 1.12)
= (11.98, 14.22)
Thus, we can be 95% confident that the interval
(11.98, 14.22) contains the true value of the
population mean.
Solution for example 2:
117
For a 95% confidence interval, 1 – ά = 0.95, so ά =
0.05. Using the z – table, we have 𝑧ά = 𝑧0.05 = 𝑧0.025 =
2
2
1.96. Substituting these values into the formula for the
confidence interval for a population mean when σ is
unknown, we have
𝑠
𝑠
(𝑥̅ − 𝑧ά ∙
,
𝑥̅ + 𝑧ά ∙
)
2 √𝑛
2 √𝑛
2.27
2.27
= (15.4 − 1.96 ∙
,
15.4 + 1.96 ∙
)
√64
√64
= (15.4 − 0.56, 15.4 + 0.56)
= (14.84, 15.96)
Thus, we can be 95% confident that the interval
(14.84, 15.96) contains the true value of the
population mean.
E. Developing Mastery.
F. Finding practical
applications of
concepts and skills in
daily living.
G. Making
generalizations and
abstraction about
lesson.
H. Evaluation
Let’s Practice:
Find and interpret a (1 – ά) 100% confidence interval
for the population mean μ given the following values:
a. ά = 0.05, 𝑛 = 64, 𝑥̅ = 14.1, 𝜎 2 = 4.32
b. ά = 0.01, 𝑛 = 36, 𝑥̅ = 7.23, 𝑠2 = 0.3047
c. ά = 0.10, 𝑛 = 98, 𝑥̅ = 66.3, 𝑠2 = 2.48
Calvin owns a water refilling station in his
neighbourhood. To assess the efficiency of his
company’s operation, he decided to do a study of the
water consumption of his costumers. He selected 45
households at random where the number of liters (L)
of water consumed by each household during the past
six months was recorded. The average consumption
was found to be 134.6 L with a standard deviation of
21.1 L. What is a 95% confidence interval for the
mean water consumption during the past six months
among his company’s customers?
How do you compute the confidence interval estimate
based on the appropriate form of the estimator for the
population mean?
How do you solve problems involving confidence
interval estimation of the population mean?
1.) Find and interpret a (1 – ά) 100% confidence
interval for the population mean μ given the
following values:
a. ά = 0.01, 𝑛 = 100, 𝑥̅ = 18.5, 𝜎 2 = 9.27
b. ά = 0.10, 𝑛 = 49, 𝑥̅ = 7.23, 𝑠2 = 4.47
c. ά = 0.05, 𝑛 = 81, 𝑥̅ = 66.3, 𝑠2 = 6.23
2.) A random sample of 10 chocolate energy bars of a
certain brand has, on the average, 230 calories
with known population standard deviation of 15
calories. Construct and interpret a 99% confidence
interval for the mean calorie content of this brand
of energy bar. Assume that the distribution of
calories is approximately normal.
3.) A commonly used IQ test is scaled to have a mean
of 100 and a standard deviation of 15. A school
118
counsellor was curious about the average IQ of the
students in her school and took a random sample of
forty students’ IQ scores. The average of these scores
was 107.9. Find a 95% confidence interval for the
mean student IQ in the school.
V. REMARKS
(Indicate special cases
including but not limited to
continuation of lesson
plan to the following day
in case of re-teaching or;
lack of time, transfer of
lesson to the following
day, in cases of class
suspension, etc.)
VI. REFLECTION
(Reflect on your teaching
and assess yourself as a
teacher. Think about your
student’s progress. What
works? What else needs
to be done to help the
students learn?)
A. No. of learners who
earned 80% in the
evaluation.
B. No. of learners who
require additional
activities for remediation
who scored below 80%
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson
D. No. of learners who
continue to require
remediation
E. Which of my teaching
strategies worked? Why
did these work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did I
use/discover which I
wish to share with other
teachers?
119
ANSWER KEY:
Let’s Practice:
a. (13.59, 14.61)
We can be 95% confident that the interval (13.59, 14.61) contains the true
value of the population mean.
b. (6.99, 7.47)
We can be 99% confident that the interval (6.99, 7.47) contains the true value
of the population mean.
c. (66.04, 66.56)
We can be 90% confident that the interval (66.04, 66.56) contains the true
value of the population mean.
Practical Problem:
A 95% confidence interval for the mean water consumption is (128.44, 140.76)
Evaluation:
1. a. (17.72, 19.28)
We can be 99% confident that the interval (17.72, 19.28) contains the true value
of the population mean.
b. (6.73, 7.73)
We can be 90% confident that the interval (6.73, 7.73) contains the true value of
the population mean.
c. (65.76, 66.84)
We can be 95% confident that the interval (65.76, 66.84) contains the true value
of the population mean.
2. A 99% confidence interval for the mean calorie content of this energy bar is
(217.92, 242.21).
We can therefore be 99% confident that the true mean calorie content of this brand
of energy bar is between 217.79 and 242.21 calories.
3. A 95% confidence interval for the mean student IQ in the school is (103.25,
112.55)
120
121
School
Teacher
Time and Date
I.
Objectives
A. Content
Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II. Topic
A. References
B. Teacher’s guide
pages
C. Learner’s material
pages
D. Textbook pages
E. Additional materials
from learning
resource portal
F. Other learning
resources
IV. Procedure
A. Reviewing previous
lesson and
presenting the new
lesson
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key concepts
of estimation of population mean and population
proportion.
The learner is able to estimate the population mean and
population proportion to make sound inferences in reallife problems in different disciplines.
1. Explain the process on how to determine the t–
distribution value.
2. Illustrate the t– distribution
3. Construct a t– distribution
M11-12SP-IIIg-2 and m11-12sp-IIIg-3
t – Distribution
Mercado, Jesus P., et al., Next Century Mathematics
Statistics and Probability, 2016
Belecina, Rene R., et al., Statistics and Probability, 2016
pen, board, calculators, visual aids, pentel pen, manila
papers
 Daily routine (opening prayer, checking of attendance)
 Recall the properties of normal distribution:
 Ask a student to illustrate the graph of normal
distribution.
 Call other set of students to label the
components of the normal curve (line of
symmetry, asymptote, location of measures of
central tendency and standard deviation)
B. Establishing the
 Emphasize that there are situations that can be treated
by normal distribution, t– distribution instead.
purpose of the lesson
 Present the learning objectives. (Write or post the
learning objectives on the board)
C. Presenting a
examples/ instances
of a new lesson
 Given a normal distribution curve, show the t distribution curve.
 Let the students compare and contrast the two curves.
122
Normal
distribution
df= 1
df= 3
-6
-4
-2
0
2
4
6
 Present and discuss the conditions when to use t–
distribution and its formula.
t= x - 
s/ n
where: x = sample mean
 = population mean
s = standard deviation of the sample
n = sample size
 Emphasize to the class that the computed t- value
needs to be compare to the critical t- tabular value.
 Guide the students on how to determine the degree of
freedom and how to find the critical t value given the
sample size (refer to the Table of t critical values).
D. Discussing new
concepts and
practicing new skills
#1
 Let the students find the critical t- value given the
following data:
1) n = 10; confidence level = 95% ; one tail
2) n = 18; confidence level = 99% ; two tails
3) n = 25; confidence level = 90% ; one and two tails
 Ask at least 4 students to illustrate t -distribution curves
of the given data above with its corresponding critical
value.
 Let others students evaluate the answers and
illustrations of their classmates.
E. Discussing new
concepts and
practicing new skills
#2
Let us try to use t- distribution to solve this problem:
A group of Grade 12 Practical Research 2 students,
conducted a survey regarding the family profile of SHS
students in terms economic status. Based from the data
collected, they are suspecting that it is significantly lower
than the national record stating that the Filipino families
average monthly income is P22,000. Is their suspicion
correct? Use 95% confidence level. Below are the monthly
income (in thousands) of 16 respondents:
123

Present to the class the steps on how to determine
the t-value.
 Ask the class to perform the following:
Step 1: Find the mean and standard deviation.
Mean = 21 and Standard deviation = 7.81
Step 2: Find the degree of freedom
df = n – 1
= 16 – 1
= 15
Step 3: Find the critical tabular t- value
Using the table of critical t- value where  = 5%
and df= 15.
The critical value is 2.131
Step 4: Compute the t- distribution value
t= x - 
s/ n
21- 22
-1
=
=
= -0. 5122
7.81/ 16
1. 9525

Ask the students to plot the critical and computed tvalues in the curve.
-2.131 -0. 5122

F. Developing Mastery
-2.131
Lead the class to interpret the result and formulate
conclusion.
- Is the suspicion of the group of Grade 12
researchers correct? Justify your answer.
Group Activity:
 Divide the class into five groups and let each group
perform the given task.
 In your respective group, calculate and illustrate the
value of t- distribution.
 Formulate appropriate conclusion.
124
Group Group Group Group Group
1
2
3
4
5
Sample
mean
Population
mean
Standard
deviation
Sample
size
Confidence
level ;
two- tails
12
21
35. 5
60
120. 5
14
18. 5
40.22
58.1
132.4
3
5
10
6.2
4. 5
25
12
16
20
10
90 %
95%
99%
90 %
95%
Gallery Walk:
 Ask each group to post their solution on the wall/
board.
 Let other groups evaluate each solution or answer.
Note: While other members are roaming around, one
member from each group should stay in their post to
present the solution and answer possible questions of
other groups.
G. Finding Practical
Applications of
Concepts and skills
in daily life


What is the importance of t- distribution in decision
making?
Let at least five students cite real- life situations where
they can apply the concept of t-distribution.
H. Making
Generalization and
abstraction about the
lesson



What new insights you gained from our lesson today?
What is t distribution?
What are the steps in construct t- distribution?
I.
The mean scores of a random sample of 22 TVL students
in General Mathematics test is 43. If the standard deviation
of the scores is 6.2 and population mean scores is 40.4,
find the t- distribution value and describe the result. Use 90
% confidence level, two-tails.
Evaluating Activities
J. Additional Activities
for Application or
Remediation
V-REMARKS
VI-REFLECTION
125
School
Teacher
Time and Date
I. OBJECTIVES
A. Content
Standards
B. Performance
Standards
C. Learning
Competencie
s/ Objectives
(Write the LC
code for
each)
II.
CONTENT
III. LEARNING
RESOURCES
A. References
1. Teacher’s
Guide
pages
2. Learner’s
Materials
pages
3. Textbook
pages
4. Additional
Learning
Materials
for
Learning
B. Other
Learning
Resources
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key
concepts of estimation of population mean and
population proportion.
The learner is able to estimate the population mean
and population proportion to make sound inferences in
real-life problems in different disciplines.
Learning Competency:
Identifies regions under the t-distribution
corresponding to different t- values. (M11/12SP – IIIg4)
Specific Objectives:
At the end of the session, the learner is able to:
1. Use the t-distribution in determining the critical
values.
2. Identify regions under the t-distribution that
corresponds to different t-values.
3. Apply some real-life situation in the concepts
learned in these lessons
IDENTFYING REGIONS UNDER T-DISTRIBUTION
CORRESPONDING TO T-VALUES
Next Century Mathematics by Mercado, Jesus P. pp.
225 – 227, 230-231
Internet (Google)
126
IV. PROCEDURES
A. Reviewing
previous
lesson or
presenting
the new
lesson
B. Establishing
a purpose for
the lesson
C. Presenting
examples/
instances of
the new
lesson
1. Using the given graph,
how do we compare tscores and z-scores for a
given level of confidence?
2. Describe a t-distribution. In what instances are we
going to use the t-distribution in determining the
confidence interval of a sample population?
3. How do we find the degrees of freedom given the
population (n)?
What is the use of degrees of freedom in using the tdistribution table?
Group Game: Find the following critical values in the
Table of t-Critical Values. Students will be provided
with the Table of t-Critical Values. The teacher will
flash t-values on the screen one-by-one. Three
points will be given to the group with the correct
answer after the given time.
1. t 0.05 for df = 8
6. t 0.10 for df = 9
2. t 0.05 for df = 25
7. t 0.01 for df = 20
3. t 0.025 for df = 15
8. t 0.01 for df = 18
4. t 0.025 for df = 27
9. t 0.025 for df = 9
5. t 0.10 for df = 12
10. t 0.025 for df = 7
A sample of size n = 20 is a simple random sample
selected from a normally distributed population. Find
the value of t such that the shaded area to the left of t
is 0.05.
D. Discussing
Find the critical values of t when the area of the rightnew
hand tail of the t-distribution is:
concepts and
a. 0.05; df = 39
practicing
b. 0.10; df = 54
new skills # 1
E. Discussing
new
concepts and
practicing
new skills # 2
Suppose you have a sample of size n = 12 from a
normal distribution. Find the critical value 𝑡2∝ that
corresponds to a 95% confidence level.
1. What is the degree of freedom df?
2. Using the confidence level of 95%, what is ∝
∝
and 2 ?
3. What is the critical value 𝑡2∝ ?
F. Developing
Mastery
(Leads to
Formative
Using the t-distribution table. Find the degrees of
freedom and the critical values of the following data.
1. n=20 at 95% confidence coefficient
2. n=10 at 99 % confidence coefficient
127
assessment
3)
G. Finding
practical
applications
of concepts
and skills in
daily living
3. n=12 at 95 % confidence coefficient
Dana wants to know the age of all entering Grade 12
for the school year 2020-2021. The mean age of a
random sample of 25 students is 18 years and
standard deviation is 1.3 years. The sample comes
from a normally distributed population. Use ∝ =0.1 to
determine the critical value of the given data.
H. Making
generalizatio
ns and
abstractions
about the
lesson
How to Calculate the Score for a t-Distribution
I.
Answer the following questions.
Evaluating
learning
J. Additional
activities for
application or
remediation
Step 1: Subtract one from your sample size. This will
be your df, or degrees of freedom.
Step 2: Look up the df in the left hand side of the tdistribution table.
Locate the column under your alpha level (the alpha
level is usually given to you in the question.
(Average Group)
1. The t distribution has
degrees of
freedom.
2. n
b. 2
c. 10
d. n -1
3. What is the df and the critical value if n=25 at
99% confidence level (use two-tail)?
4. For a t-distribution with 25 degrees of freedom,
find the values of t such that the area to the
right of t is 0.05.
(Advance Group)
1. Jose took a random sample of n = 12 giant squid
and tracked them to calculate their mean
lifespan. Their lifespans were roughly symmetric
with a mean of 𝑥̅ = 4 years and a standard
deviation of s = 0.5 years. He wants to use this
data to construct a t interval for the mean
lifespan of this type of squid with 90%
confidence. What critical value 𝑡2∝
Should Jose use?
2. For a t-distribution with 14 degrees of freedom,
find the values of t such that the area between –
t and t is 0.90.
3. What is the critical value 𝑡2∝ for constructing a
98% confidence interval for a mean from a
sample size of n = 15 observation?
On Math journal:
1. After the lesson, I have learned that
________________________.
128
2. I want to clarify
____________________________________
____.
V. REMARKS
VI. REFLECTION
A. No. of learners
who earned 80%
on the formative
assessment
B. No. of learners
who require
additional
activities for
remediation who
earned below
80%
C. Did the remedial
lessons work?
No. of learners
who have caught
up with the
lesson.
D. No. of learners
who continue to
require
remediation
E. Which of my
teaching
strategies worked
well? Why did
this work?
F. What difficulties
did I encounter
which my
principal or
supervisor can
help me solve?
G. What innovation
or localized
materials did I
use/ discover
which I wish to
share with other
teachers?
129
Daily Lesson Plan in Statistics and Probability
Grade 11
I.
OBJECTIVES
A. Content Standard
B. Performance Standard
C. Learning
Competency/Objectives
Write the LC code for
each
II.
CONTENT
III.
LEARNING
RESOURCES
A. Reference
1. Teacher’s Guide pages
2. Learner’s Material pages
3. Textbook pages
4. Additional Materials from
Learning
Resource(LR) Portal
B. Other Learning
Resources
IV.
PROCEDURES
A. Reviewing previous
lesson or presenting the
new lesson
The learner demonstrates understanding of key
concepts of estimation of population mean and population
proportion
The learner is able to estimate the population mean
and population proportion to make sound inferences in
real-life problems in different disciplines.
The learner identifies percentiles using the t-table.
M11/12SP-IIIg-5
Estimation of Parameters
Statistic and Probability by Belencina, Baccay & Mateo
Senior High Conceptual Math and Beyond Statistics and
Probability
pp.168-171
pp.138-140
https://www.dummies.com/education/math/statistics/howto-find-t-values-for-confidence-intervals/
, Manila paper, pentel pen, t-table, improvised bingo card,
projector, laptop
Preliminary Activity:
The learners will be ask to determine whether the
statement is True or False and if it is false modify the
statement to make it true.
1. The shape of a normal curve is symmetrical.
2. When the confidence level is 90%, 𝛼 = 0.01
3. All confidence levels have the same confidence
coefficient.
4. The confidence coefficient for a 95% confidence
interval are ± 2.58.
5. The Central Limit Theorem states that as sample
size increases, the shape of the distribution
approximates the normal curve.
Answer:
1. TRUE
2.TRUE
3.
FALSE(Different)
4.FALSE (±1.96)
5.T
130
B.
Establishing a purpose for
the lesson
C. Presenting
examples/instances of the
new lesson
Present the objective of the lesson through Power Point
Presentation
Show the illustration of a Normal Distribution when the
z
sample size becomes small.
t-for
n=21
t-for n==6
D.Discussing new concepts and
practicing new skills # 1
0
Start the lesson by telling the history of t-distribution
originated.
The t-distribution was formulated in 1908 by WS Gosset
an Irish brewing employee.
Discuss that t-distribution is a type of probability
distribution that is similar to the normal distribution with its
bell shape, but has heavier tails
Explain that when 𝜎 is not known, it must be
estimated by s, the sample standard deviation and when
the sample size is small the critical values greater than
the values for 𝑧𝛼/2 and with small sample size, more
𝑠
standard errors
are needed to span the 0.95
√𝑛
confidence intervals and the tail of the normal curve
becomes heavier.
This number of standard error values is called t.
D. Discussing new concepts
and practicing new skill
#2
 Discuss the steps to determine the percentile of the
distribution using t-table.
The t-values found in the reproduced t-table are the
proportion of the areas in two tails of the t-curve.
 Define the percentile and give examples
A percentile is a number on a statistical distribution whose
less-than probability is the given percentage;
Example 1,
The 95th percentile of the t-distribution with n – 1 degrees
of freedom is that value of whose left-tail (less-than)
probability is 0.95 (and whose right-tail probability is 0.05).
 What is a degree of freedom?
A degree of freedom are the number of values that are
free to vary after a sample statistic has been computed.
A degree of freedom also suggest the specific curve
applicable when a distribution consist of family curve.
Step 1. Find the degree of freedom if n = 5
df = n-1
= 5-1
df = 4
 What does this means of having a degree of freedom
which is 4?
131
Answer: 4 values are free to vary and one must be a fixed
value.
Step 2 :Using the t-table, you look at the row for df = 4.
The 95th percentile is the number where 95% of the
values lie below it and 5% lie above it, so you want the
right-tail area to be 0.05. Move across the row, find the
column for 0.05, and you get 3.182
This is the 95th percentile of the t-distribution with 4
degrees of freedom.
E. Developing mastery
leads to Formative
Assessment
F. Finding practical
application of concept
and skill in daily living.
To determine the percentile divide the class into three
groups and distribute the worksheets, and the t-table
Group 1: and Group 2 -TRIVIA
Group 3. and Group 4 - BINGO
The t-values found in the reproduced t-table are the
proportion of the areas in two tails of the t-curve.
They are critical values of t in the sense that they are the
boundaries of the middle area where the true mean lies.
Like the z they are also called confidence coefficient.
Example:
The 95th percentile of the t-distribution
with 6 degrees of freedom is that value of
whose left-tail (less-than) probability is 0.95
(and whose right-tail probability is 0.05).
Using t-table the value is 2.447 and is
located as shown in the graph.
G. Making generalization
and abstraction about the
lesson
𝛼 = 0.05
95
% 0 2.447
Give the summary by asking the students what are the
steps to determine the percentile using the t-table
Step 1.Solve the degree of freedom using the formula df –
n-1.
Step 2. Look at the row for the degree of freedom (df)
Srep3. Move across the row, and find the column for the
confidence level to get the value of the percentile of the tdistribution with the degree of freedom.
H. Evaluating Learning
Determine the percentile using the t-table
1.The sample size n is 6 and 90% confidence level
2.The sample size n is 12 and 90% confidence level
3. The sample size n is 17 and 95% confidence level
4. The sample size n is 8 and 99% confidence level
5. The sample size n is 17 and 90% confidence level
132
Answers:
1) 2.015
2) 1.796
3)2.120
4)3.499
5)1.746
J. Additional activities for
application or
remediation
V.
REMARKS
VI.
REFLECTION
A. No of learners who earned
80% in the
evaluation
B. No of learners who require
additional
activities for remediation who
scored
below 80%
C. Did the remedial lessons
work? No. of
learners who have caught up
with the
lesson
D. No of learners who continue
to require remediation
E. Which of my teaching
strategies worked well? Why did
these work?
F. What difficulties did I
encounter which
my principal or supervisor can
help me
solve?
G. What innovation or localized
materials did I use/discover
which I wish to share with other
teachers?
133
Group 1
TRIVIA
A former military captain in the Philippines who died in the sinking of
Titanic.
Determine the percentile of the distribution using the t -table.
A. The sample size n is 4 and 95% confidence level
B. The sample size n is 8 and 95% confidence level
.C.The sample size n is 11 and 90% confidence level
D. The sample size n is 7 and 90% confidence level
E. The sample size n is 18 and 90% confidence level
F.The sample size n is 12 and 90% confidence level
G.The sample size n is 14 and 99% confidence level
H.The sample size n is 10 and 90% confidence level
I.The sample size n is 17 and 99% confidence level
J.The sample size n is 8 and 99% confidence level
K.The sample size n is 13 and 95% confidence level
L. The sample size n is 9 and 99% confidence level
M.The sample size n is 21 and 99% confidence level
N The sample size n is 3 and 95% confidence level
O. The sample size n is 10 and 99% confidence level
P. The sample size n is 7 and 95% confidence level
R. The sample size n is 20 and 99% confidence level
S The sample size n is 11 and 99% confidence level
T. The sample size n is 18 and 95% confidence level
U. The sample size n is 12 and 90% confidence level1.
Write the letter of the correct answer in the box to find the answer
3.1
82
2.8
61
1.8
12
1.8
33
2.9
21
2.3
65
3.1
82
3.3
55
1.9
43
2.3
65
1.7
96
2.1
10
2.1
10
3.1 2.8
82
61
A
R
Answer:
1.8
12
C
1.8
33
H
2.9
21
I
2.3
65
B
3.1
82
A
3.3
55
L
1.9
43
D
2.3
65
B
1.7
96
U
2.1
10
T
2.1
10
T
134
Group 2
TRIVIA
What was the name of the philosopher who once stated "Children today are
tyrants. They contradict their parents, gobble their food, and tyrannize their
teachers"?
Determine the percentile of the distribution using the t -table.
A. The sample size n is 4 and 95% confidence level
B. The sample size n is 8 and 95% confidence level
.C.The sample size n is 11 and 90% confidence level
D. The sample size n is 7 and 90% confidence level
E. The sample size n is 18 and 90% confidence level
F.The sample size n is 12 and 90% confidence level
G.The sample size n is 14 and 99% confidence level
H.The sample size n is 10 and 90% confidence level
I.The sample size n is 17 and 99% confidence level
J.The sample size n is 8 and 99% confidence level
K.The sample size n is 13 and 95% confidence level
L. The sample size n is 9 and 99% confidence level
M.The sample size n is 21 and 99% confidence level
N The sample size n is 3 and 95% confidence level
O. The sample size n is 10 and 99% confidence level
P. The sample size n is 7 and 95% confidence level
R. The sample size n is 20 and 99% confidence level
S The sample size n is 11 and 99% confidence level
T. The sample size n is 18 and 95% confidence level
U. The sample size n is 12 and 90% confidence level1.
Write the letter of the correct answer in the box to find the answer
3.1 3.3 2.3 1.7 2.8 2.1 1.7 2.9 4.3 3.1
2.1 1.7 2.9
82 55 65 40 61 10 40 21 03 69
10 40 21
4.3
03
Answer
3.1
82
A
3.3
55
L
2.3
65
B
1.7
40
E
2.8
61
R
2.1
10
T
1.7
40
E
2.9
21
I
135
4.3
03
N
3.1
69
S
2.1
10
T
1.7
40
E
2.9
21
I
4.3
03
N
Group 3 and Group 4
BINGO
Each bingo card contains 24 numbers in decimal form and a blank square,
situated on a 5 by 5 grid. When the game starts, random problems are drawn in
determining the percentile of the distribution using t- tables whoever of the players
participating in the game completes a bingo pattern first, wins the prize (a line with
five numbers in diagonal, horizontal or vertical row).
Problems to be drawn in a box
.1.The sample size n is 4 and 95% confidence level
2. The sample size n is 8 and 95% confidence level
.3. .The sample size n is 11 and 90% confidence level
4. The sample size n is 7 and 90% confidence level
5. The sample size n is 18 and 90% confidence level
6. The sample size n is 12 and 90% confidence level
7. The sample size n is 14 and 99% confidence level
8. The sample size n is 10 and 90% confidence level
9. The sample size n is 17 and 99% confidence level
10.The sample size n is 8 and 99% confidence level
11.The sample size n is 13 and 95% confidence level
12. The sample size n is 9 and 99% confidence level
13.The sample size n is 21 and 99% confidence level
14. The sample size n is 3 and 95% confidence level
15. The sample size n is 10 and 99% confidence level
16. The sample size n is 7 and 95% confidence level
17. The sample size n is 20 and 99% confidence level
18. The sample size n is 11 and 99% confidence level
19. The sample size n is 18 and 95% confidence level
20. The sample size n is 12 and 90% confidence level
21. The sample size n is 10 and 90% confidence level
22. The sample size n is 7 and 99% confidence level
23. The sample size n is 20 and 95% confidence level
24. The sample size n is 9 and 99% confidence level
25. The sample size n is 18 and 95% confidence level
26. The sample size n is 12 and 90% confidence level
136
3.182 2.365 1.812 1.943 1.740
1.833 1.796 3.012 2.921 3.499
2.179 3.355 FREE 2.845 4.303
2.447 2.861 3.169 2.110 1.796
1.833 3.707 2.093 3.355 5.841
2.845 3.012
3.169
1.796
3.355
1.943 3.499
3.182
2.110
1.833
2.861 1.796 FREEE 1.740
3.707
1.812 2.179
2.447
2.921
4.303
3.355 2.365
1.833
5.841
2.093
3.707 1.833 5.841 1.796 2.447
2.160 2.110 1.796 2.365 3.499
1.812 3.182 FREE 3.012 2.845
3.355 2.921 2.093 3.355 1.833
2.179 2.861 1.740 4.303 3.169
137
138
School
Teacher
Time and Date
I.
OBJECTIVES
A. Content
Standards
B. Performance
Standards
C. Learning
Competencies/
Objectives
(Write the LC
code for each)
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key
concepts of estimation of population mean and
population proportion.
The learner is able to estimate the population mean
and population proportion to make sound
inferences in real-life problems in different
disciplines.
Learning Competency:
Computes for the confidence interval estimate
based on the appropriate form of the estimator for
the population mean. (M11/12SP – IIIh- 1)
Specific Objectives:
At the end of the session, the learner is able to:
4. Define confidence level and confidence
interval
5. Solve for the margin of error and confidence
interval (interval estimate).
6. Appreciate the importance of confidence
interval as one of the statistical techniques.
II.
CONTENT
LEARNING
RESOURCES
A. References
5. Teacher’s
Guide pages
6. Learner’s
Materials
pages
7. Textbook
pages
8. Additional
Learning
Materials for
Learning
B. Other Learning
Resources
IV.
PROCEDURES
A. Reviewing
previous
lesson or
presenting
COMPUTING INTERVAL ESTIMATES FOR THE
POPULATION MEAN
III.
Statistics and Probability by Belecina, Rene R.,
et.al. pp. 153 - 166
Internet (Google)
Recall solving point estimate: “Think – Group –
Share”. Using the “show me board” ask each group
to give the point estimate of the following data
(given will be flashed on the screen one by one).
139
the new
lesson
After the allotted time they will raise their answer.
For every correct answer, the group will receive 2
points.
1. 4, 3.5, 3.3, 3, 3.8
2. 25, 34, 45, 30, 53, 55
3. 80, 75,86,81,77, 76, 82, 90
4. The height of the group
5. The daily allowance of the group
B. Establishing a
purpose for the
lesson
Suppose we want to know the “true” average
weight of all the students in our class. Form five
groups and assume that these groups are random
samples. Let the students list their weights
carefully.
1. What are the different ways to find the
weights of the members per group?
2. What is the “true” average weight of the
students?
3. How would you describe your group based
on the result of the computation?
4. If you are going to enhance the precision of
your guess, what is needed to be modified?
5. What is your estimate of the mean of the
population where your group seems to
belong?
6. Reflect on your estimation. Are you confident
about it? To what extent are you confident?
Express your confidence as a percentage.
7. What do we call this value (percentage or
confidence)?
C. Presenting
examples/
instances of
the new lesson
Suppose we state a confidence of 95%, given that
𝑥̅ = 18, n = 50 and 𝜎 = 4
1. What is the best point estimate of the
population mean?
2. What is the margin of error (E)?
3. What is the 95% confidence interval of the
population mean?
D. Discussing
new concepts
and practicing
new skills # 1
Group activity: Applying the steps in calculating
the interval estimates, let the students solve the
following, then let them present and discuss their
work in front.
Average students: Sample population is
normally distributed; 𝑥̅ = 42, n = 40 and 𝜎 = 3. Find
the 95% confidence interval estimate for 𝜇.
Advanced students: Sample population is
normally distributed; n = 100, mean = 16 and
population variance = 16. What is the 99%
confidence interval estimate for 𝜇.
140
E. Discussing
new concepts
and practicing
new skills # 2
K. Developing
Mastery
(Leads to
Formative
assessment 3)
The mean score of a random sample of 49 Grade
11 students who took the first periodic test is
calculated to be 78. The population variance is
known to be 0.16.
a. Find the 99% confidence interval for the
mean of the entire Grade 11 students.
b. Find the lower and the upper confidence
limits.
Solve Me! The teacher will distribute the worksheet
per group with word problem and template on it. The
students will brainstorm how to solve the given
problem.
Average students: A researcher wants to
estimate the number of hours that a senior high
school spend studying their lessons. A sample of 50
Grade 11 students was observed to have a mean
studying time of 3 hours. The population is normally
distributed with a population standard deviation of
0.5 hours. Find:
a. The best point estimate of the population
mean
b. The 95% confidence interval of the
population mean
Advance students: A random selection of 40
entering Grade 11 GAS has the following GWAs
(general weighted average). Assume that 𝜎 = 0.46.
96
89
86
89
93
87
89
96
97
90
87
86
90
87
86
87
97
86
90
86
86
83
98
86
87
90
87
83
89
87
87
82
83
82
98
92
86
83
89
86
Estimate the true mean GWA with 99% confidence
then describe the result.
L. Finding
practical
applications of
concepts and
skills in daily
living
Think-Pair-Share: Using the confidence levels
90%, 95%, and 99%. Let each group construct a
situation or a problem consisting a mean (𝑥̅ ),
number of sample (n), and population standard
deviation (𝜎). Exchange this problem to other group
then solve for the confidence interval.
M. Making
generalizations
and
abstractions
about the
lesson
1. What is a confidence level?
2. What is an interval estimate?
3. What is the computing formula for margin of
error (E)?
141
4. What is the general formula for confidence
intervals for large samples?
5. How do you compute the interval estimate?
N. Evaluating
learning
Solve: You asked 50 students how satisfied they
were with their track in Senior High School with a
10-point scale, with 1 = not at all satisfied and 10 =
extremely satisfied. It was found out that the mean
point was 7.5 with the standard deviation of 3.5.
Use 90% confidence to compute the interval
estimate of the population mean.
O. Additional
activities for
application or
remediation
The mean and the standard deviation of the blood
sugar level of randomly selected 50 patients in a
hospital are 130 mg/dl and 4.6 mg/dl, respectively.
a. Find the 90% confidence interval for the
mean of all patients in the hospital.
b. Find the lower and upper confidence limits.
V.
REMARKS
VI.
REFLECTION
A. No. of learners who
earned 80% on the
formative
assessment
B. No. of learners who
require additional
activities for
remediation who
earned below 80%
C. Did the remedial
lessons work? No.
of learners who
have caught up
with the lesson.
D. No. of learners who
continue to require
remediation
E. Which of my
teaching strategies
worked well? Why
did this work?
F. What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
G. What innovation or
localized materials
did I use/ discover
which I wish to
share with other
teachers?
142
School
Teacher
Time and Date
I.
OBJECTIVES
A. Content
Standards
B. Performance
Standards
C. Learning
Competencies/
Objectives
(Write the LC
code for each)
II.
CONTENT
LEARNING
RESOURCES
C. References
1. Teacher’s
Guide
pages
2. Learner’s
Materials
pages
3. Textbook
pages
4. Additional
Learning
Materials for
Learning
D. Other Learning
Resources
IV.
PROCEDURES
A. Reviewing
previous
lesson or
presenting
the new
lesson
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key concepts
of estimation of population mean and population
proportion.
The learner is able to estimate the population mean and
population proportion to make sound inferences in real-life
problems in different disciplines.
Learning Competency:
Solves problems involving confidence interval
estimation of the population mean. (M11/12SP – IIIh- 2)
Specific Objectives:
At the end of the session, the learner is able to:
7. Recall the steps in calculating the confidence
interval
8. Solve word problems involving confidence interval
estimates
9. Show perseverance and active participation during
group activities
SOLVING WORD PROBLEMS INVOLVING
CONFIDENCE INTERVAL ESTIMATION FOR THE
POPULATION MEAN
III.
Next Century Mathematics (Statistics & Probability for
Senior High School) by Jesus P. Mercado pp. 233 – 242
Group Work: Fish bowl method. Ask a representative to
pick 1 question from the bowl then discuss the answer with
the group. A representative will give the idea of the group
after the allotted time.
a. What is an interval estimate? How does it differ from
point estimate?
b. What is the advantage of interval estimate over point
estimate?
143
B. Establishing a
purpose for the
lesson
c. What are the variables required in order to compute
for the margin of error (E)? Give the formula.
d. What are the four-step process in calculating the
interval estimate?
Think – group – share: Group numbers 1 & 3 to answer
number 1 problem, and group numbers 2 & 4 to solve
number 2. They will discuss in front their solution after the
given time.
1. Given data: 99% confidence level; n=50, 𝑥̅ =
18,000 and 𝜎= 2,500. Assuming normality, use the
given data to find the following:
a. Margin of error
b. Confidence interval for estimating the
population parameter.
2. Find the minimum sample size required to estimate
an unknown population mean 𝜇 using the following
data: confidence level = 90%; margin of error =
0.891; 𝜎 2 = 9
C. Presenting
examples/
instances of
the new lesson
The mean and the standard deviation of the content of a
sample of 10 similar containers are 10.5 liters and 0.352,
respectively. Assume that the containers are
approximately normally distributed.
a. Find a 95% confidence interval for the actual mean
content.
b. Find the lower and upper confidence limits.
D. Discussing
new concepts
and practicing
new skills # 1
Group activity: Applying the steps and formulas in
calculating the confidence interval for estimating the
population mean when the population variance and
population standard deviation are unknown and the
sample size n is less than 30 (n<30), let the students solve
the following, then let them present and discuss their work
in front.
Average students: The following are randomly
selected scores in Statistics and Probability test of twelve
Grade 11 students:
75 65 76 80 85 77 81 83 80
70 71 69
a. Find a 99% confidence interval for the mean score
of all grade 11 students, assuming that the
students’ score is approximately normally
distributed.
b. Find the lower and upper confidence limits
Advanced students: The following data are selected
randomly from a population of normally distributed values.
36 38 43 40 46 45 46 43 47
49
44 48 47 43 51 50 52 53 54
56
a. Construct a 95% confidence interval to estimate 𝜇.
b. Find the lower and the upper confidence limits.
144
E. Discussing
new concepts
and practicing
new skills # 2
F. Developing
Mastery
(Leads to
Formative
assessment
3)
Try Me! : Suppose the following data are selected
randomly from a population of normally distributed values
with unknown variance:
40
39
45
45
49
41
47
44
56
53
39
42
44
38
57
58
a. Compute the mean and standard deviation
b. Construct a 90% confidence interval to estimate
the population mean 𝜇.
c. Find the lower and upper confidence limits.
d. Interpret the interval.
Solve Me! The teacher will let the students pick from a bowl
their favorite color of candies. Students with the same color
of candies will be grouped. Each group will be given a
worksheet (and template) with word problem. The students
will brainstorm how to solve the given problem.
Yellow Group: A random sample of n = 24 data from a
normal distribution with unknown variance produced 𝑥̅ =
42.5 and s = 2.6.
c. Find a 90% confidence interval for the population
mean.
d. Interpret the interval.
Red Group: A random sample of n = 19 data from a
normal distribution with unknown variance produced 𝑥̅ =
32.7 and s = 2.5.
a. Find a 90% confidence interval for the population
mean.
b. Interpret the interval.
Blue Group: A random sample of n = 16 data from a
normal distribution with unknown variance produced 𝑥̅ =
25.7 and s2 = 5.29.
a. Find a 99% confidence interval for the population
mean.
b. Interpret the interval.
Green Group: A random sample of n = 14 data from a
normal distribution with unknown variance produced 𝑥̅ =
35.7 and s2 = 6.76.
a. Find a 99% confidence interval for the population
mean.
b. Interpret the interval.
G. Finding
practical
applications of
concepts and
skills in daily
living
Jeric observed that the mean age of 25 Red Cross
volunteers of Sorsogon city is 18 years with standard
deviation of 5 years. What is the interval estimate of the
population mean? Adapt 95% confidence level.
145
H. Making
generalizations
and
abstractions
about the
lesson

If n<30 and 𝜎 is unknown, the confidence interval for
population mean 𝜇 is :
𝑠
𝑠
𝑠
(𝑥̅ − 𝑡2∝
, 𝑥̅ + 𝑡2∝
) 𝑜𝑟 𝑥̅ − 𝑡2∝
< 𝜇
√𝑛
√𝑛
√𝑛
𝑠
< 𝑥̅ + 𝑡2∝
√𝑛
Where:
𝑥̅ = mean of a random sample of size n
n = sample size
s = sample standard deviation
∝
𝑡2∝ = t-value at (1 − ) 100% confidence level

The confidence interval can be written as:
( 𝑥̅ – E , 𝑥̅ + E) or 𝑥̅ – E < 𝜇 < ̅𝑥 + E
To find the margin of error (E), use the formula:
𝑠
E = 𝑡2∝
where 𝑡2∝ has n-1

2
√𝑛

I.
Evaluating
learning
J. Additional
activities for
application or
remediation
V.
VI.
degrees of freedom
To solve problems involving confidence interval of the
population mean, the following steps should be
followed:
a. Find the sample mean 𝑥̅ and the sample standard
deviation s (if sample mean and standard deviation
is unknown)
b. Find the degrees of freedom df
c. Find ∝ in (1 - ∝)100% confidence level
d. Find the critical value of 𝑡2∝ using the Table of tCritical Values
e. Find the margin of error E
f. Find the confidence interval
g. Find the lower and upper confidence limits
The following were scores in a General Mathematics test
selected by a teacher from all the test scores of Grade 11
students.
74
78
86
88
69
83
70
83
87
62
92
66
65
75
74
90
76
57
72
83
Assume that the above scores were randomly selected by
the teacher from a normal population and the variance is
unknown.
a. Calculate the mean and standard deviation
b. Construct a 95% confidence interval to estimate
the population mean score 𝑥.
̅
c. Find the lower and upper confidence limits.
d. Interpret the interval.
On your journal, copy and complete the following
sentences:
1. I found the lesson ________.
2. I have to know more about ________________.
3. I am still confused about __________________.
REMARKS
REFLECTION
146
B.
C.
D.
E.
F.
G.
A. No. of
learners
who earned
80% on the
formative
assessment
No. of learners who
require additional
activities for
remediation who
earned below 80%
Did the remedial
lessons work? No.
of learners who
have caught up
with the lesson.
No. of learners who
continue to require
remediation
Which of my
teaching strategies
worked well? Why
did this work?
What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
What innovation or
localized materials
did I use/ discover
which I wish to
share with other
teachers?
147
School
Teacher
Time and Date
I.
OBJECTIVES
A. Content
Standards
B. Performance
Standards
C. Learning
Competencies/
Objectives
(Write the LC
code for each)
II.
CONTENT
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide pages
2. Learner’s
Materials
pages
3. Textbook
pages
4. Additional
Learning
Materials for
Learning
B. Other Learning
Resources
IV.
PROCEDURES
A. Reviewing
previous
lesson or
presenting
the new
lesson
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key concepts
of estimation of population mean and population
proportion.
The learner is able to estimate the population mean and
population proportion to make sound inferences in reallife problems in different disciplines.
Learning Competency:
Draws conclusion about the population mean based
on its confidence interval estimate. (M11/12SP – IIIh- 3)
Specific Objectives:
At the end of the session, the learner is able to:
10. Apply the steps in solving confidence interval
estimation of the population mean
11. Describe and interpret the population mean
based on its confidence interval estimate.
12. Value the importance of confidence interval as
one of the statistical tool in analyzing the data.
DRAWING CONCLUSION ABOUT THE POPULATION
MEAN BASED ON ITS CONFIDENCE INTERVAL
ESTIMATE
III.
Statistics and Probability by Belecina, Rene R., et.al. pp.
172 - 177
Internet (Google)
Thinking Skills: By pair, let the students reflect on the
different steps on how to solve problems involving
confidence interval estimation of the population mean by
completing the following sentences. (For advance
learners, answers will not be provided. For average
learners, answers can be selected from the meta strips
posted on the board:
1. The parameter of interest is the ________ of
the population where the sample comes from.
148
2. Specify the ______________ criteria. This
information includes the ______ size and the
_________; the ____ of confidence and the
______ value.
3. Collect _______ and find the ______ estimate.
4. Determine the ______ interval. Find the
confidence ________ and compute the
_______. Solve for the upper and lower
_________, then ________ the results.
Answers:
1. Mean
2. Confidence interval; sample; standard deviation;
level; critical
3. Evidences; point
4. Confidence; coefficient; maximum error E;
confidence limits; describe
B. Establishing a
purpose for the
lesson
(Show a picture of a farmer harvesting rice and entail a
story) After applying a certain kind of fertilizer, Mang Tino
harvested “palay” which yield a population standard
deviation to be 4 sacks and there are 30 sacks of “palay”
harvested. Find the interval estimate of the population
mean per sack, using 95% confidence.
8. The parameter of interest is the _____ of the
____ where the sample comes from.
9. The ____ test is applicable to this kind of
problem.
10. The level of confidence is ____ the degree of
freedom is ____ and the confidence coefficients
are ____.
11. The point estimate is _____.
12. The margin of error is _____.
13. The interval estimate ranges from ____ to ____.
14. The result shows that
____________________________________.
C. Presenting
examples/
instances of
the new lesson
The 15 residents of Brgy. Cambulaga attempted to add
a channel on their cable TV to a list of favorites. After
the task, they rated the difficulty using a 7-point scale (7
being the most difficult task). The responses are shown
below:
2, 1, 3, 6, 7, 1, 2, 3, 5, 7, 2, 2, 1, 3, 2
Using the 95% confidence interval, what will be your
conclusion pertaining to this result?
The mean score of a random sample of 17 students who
took a special test is 83.5. If the standard deviation of
the scores is 4.1 and the sample comes from an
approximately normal distribution, what are the point
and interval estimates of the population mean adopting
a confidence level of 99%.
1. What is the parameter of interest?
2. Is t-test applicable to this kind of problem?
D. Discussing
new concepts
and practicing
new skills # 1
149
3. What are the confidence level and the critical tvalues?
4. What is the maximum error(E)?
5. What is the point estimate?
6. At what values does the interval estimate
ranges from?
7. What does the results tell us?
E. Discussing
new concepts
and practicing
new skills # 2
In Sorsogon provincial hospital, a sample of 10 weeks
was selected and found out that the average of 10
babies were born per week. Find the 90% confidence
interval of the true mean if the standard deviation of the
sample was 3. Draw inferences regarding the result of
the data.
F. Developing
Mastery
(Leads to
Formative
assessment
3)
Solve Me! The teacher will distribute the activity sheet
based on the performance level of the students. This will
be done by triad and the students will brainstorm how to
solve the given problem.
Average students: We can say with 99% confidence
that the interval between 17. 33 and 18.67 contains the
true mean age of the population of entering Grade 12
students based on the sample size of 25. The point
estimate for the population mean is 18 years. How will you
interpret the result of this problem?
Advance students: A sample of 60 Grade 9 students’
ages was obtained to estimate the mean age of all grade
9 students. Mean age = 15.3 years and the population
variance is 16.
a. Find the 95% confidence interval for 𝜇.
b. What conclusions can you make based on each
estimate?
G. Finding
practical
applications of
concepts and
skills in daily
living
H. Making
generalizations
and
abstractions
about the
lesson
Suppose observation on 25 chocolate bars selected from
a normally distributed population yield an average weight
of 200g with a standard deviation of 10 g, what is the
interval estimates using 95% confidence level? Draw
conclusion about the result.
I.
Evaluating
learning


How will you come up with a conclusion in a given
problem?
In describing the results, the statement reflects our
confidence in the interval estimation process rather
than the value computed from the sample data. This is
because repeated application may yield different lower
and upper limits of the interval. However, interval
estimates are more reliable than point estimates
because of the confidence coefficients associated with
the range of values. This is also the reason why they
are generally preferred to point estimates.
Draw me close! In an interview held among 20
randomly selected senior high school students, the
mean daily allowance of the students was found out to
150
J. Additional
activities for
application or
remediation
be Php20 with a standard deviation of Php3. Use 90%
confidence to compute the interval estimate of the
population mean. Come up with a clear and specific
interpretation of the results of the data.
You have learned how to compute means and standard
deviations. You have also learned how to estimate the
population mean 𝜇. Do you think that the t-test is a
versatile test in the sense that it may be used for
both small and large samples? Explain your answer
V.
REMARKS
VI.
REFLECTION
A. No. of learners who
earned 80% on the
formative
assessment
B. No. of learners who
require additional
activities for
remediation who
earned below 80%
C. Did the remedial
lessons work? No.
of learners who
have caught up
with the lesson.
D. No. of learners who
continue to require
remediation
E. Which of my
teaching strategies
worked well? Why
did this work?
F. What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
G. What innovation or
localized materials
did I use/ discover
which I wish to
share with other
teachers?
151
School
Teacher
Time and Date
I. OBJECTIVES
A. Content Standards
B. Performance
Standards
C. Learning
Competencies/
Objectives
II. CONTENT
III. LEARNING
RESOURCES
A. References
1. Teacher’s Guide
Pages
2. Learner’s Material
Pages
3. Textbook Pages
4. Additional Materials
from Learning
Resources
B. Other Learning
Resources
IV. PROCEDURES
A. Reviewing previous
lesson or presenting
the new lesson
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key
concepts of estimations of population mean and
population proportion.
The learner is able to estimate the population mean
and population proportion to make sound inferences in
real-life problems in different disciplines.
1. The learner identifies point estimator for the
population proportion. (M11/12SP-IIIi-1)
2. The learner computes for the point estimate of the
population proportion. (M11/12SP-IIIi-2)
Estimation of Parameters
Statistics and Probability 344-346
1. Statistics and Probability 146-151
2. Statistics and Probability by Danilo De Guzman
pages 133-135
Activity:
Jason noted that every weekend, a group of
athletes jog in a nearby park. On a particular day, he
observed that there were 20 runners and that 8 of
them were female. He observed the joggers for five
weeks, noting their number and their gender.
Fill-in the blanks in the table to complete the
data and answer question below:
Week
1st
2nd 3rd 4th 5th
Male
12
12
10
Female
8
10
6
15
Total
20
26
20
Guide Question:
What proportions of runners are female
a. in fraction form?
b. in decimal form?
c. in percentage form?
152
B. Establishing a
purpose for the
lesson
Solution:
To determine the proportion of the female
runners, simply compute as follows:
Week
1st
2nd
3rd
4th
5th
Male
12
16
12
10
9
Female
8
10
6
10
15
Total
20
26
18
20
24
8
10
6
10
15
Proportion
20
26
18
20
24
Decimal
0.4
0.38 0.33
0.5
0.625
Percentag 40% 38% 33% 50% 62.5%
e
C. Presenting
examples/instances
of the new lesson
The percentage expression 40% is also called
a proportion. Percentages are commonly used when
reporting frequencies of subsets of populations. In
computing, percentages are first converted to
proportions in decimal form. Proportions represent
probabilities. If Jason makes a probability statement
that all joggers in the park on the first week are
female, he may report 40% or 0.40.
D. Discussing new
concepts and
practicing new skill
#1
We denote 𝑝 as the population proportion and
𝑝̂ “p hat” as the point estimate of sample proportion.
𝒙
̂ = , where 𝑥 is the number
The formula is 𝒑
𝒏
of sample elements that possess the desired
characteristics and 𝑛 is the sample size.
Example 1:
If 350 students from the graduates of a batch
were surveyed and 30 of them answered that they
took up BS Mechanical Engineering (BSME), what is
the estimated proportion of those who took up BS ME
out of the whole batch?
Solution:
Let 𝑝̂ as the sample proportion of BSME
graduates, 𝑥=30 (number of BS ME graduates), and
𝑛=350 (total number of surveyed graduates).
𝑥
30
𝑝̂ = =
= 0.086 or 8.6%
𝑛
E. Discussing new
concepts and
practicing new skill
#2
350
We denote 𝑞 as the sample proportion of
“not 𝑝”, and 𝑞̂ “q hat” as the point estimate of
proportion of “not 𝑝”.
̂ =𝟏− 𝒑
̂
The formula is 𝒒
Example 2:
From example 1, what is the estimated
proportion of graduates who didn’t take up BSME?
153
Solution:
Let 𝑞̂ as the sample proportion of non-BSME
graduates.
𝑞̂ = 1 − 𝑝̂ = 1 − 0.086 = 0.914 or 91.4%
F. Developing mastery
(Leads to Formative
Assessment)
G. Finding practical
applications of
concepts and skills
in daily living
Firm Up:
Find 𝑝̂ and 𝑞̂, given 𝑥 and 𝑛.
a. 𝑥 = 56 ; 𝑛 = 80
b. 𝑥 = 35 ; 𝑛 = 96
c. 𝑥 = 420 ; 𝑛 = 1000
Answers:
a. 𝑝̂ = 0.7 or 70%
; 𝑞̂ = 0.3 or 30%
b. 𝑝̂ = 0.364 or 36.4% ; 𝑞̂ = 0.636 or 63.6%
c. 𝑝̂ = 0.42 or 42%
; 𝑞̂ = 0.58 or 58%
Restaurants regularly ask customers to
accomplish questionnaires on the kind of service that
their staff renders. Suppose 1 200 people are
randomly chosen from a target population and are
asked if they like the services of a specific restaurant.
There are 908 respondents who like the service. What
is the estimate of true proportion of all customers who
like the service of the restaurant?
Solution:
Let 𝑝 the population proportion.
We can estimate 𝑝 by computing
𝑝̂ =
𝑁𝑜. 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 𝑤ℎ𝑜 𝑙𝑖𝑘𝑒 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑖𝑐𝑒
𝑁𝑜. 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 𝑠𝑎𝑚𝑝𝑙𝑒𝑑
So, 𝑝̂ =
908
1200
= 0.757
Only 76% of all customers like the services of
the restaurant. The owner might want to improve the
service to increase the revenue of the restaurant.
H. Making
generalizations and
abstractions about
the lesson
A point estimate is a single value used to
approximate a population parameter.
̂=
The point estimator of sample population is 𝒑
𝒙
̂ =𝟏− 𝒑
̂ as the estimator of “not 𝑝̂ ”.
, and 𝒒
𝒏
154
I.
Evaluating Learning
1. What is the point estimator of the population
proportion?
2. Nora conducted a survey for a fast food
restaurant owner who wanted to know the level of
acceptability (Acceptable or Not Acceptable) of a new
food combo among customers. There were 182
respondents found the new food combo was
acceptable. If there were 256 in the sample, what is 𝑝̂
and 𝑞̂ ?
Answer:
𝑥
1. 𝑝̂ =
𝑛
2. a. There are 182 who find the new food combo
acceptable. In terms of proportion:
𝑥 182
𝑝̂ = =
= 0.71 = 71%
𝑛 256
2. b. There are 74 who find the new food combo
not acceptable. This proportion is
𝑞̂ = 1 − 𝑝̂ = 1 − 7.1 = 0.289 = 28.9% or 29%
J. Additional activities
for application or
remediation
V. REMARKS
VI. REFLECTION
A. No. of learners who
earned 80% in the
evaluation
B. No. of leaners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lessons work? No.
of learners who
have caught up with
the lesson
D. No. of leaners who
continue to require
remediation
E. Which of my
teaching strategies
worked well? Why
did these work?
F. What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
155
G. What innovation or
localized materials
did I use/discover
which I wish to
share with other
teachers?
156
School
Teacher
Time and Date
I. OBJECTIVES
A. Content Standards
B. Performance
Standards
C. Learning
Competencies/
Objectives
II. CONTENT
III. LEARNING
RESOURCES
C. References
1. Teacher’s Guide
Pages
2. Learner’s Material
Pages
3. Textbook Pages
4. Additional Materials
from Learning
Resources
D. Other Learning
Resources
IV. PROCEDURES
A. Reviewing
previous lesson or
presenting the new
lesson
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key
concepts of estimations of population mean and
population proportion.
The learner is able to estimate the population mean
and population proportion to make sound inferences in
real-life problems in different disciplines.
3. The learner identifies the appropriate form of the
confidence interval estimator for the population
proportion based on Central Limit Theorem.
(M11/12SP-IIIi-3)
4. The learner computes for the confidence interval
estimate of the population proportion. (M11/12SPIIIi-4)
Estimation of Parameters
Statistics and Probability 346-348
1. Statistics and Probability 146-151
2. Statistics and Probability by Danilo De Guzman
pages 134-138
Review the previous lesson.
If 350 students from the graduates of a batch
were surveyed and 30 of them answered that they
took up BS Mechanical Engineering (BSME), what is
the estimated proportion of those who took up BS ME
out of the whole batch?
Solution:
𝑥
30
a. 𝑝̂ = =
= 0.086 or 8.6%
𝑛
350
b. 𝑞̂ = 1 − 𝑝̂ = 1 − 0.086 = 0.914 or 91.4%
Hence, the point estimate of the population
proportion of those who took up BSME out of the
whole batch is 0.086 or 8.6%.
157
B. Establishing a
purpose for the
lesson
Estimates about a population proportion may be
made just like estimates of the population mean. Just
like means, we can also form confidence intervals
about the population proportion.
C. Presenting
examples/instances
of the new lesson
If 350 students from the graduates of a batch
were surveyed and 30 of them answered that they
took up BS Mechanical Engineering (BSME), construct
90% confidence interval for the estimated proportion of
those who took up BS ME out of the whole batch?
D. Discussing new
concepts and
practicing new skill
#1
Solution:
30
Step 1: Given 𝑥 = 30, and 𝑛 = 350, 𝑝̂ =
=
350
0.086. Thus, 𝑞̂ = 1 − 𝑝̂ = 0.914.
Step 2: Since 𝑛𝑝̂ and 𝑛𝑞̂ are greater than 5, and
𝑛 ≥ 30, the sample size is normally distributed
(according to Central Limit Theorem). Thus, use the
confidence interval estimate:
𝑝̂ 𝑞̂
𝑝̂ 𝑞̂
𝑝̂ − 𝑧𝛼 √ < 𝑝 < 𝑝̂ + 𝑧𝛼 √
𝑛
𝑛
2
2
Step 3: Recall that for a confidence level of 90%,
𝛼 = 10% and 𝑧𝛼 = 1.645 (use z-table).
2
Step 4: Substituting the given values, we get:
0.086−1.645√
(0.086)(0.914)
350
0.086+1.645√
<𝑝<
(0.086)(0.914)
350
Step 5: With 90% confidence, we state that the
interval estimate from 0.061 to 0.111 or 6.1% to 11.1%
contains the true percentage of graduates of a batch
who answered that they took up BS Mechanical
Engineering (BSME).
158
E. Discussing new
concepts and
practicing new skill
#2
To check if the confidence interval estimate is
true, we let:
𝑝̂𝑞̂
𝑛
as the lower boundary, and
𝑝̂𝑞̂
𝑛
as the upper boundary.
𝑝̂ − 𝑧𝛼 √
2
𝑝̂ + 𝑧𝛼 √
2
Then, get the average of the lower and upper
boundaries as shown below:
𝐿𝐵+𝑈𝐵
𝑝̂ =
2
0.061+ 0.111
=
2
𝑝̂ = 0.086
It shows that 𝑝̂ = 0.086 or 8.6% lies between the
confidence interval estimate from 0.061 to 0.111 or
6.1% to 11.1%.
F. Developing mastery
(Leads to Formative
Assessment)
Firm Up:
Compute the confidence interval of the
population proportion given 𝑛, 𝑝̂ , and the confidence
level.
a. 𝑛 = 210, 𝑝̂ = 0.60, 95%
b. 𝑛 = 460, 𝑝̂ = 0.53, 90%
c. 𝑛 = 678, 𝑝̂ = 0.28, 99%
Use this table as reference for confidence interval
calculation.
Confidence
Critical (𝑧)
Level
value
50%
0.67449
75%
1.15035
90%
1.64485
95%
1.95996
97%
2.17009
99%
2.57583
99.9%
3.29053
G. Finding practical
applications of
concepts and skills
in daily living
Answers:
a. from 0.534 to 0.666 or 53.4% to 66.6%
b. from 0.492 to 0.568 or 49.2% to 56.8%
c. from 0.235 to 0.324 or 23.5% to 32.4%
In a survey of 500 random households in a
particular village, mothers have been asked if they use
cell phones to communicate. There are 376 who have
said YES. Use a 95% confidence to estimate the
proportion of all mothers who use the cell phone to
communicate.
Solution:
159
376
Step 1: Given 𝑥 = 376, and 𝑛 = 500, 𝑝̂ =
=
500
0.752. Thus, 𝑞̂ = 1 − 𝑝̂ = 0.248.
Step 2: Since 𝑛𝑝̂ and 𝑛𝑞̂ are greater than 5, and
𝑛 ≥ 30, the sample size is normally distributed
(according to Central Limit Theorem). Thus, use the
confidence interval estimate:
𝑝̂ 𝑞̂
𝑝̂ 𝑞̂
𝑝̂ − 𝑧𝛼 √ < 𝑝 < 𝑝̂ + 𝑧𝛼 √
𝑛
𝑛
2
2
Step 3: Recall that for a confidence level of 95%,
𝛼 = 5% and 𝑧𝛼 = 1.96 (use z-table).
2
Step 4: Substituting the given values, we get:
(0.752)(0.248)
500
0.752−1.96√
<𝑝<
(0.752)(0.248)
0.752+1.96√
H. Making
generalizations and
abstractions about
the lesson
500
Step 5: With 95% confidence, we state that the
interval estimate from 0.714 to 0.79 or 71.4% to 79%
contains the true percentage of households in a
particular village that uses cell phones to
communicate.
Since 𝑛𝑝̂ and 𝑛𝑞̂ are greater than 5, and 𝑛 ≥ 30,
the sample size is normally distributed (according to
Central Limit Theorem). Thus, use the confidence
interval estimate:
𝑝̂ 𝑞̂
𝑝̂ 𝑞̂
𝑝̂ − 𝑧𝛼 √ < 𝑝 < 𝑝̂ + 𝑧𝛼 √
𝑛
𝑛
2
2
Then, to check if the confidence interval estimate
is true get the average of the lower and upper
boundaries as shown below:
𝐿𝐵 + 𝑈𝐵
𝑝̂ =
2
I.
Evaluating Learning
1. What is the confidence interval estimator for
the population proportion based on Central Limit
Theorem?
2. Nora conducted a survey for a fast food
restaurant owner who wanted to know the level of
acceptability (Acceptable or Not Acceptable) of a new
food combo among customers. Get the 97%
confidence interval that 182 respondents found the
new food combo was acceptable if there were 256 in
the sample.
Answer:
𝑝̂𝑞̂
𝑛
1. 𝑝̂ − 𝑧𝛼 √
2
𝑝̂𝑞̂
𝑛
< 𝑝 < 𝑝̂ + 𝑧𝛼√
2
2. With 97% confidence, we state that the interval
estimate from 0.648 to 0.772 or 64.8% to 77.2%
contains the true percentage of respondents who
found the new food combo was acceptable.
160
J. Additional activities
for application or
remediation
V. REMARKS
VI. REFLECTION
A. No. of learners who
earned 80% in the
evaluation
B. No. of leaners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lessons work? No.
of learners who
have caught up with
the lesson
D. No. of leaners who
continue to require
remediation
E. Which of my
teaching strategies
worked well? Why
did these work?
F. What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
G. What innovation or
localized materials
did I use/discover
which I wish to
share with other
teachers?
161
Daily Lesson Plan in Statistics and Probability
Grade 11/12
Quarter 3 Week 9
I.
OBJECTIVES
A. Content Standard
B. Performance Standard
C. Learning
Competency/Objectives
II.
CONTENT
The learner demonstrates understanding of key
concepts of estimation of population mean and
population proportion.
The learner is able to estimate the population mean
and population proportion to make sound inferences
in real-life problems in different disciplines.
The learner solves problems involving confidence
interval estimation of the population proportion.
M11/12SP-IIIi-5
Problem Solving on Confidence Interval
Estimation of Population Proportion
III.
LEARNING RESOURCE
References
Statistics and Probability by Belencina, Baccay &
Mateo pp. 195-199
Other Learning Resource
Manila paper, calculator, permanent markers
IV.
PROCEDURES
A. Reviewing previous
This activity can be done orally, or given as a
lessons or presenting
written task which can be accomplished either
the new lesson
individually, dyad, triad, or in a small group.
A. Compute 𝑞̂ given 𝑝̂ . (Note that 𝑞̂ = 1 − 𝑝̂ )
a. 𝑝̂ = 0.02
(Answer: 𝑞̂ = 0.98)
b. 𝑝̂ = 0.50
(Answer: 𝑞̂ = 0.50)
c. 𝑝̂ = 0.80
(Answer: 𝑞̂ = 0.20)
d. 𝑝̂ = 0.62
(Answer: 𝑞̂ = 0.38)
e. 𝑝̂ = 0.60
(Answer: 𝑞̂ = 0.40)
𝑋
B. Find 𝑝̂ , the point estimate 𝑝. (Note that 𝑝̂ = )
𝑛
1. X = 12; n = 30
( Answer: 𝑝̂ =0.4)
2. X = 111; n = 300
( Answer: 𝑝̂ =0.37)
3. X = 250; n = 500
( Answer: 𝑝̂ =0.50)
4. X = 360; n = 900
( Answer: 𝑝̂ =0.4)
5. X = 150; n = 600
( Answer: 𝑝̂ =0.25)
C. Determine the 𝑍𝛼 given the following
2
confidence level.
1. 90% (Answer: 𝑍𝛼=1.64)
2
2. 95%
(Answer: 𝑍𝛼=1.96)
3. 97%
(Answer: 𝑍𝛼=2.17)
4. 98%
(Answer: 𝑍𝛼=2.33)
5. 99%
(Answer: 𝑍𝛼=2.58)
2
2
2
2
Note: For slow learners, give only 2 items in Parts
A and B while.
162
D. Establishing a purpose
for the lesson
Inform the learners that throughout the lesson, they
will be asked to solve word problems involving
confidence interval estimation of population
proportion.
Inform the learners of the steps in solving word
problem that they can use later on while solving
word problems.
E. Presenting
Examples/Instances of
the Lesson
Step 1. Describe the population parameter of
interest
Step 2. Specify the confidence Interval Criteria
a. Check the assumptions
b. Determine the test statistic to be used to
calculate the interval
c. State the level of confidence
Step 3. Collect and preset sample evidence
a. Collect the sample information.
b. Find the point estimate
Step 4. Compute the interval estimate
a. Find 𝑞̂.
b. Find the maximum error E.
c. Find the limits.
d. Describe the results.
Present the problem below to the learners.
In a graduate teacher college, a survey was
conducted to determine the proportion of students
who want to major in Science. If 368 out of 850
students said Yes, with 95% confidence, what
interpretation can we make regarding the probability
that all students in the teacher graduate college
want to major in Science?
After presenting the word problem, the learners will
be grouped into small groups with a minimum of 45 learner in every group to solve the given problem.
Provide the learners will materials such as manila
paper and permanent markers. Learners may use
calculators in doing this task.
F. Discussing New
concepts and
Practicing New Skills #
1
The learners will be asked to post their outputs and
share their solution to the class. (Note: If there are
groups which arrived at the same answer, choose
only one from among this groups to present the
output.)
The learners should be able to arrive at the following
answer.
Steps
Describe the population
parameter of interest
163
Solution
The parameter of
interest is the mean
proportion p of all
students in the teacher
graduate college who
want to major in
Science.
Specify the confidence
Interval Criteria
1. Check
the By CLT, the sample
assumptions
size of 850 is normally
distributed.
2. Determine the
test statistic to The test statistic is the
be
used
to p.
calculate
the
interval
3. State the level of
confidence
Confidence level: 95%
and α = 0.05
Confidence coefficient
= 1.96
Collect
and
preset
sample evidence
1. Collect
the X = 368 and n=850.
sample
information.
2. Find the point 𝑝̂ = 𝑋 = 368 = 0.432 ≈
𝑛
850
estimate
0.43
The point estimate of
the population is 0.43.
Compute the interval
estimate
1. Find q ̂.
𝑞̂= 1-𝑝̂ = 1 – 0.43 =
0.57
2. Find
the
maximum error
𝑝̂𝑞̂
(0.43)(0.57)
√
= √
=
E.
𝑛
850
0.017
3. Find the limits.
For the lower limit
𝑝̂𝑞̂
𝑛
𝑝̂ - 𝑍𝛼 √
2
=0.43 – 1.96(0.017)
=0.43 – 0.034
= 0.396 or 39.6%
For the upper limit
𝑝̂ + 𝑍𝛼 √
2
𝑝̂𝑞̂
𝑛
=0.43 + 1.96(0.017)
= 0.463 or 46.4%
164
Describe the results.
Thus,
with
95%
confidence, we can
assert that the interval
from 39.6% to 46.4%
contains
the
true
percentage
of
all
graduate students who
want to major in
Science.
For slow and average learners, a guided worksheet
may be used. (see Attached Practice Exercise 1).
G. Developing Mastery
This task will be accomplished individually.
Problem:
In a certain food stall, 278 out of 500 randomly
selected consumers indicate their preference for
new kind of food combination. Use 99% confidence
interval to estimate the true proportion p who like
new food combination.
Distribute Practice Exercise 2. (see attached)
Practice Exercise 2A for slow and average
learners and Practice Exercise 2B for advanced
learners.
H. Making generalization
and abstraction about
the lesson
What are the steps in solving word problems
involving confidence interval estimation of
population proportion?
I.
What are the essential formulae/formulas that we
need to solve this types of problems?
For Slow learners, distribute Worksheet 1.
Evaluating Learning
For average and advance learners, distribute
Worksheet 2.
165
Practice Exercise 1
Problem. In a graduate teacher college, a survey was conducted to determine the
proportion of students who want to major in Science. If 368 out of 850 students said
Yes, with 95% confidence, what interpretation can we make regarding the probability
that all students in the teacher graduate college want to major in Science?
Steps
Solution
Describe the population parameter of The parameter of interest is the mean
interest
proportion p of all students in the teacher
graduate college who want to major in
Science.
Specify the confidence Interval
Criteria
By CLT, the sample size of 850 is
1. Check the assumptions
normally distributed.
2. Determine the test statistic to be
used to calculate the interval
3. State the level of confidence
Collect and preset sample evidence
1. Collect the sample information.
2. Find the point estimate
The test statistic is the p.
Confidence level: 95% and α = 0.05
Confidence coefficient = _______
X = 368 and n=850.
𝑝̂ =
𝑋
𝑛
=
368
850
= ________
The point estimate of the population is
___.
Compute the interval estimate
1. Find q ̂.
2. Find the maximum error E.
3. Find the limits.
𝑞̂= 1-𝑝̂ = 1 – 0.43 = _______
𝑝̂𝑞̂
𝑛
√
(0.43)(0.57)
=
850
=√
_______
For the lower limit
𝑝̂𝑞̂
𝑛
𝑝̂ - 𝑍𝛼 √
2
=0.43 – 1.96(0.017)
=0.43 – _______
= __________
For the upper limit
𝑝̂ + 𝑍𝛼 √
2
Describe the results.
𝑝̂𝑞̂
𝑛
=0.43 + 1.96(0.017)
= 0.43 + _______
= _________
Thus, with 95% confidence, we can
assert that the interval from ____% to
____% contains the true percentage of
all graduate students who want to major
in Science.
166
Practice Exercise 2A
Problem: In a certain food stall, 278 out of 500 randomly selected consumers indicate
their preference for new kind of food combination. Use 99% confidence interval to
estimate the true proportion p who like new food combination.
Steps
Solution
Describe the population parameter of The parameter of interest is the mean
interest
proportion p of ____________________
Specify the confidence Interval
Criteria
1. Check the assumptions
2. Determine the test statistic to be
used to calculate the interval
3. State the level of confidence
Collect and preset sample evidence
1. Collect the sample information.
2. Find the point estimate
By CLT, the sample size of _____ is
normally distributed.
The test statistic is the p.
Confidence level: 99% and α = 0.01
Confidence coefficient = _______
X = 278 and n=500
𝑋
𝑝̂ = = = ________ (round to the nearest Thousandths)
𝑛
The point estimate of the population is
___.
Compute the interval estimate
1. Find q ̂.
2. Find the maximum error E.
3. Find the limits.
𝑞̂= 1-𝑝̂ = 1 – ______ = _______
𝑝̂𝑞̂
𝑛
√
(
=√
)(
500
)
= _______
For the lower limit
𝑝̂𝑞̂
𝑝̂ - 𝑍𝛼 √
2
𝑛
=0.556 – 2.58(______)
=0.556 – _______
= __________
For the upper limit
𝑝̂ + 𝑍𝛼 √
2
Describe the results.
𝑝̂𝑞̂
𝑛
=0.556 + 2.58(_____)
= 0.556 + _______
= _________
Thus, with 99% confidence, we can
assert that the interval from ____% to
____% contains the true proportion of all
consumers who like the new food
combination.
167
Practice Exercise 2B
Problem: In a certain food stall, 278 out of 500 randomly selected consumers indicate
their preference for new kind of food combination. Use 99% confidence interval to
estimate the true proportion p who like new food combination.
Steps
Describe the population parameter of
interest
Solution
Specify the confidence Interval
Criteria
4. Check the assumptions
1. Determine the test statistic to be
used to calculate the interval
2. State the level of confidence
Collect and preset sample evidence
1. Collect the sample information.
2. Find the point estimate
Compute the interval estimate
1. Find q ̂.
2. Find the maximum error E.
3. Find the limits.
Describe the results.
Thus, with 99% confidence, we can
assert that the interval from ____% to
____% contains the true proportion of all
consumers who like the new food
combination.
168
Worksheet 1
Problem: In a survey, 1000 Grade 7 students were asked if they read storybooks.
There were 318 who said Yes. Use 95% confidence interval to determine the
population proportion p of all grade 7 students who read story books.
Steps
Solution
Describe the population parameter of The parameter of interest is the mean
interest
proportion p of all Grade 7 students who
read story books.
Specify the confidence Interval
Criteria
By CLT, the sample size of 1000 is
1. Check the assumptions
normally distributed.
The test statistic is the p.
2. Determine the test statistic to be
used to calculate the interval
Confidence level: 95%, α = 0.05.
Confidence coefficient: _________
3. State the level of confidence
Collect and preset sample evidence
1. Collect the sample information.
X = _______ n =1000
2. Find the point estimate
𝑋
𝑝̂ = = = ___________
𝑛
The point estimate of the population
proportion is ___________
Compute the interval estimate
1. Find q ̂.
𝑞̂= 1-𝑝̂ = 1 – ______ = _______
2. Find the maximum error E.
𝑝̂𝑞̂
𝑛
√
(
)(
500
=√
)
= _______
3. Find the limits.
For the lower limit
𝑝̂𝑞̂
𝑛
𝑝̂ - 𝑍𝛼 √
2
= _______ – 1.96(______)
=_______ – _______
= __________
For the upper limit
𝑝̂ + 𝑍𝛼 √
2
𝑝̂𝑞̂
𝑛
= _______ + 1.96(______)
=_______ + _______
= __________
Describe the results.
Thus, with 99% confidence, we can
assert that the interval from ____% to
____% contains the true proportion of all
Grade 7 students who read story books.
169
Worksheet 1
Problem: In a survey, 1000 Grade 7 students were asked if they read storybooks.
There were 318 who said Yes. Use 95% confidence interval to determine the
population proportion p of all grade 7 students who read story books.
Steps
Solution
Describe the population parameter of The parameter of interest is the mean
interest
proportion p of all Grade 7 students who
read story books.
Specify the confidence Interval
Criteria
By CLT, the sample size of 1000 is
4. Check the assumptions
normally distributed.
The test statistic is the p.
5. Determine the test statistic to be
used to calculate the interval
Confidence level: 95%, α = 0.05.
Confidence coefficient: _________
6. State the level of confidence
Collect and preset sample evidence
3. Collect the sample information.
X = _______ n =1000
4. Find the point estimate
𝑋
𝑝̂ = = = ___________
𝑛
The point estimate of the population
proportion is ___________
Compute the interval estimate
4. Find q ̂.
𝑞̂= 1-𝑝̂ = 1 – ______ = _______
5. Find the maximum error E.
𝑝̂𝑞̂
𝑛
√
(
)(
500
=√
)
= _______
6. Find the limits.
For the lower limit
𝑝̂𝑞̂
𝑛
𝑝̂ - 𝑍𝛼 √
2
= _______ – 1.96(______)
=_______ – _______
= __________
For the upper limit
𝑝̂ + 𝑍𝛼 √
2
𝑝̂𝑞̂
𝑛
= _______ + 1.96(______)
=_______ + _______
= __________
Describe the results.
Thus, with 99% confidence, we can
assert that the interval from ____% to
____% contains the true proportion of all
Grade 7 students who read story books.
170
School
Teacher
Time and Date
I. OBJECTIVES
A. Content Standards
B. Performance
Standards
C. Learning
Competencies/
Objectives
II.CONTENT
III. LEARNING
RESOURCES
A. References
1. Teacher’s Guide
2. Learner’s
Materials Page
3. Textbook Pages
IV.PROCEDURES
A. Reviewing previous
lesson or motivation
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key
concepts of tests of hypotheses on the population
mean and population proportion
The learner is able to perform appropriate tests of
hypotheses involving the population mean and
population proportion to make inferences in real-life
problems in different disciplines.
M11/12SP-IVa-1
Formulates the appropriate null and alternative
hypotheses on a population mean.
The learner illustrates (a) null hypothesis (b)
alternative hypothesis and (c) types of errors in
hypothesis testing.
Tests of Hypothesis
Commission on Higher Education and Philippine
Normal University (2016). Teaching Guide for
Senior High School: Statistics and Probability. pp
375-378
Belecina, R. R.,Baccay E. S.,Mateo E. B. Statistics
and Probability pp 216-225
Ask a students to react on the government
pronouncement about El Niño phenomenon.
Describe the El Niño phenomenon and its possible
consequences further.
“The country will experience El Niño phenomenon in
the next few months.”
Write learners’ reactions on the board. Their
reactions may include the following:
1. The occurrence of El Niño phenomenon is not
sure.
2. There is a possibility that El Niño phenomenon
may not occur.
3. The effects of El Niño phenomenon are
devastating to the country.
4. Some of the consequences of the El Niño
phenomenon are tolerable while other
consequences are not.
5. The validity of the statement could be tested
based on some empirical facts
171
B. Establishing a
purpose for the
lesson
How can you determine if your conjecture is either
true or false? What is your basis in making
decisions?
In decision-making, we usually follow certain
processes; weigh alternatives, collect evidence and
make a decision. After a decision is made, an
appropriate interpretation is made (or an action is
undertaken). We follow these basic processes in
testing hypothesis in statistics.
C. Presenting
examples/ instances
of the new lesson
Discuss the results of the motivational activity with
emphasis on the following points:
A statistical hypothesis is a claim or a conjecture
that may either be true or false
There are two possible actions that one can do with
the statement. These actions are either to accept
the statement or to reject it
The degree of the possible consequence is the
basis in making the decision. If the consequences of
accepting the claim that El Niño phenomenon is
going to happen are tolerable, then we may not
reject the pronouncement. However, if the
consequences are severe, then we reject the claim.
D. Discussing new
concepts and
practicing new skills
#1
A test of hypothesis is a procedure based on a
random sample of observations with a given level of
probability of committing an error in making the
decision, whether the hypothesis is true or false.
There are two kinds of a statistical hypothesis: the
null and the alternative hypothesis.
• Null hypothesis
-Denoted by Ho
-This is a statement or claim or conjecture to be
tested
• Alternative hypothesis
- Denoted by Ha
- the claim that is accepted in case the null
hypothesis is rejected
The null and alternative hypotheses are
complementary and must not overlap. The usual
pairs are as follow:
a. Ho: Parameter = Value versus Ha: Parameter ≠
Value;
b. Ho: Parameter = Value versus Ha: Parameter <
Value;
c. Ho: Parameter = Value versus Ha: Parameter >
Value;
172
d. Ho: Parameter ≤ Value versus Ha: Parameter >
Value; and
e. Ho: Parameter ≥ Value versus Ha: Parameter <
Value
Example: Consider the average number of text
messages that a Grade 11 student sends in a day.
The statement could be stated as follows:
“The average daily number of text messages that a
Grade 11 student sends is equal to 100.”
•
The statement “The average daily number
of text messages that a Grade 11 student sends is
equal to 100” is considered as the Null
hypothesis(Ho)
•
The statement “The average daily number of
text messages that a Grade 11 student sends is not
equal to 100” is the alternative statement(Ha)
E. Discussing new
concepts and
practicing new skills
#2
Types of error in hypothesis testing
Type I error
- an error is committed when we reject a true
hypothesis
Type II error
- an error is committed when we fail to reject
(accept) a false hypothesis
Action
Reject the hypothesis
Fail to reject(Accept)
the hypothesis
Hypothesis is
TRUE
Error
Committed
No Error
Committed
Hypothesis is
FALSE
No Error
Committed
Error
Committed
Example:
Alden is exclusively dating Maine. He remembers
that on their first date, Maine told him that her
birthday was this month. However, he forgot the
exact date. Ashamed to admit that he did not
remember, he decides to use hypothesis testing to
make an educated guess that today is Maine’s
birthday. Help Alden do it.
Answer:
•
The null hypothesis can be stated as Ho:
Today is Maine’s birthday while the alternative
hypothesis is Ha: Maine’s birthday is on another day
and not today.
•
Type I error is committed when Alden’s
guess of Maine’s birthday is not on this day and a
possible consequence is that Alden failed to greet
or give Maine a birthday gift today.
173
F. Developing mastery
(Leads to Formative
Assessment 3)
•
On the other hand, Type II error is
committed when Alden guessed that today is
Maine’s birthday. A possible consequence of this
Type II error is that Alden made the mistake of
greeting Maine a happy birthday on that day.
Group Activity
The class will be divided into 4 groups. Provide
each group with the materials needed in
accomplishing the tasks as manila paper, pentel
pen and hand out.
Direction: Formulate the appropriate null and
alternative hypotheses and identify situations where
Type I and Type II errors are committed. Have them
state its possible consequences.
Group 1
A manufacturer of IT gadgets recently
announced they had developed a new battery for
a tablet and claimed that it has an average life of
at least 24 hours. Would you buy this battery?
Answer:
The null hypothesis can be stated as Ho: The
average life of the newly developed battery for a tablet is
at least 24 hours while the alternative hypothesis is Ha:
The average life of the newly developed battery for a tablet
is less than 24 hours. Type I error is committed when you
did not buy the battery and a possible consequence is you
lost the opportunity to have a battery that could last for at
least 24 hours. On the other hand, Type II error is
committed when you did buy the battery and found out
later that the battery’s life was less than 24 hours. A
possible consequence of this Type II error is that you
wasted your money in buying the battery.
Group 2
A teenager who wanted to lose weight is
contemplating on following a diet she read about
in the Facebook. She wants to adopt it but,
unfortunately, following the diet requires buying
nutritious, low calorie yet expensive food. Help
her decide.
Answer:
The null hypothesis can be stated as Ho: The
diet will not result to a change in her weight while the
alternative hypothesis is Ha: The diet will induce a
reduction in her weight. Type I error is committed when the
teenager did follow the diet and a possible consequence
is that she spent unnecessarily for a diet that did not help
her reduce weight. On the other hand, Type II error is
committed when the teenager did not follow the diet. A
possible consequence of this Type II error is that the
teenager lost the opportunity to attain her goal of weight
reduction.
174
Group 3
After senior high school, Lily is pondering
whether or not to pursue a degree in Statistics.
She was told that if she graduates with a degree
in Statistics, a life of fulfilment and happiness
awaits her. Assist her in making a decision.
Answer:
The null hypothesis can be stated as Ho: Life
of fulfillment and happiness awaits her after obtaining a
degree in Statistics while the alternative hypothesis is Ha:
Life of fulfillment and happiness does not happen after
obtaining a degree in Statistics. Type I error is committed
when Lily does not pursue a degree in Statistics and a
possible consequence is that she’ll miss the promised life
of fulfilment and happiness after obtaining a degree in
Statistics. On the other hand, Type II error is committed
when Lily decides to obtain a degree in Statistics. A
possible consequence of this Type II error is that Lily will
miss the opportunity to experience a life of fulfilment and
happiness after obtaining a degree in Statistics.
Group 4
An airline company regularly does quality control
checks on airplanes. Tire inspection is included
since tires are sensitive to the heat produced
when the airplane passes through the airport’s
runway. The company, since its operation, uses
a particular type of tire which is guaranteed to
perform even at a maximum surface temperature
of 107oC. However, the tires cannot be used and
need to be replaced when surface temperature
exceeds a mean of 107oC. Help the company
decide whether or not to do a complete tire
replacement.
Answer:
The null hypothesis can be stated as Ho: The
surface temperature of the tires is at most 107 oC while
the alternative hypothesis is Ha: The surface temperature
of the tires is greater than 107 oC. Type I error is
committed when the airline company orders a tire
replacement when in fact it is not needed. A possible
consequence of this is that the company will waste money
in replacing the tires. On the other hand, Type II error is
committed when the airline company does not order tire
replacement. A possible consequence of this Type II error
is an accident that may happen because of no
replacement of the tires.
G. Making
generalizations and
abstractions about
the lesson
To generalize the lesson, the learners should be
able to answer the following:
1. How do you differentiate null hypothesis and
alternative hypothesis?
2. What is a statistical hypothesis?
3. How do you illustrate the two types of error in
hypothesis testing?
175
H. Evaluating learning
Choose the letter of the correct answer.
1. A Type II error is committed when
a. we reject a null hypothesis that is true.
b. we don't reject a null hypothesis that is
true.
c. we reject a null hypothesis that is false.
d. we don't reject a null hypothesis that is
false.
2. A Type I error is committed when
a. we reject a null hypothesis that is true.
b. we don't reject a null hypothesis that is
true.
c. we reject a null hypothesis that is false.
d. we don't reject a null hypothesis that is
false.
3. Selecting the significance level 𝛼 will
determine
a. the probability of a type I error
b. the probability of a type II error
c. power
4. Which of the following would be an
appropriate null hypothesis?
a. The mean of a population is equal to 50.
b. The mean of a sample is equal to 50.
c. The mean of a population is greater than
50.
d. Only (a) and (c) are true.
5. Which of the following would be an
appropriate alternative hypothesis?
a. The mean of a population is equal to 50.
b. The mean of a sample is equal to 50.
c. The mean of a population is greater than
50.
d. The mean of a sample is greater than 50
Answer: 1) d 2) a
V.REMARKS
VI. REFLECTION
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require additional
activities for
remediation
C. Did the remedial
lessons work? No. of
176
3) a
4) a
5) c
learners who caught up
with the lesson
D. No. of learners who
continue to require
remediation
E.
Which of my
teaching strategies
worked well? Why did
these work?
F.
What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
177
School
Teacher
Time and Date
I.
OBJECTIVES
A. Content
Standards
B. Performance
Standards
C. Learning
Competencies/
Objectives
II.CONTENT
III. LEARNING
RESOURCES
A. References
1. Teacher’s Guide
2. Learner’s Materials
Page
3. Textbook Pages
IV.PROCEDURES
A. Reviewing previous
lesson or motivation
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key
concepts of tests of hypotheses on the population
mean and population proportion
The learner is able to perform appropriate tests of
hypotheses involving the population mean and
population proportion to make inferences in reallife problems in different disciplines.
The learners
•
Illustrate level of significance and
corresponding rejection region
•
Calculate the probabilities of committing a
Type I and Type II error.
M11/SP-Iva-2
Tests of Hypothesis
Commission on Higher Education and Philippine
Normal University (2016). Teaching Guide for
Senior High School: Statistics and Probability. pp
375-378
Belecina, R. R.,Baccay E. S.,Mateo E. B.
Statistics and Probability pp 226-235
Ask learners how a court trial proceeds based on
their knowledge. Guide them by citing a popular
case and letting them identify the steps to come
up with a verdict for the case.
For example, take the case of former President
Marcos’ ill-gotten wealth case.
List the steps that the learners identified.
B. Establishing a
purpose for the lesson
They may mention the following:
1. State the accusation against the family of
former President Marcos.
2. Choose the jury. Set or review the guidelines to
be used in the decision-making process.
3. Present the evidences
4. Decide on the matter, based on the evidences.
5. State the verdict, based on the decision made.
How can we relate the steps in conducting
hypothesis testing on the steps in a court
proceedings? How can we make a decision
whether to reject or fail to reject (accept) the null
178
hypothesis? How can we determine the probability
of making a correct decision in accepting or
rejecting a true null hypothesis?
C. Presenting examples/
instances of the new
lesson
The teacher will inform the students that the
answers to those questions will be answered as
the discussions go on.
Emphasize that the steps in a court proceeding
are similar if one has to conduct a test of
hypothesis.
Steps in Testing Hypothesis
1. Formulate the
hypotheses to be
tested.
2. State the decision rule
that we will follow in
making a decision on
whether to reject or
fail to reject (accept)
the null hypothesis.
D. Discussing new
concepts and
practicing new skills
#1
Court proceedings
1. State the accusation or
the statement of what
will be evaluated as
true or false
2. It is a guideline that the
court uses to evaluate
the quantity and quality
of evidences to be
presented. And based
on this guideline, the
court decides whether
to reject or accept the
hypothesis that the
accused is not guilty.
3. Compute the value of
the test statistic using
a random sample of
observations gathered
or collected for the
purpose of the test of
hypothesis.
4. Use the decision rule
to make a decision
whether to reject or
fail to reject (accept)
the null hypothesis.
3. This is the time that the
gathered
evidences
are presented.
5. State the conclusion.
5. This is the time when
the court gives its
verdict on the accused.
In both scenarios, this
last step is the most
awaited part of the
procedure.
4. The jury or the court
will decide whether the
accused is guilty or not
based
on
the
evidences presented.
To be able to specify the decision rule in a
hypothesis testing procedure, there is a need to
specify the components of the rule. These
components include the following:
1.
Specify a level of significance, which is
usually denoted as α in doing the test
of hypothesis. A level of significance is
179
2.
3.
the probability of rejecting a true null
hypothesis or committing a Type I error
in the test of hypothesis.
Identify the test statistic to use in the
decision rule.
Part of the decision rule is the
specification of the rejection region.
The rejection region is that part of the
distribution of the test statistic where
we reject the null hypothesis.
An example of a decision rule is stated as follows:
“At a given α = 0.05, we reject Ho if the computed
test statistic (denoted as tc) is greater than a
tabular value of the t distribution with n-1 degrees
of freedom. Otherwise, we fail to reject Ho.”
E. Discussing new
concepts and practicing
new skills #2
The probability of committing an error is a
conditional probability problem. It is the probability
of making a decision based on the uncertainty of
the true state of nature of the hypothesis being
tested.
Example:
In testing the null hypothesis “The average daily
number of text messages that a Grade 11 student
sends is equal to 100” against an alternative
hypothesis stated as “The average daily number of
text messages that a Grade 11 student sends is
greater than 100”. A random sample of 16 students
were selected and interviewed. The daily number
of text messages she sends is obtained. The null
hypothesis is said to be rejected if the sample mean
is at least 102, otherwise the null hypothesis will be
accepted or we fail to reject Ho. It is assumed that
the number of text messages that a Grade 11
student sends in a day follows a normal distribution
with standard deviation equal to 5 text messages.
Computing for the probability of committing Type I
error, we have
𝛼 = 𝑃[𝑐𝑜𝑚𝑚𝑖𝑡𝑡𝑖𝑛𝑔 𝑎 𝑇𝑦𝑝𝑒 𝐼 𝑒𝑟𝑟𝑜𝑟]
= 𝑃 [𝑅𝑒𝑗𝑒𝑐𝑡 𝐻𝑜I𝐻𝑜 𝑖𝑠 𝑇𝑟𝑢𝑒]
𝑋 − 100
] = 𝑃[𝑍 ≥ 1.60]
5
√16
= 1 − 𝑃[𝑍 < 1.60] = 1 − 0.9452 = 0.0058
= 𝑃 [𝑋 ≥ 102 I 𝜇 = 100] = 𝑃 [
180
Thus, probability of rejecting a true null hypothesis
is 0.0058 or with 94.52% (1-0.0058 = 0.9452)
confidence that we are making a correct decision
in accepting a true null hypothesis.
𝛽 = 𝑃[𝑐𝑜𝑚𝑚𝑖𝑡𝑡𝑖𝑛𝑔 𝑎 𝑇𝑦𝑝𝑒 𝐼𝐼 𝑒𝑟𝑟𝑜𝑟]
= 𝑃[𝐴𝑐𝑐𝑒𝑝𝑡 𝐻𝑜I𝐻𝑜 𝑖𝑠 𝐹𝑎𝑙𝑠𝑒]
= 𝑃[𝐴𝑐𝑐𝑒𝑝𝑡 𝐻𝑜I𝐻𝑎 𝑖𝑠 𝑇𝑟𝑢𝑒]
𝑋 − 103 102 − 103
]
<
5
5
√16
√16
= 𝑃[𝑍 < −0.80]
= 1 − 𝑃[𝑍 < 0.80] = 1 − 0.7881 = 0.2119
= 𝑃[𝑋 < 102 I 𝜇 = 103] = 𝑃 [
F. Developing mastery
(Leads to Formative
Assessment 3)
Group Activity:
(The class will be divided into 4 groups. Each group
will be given the same problem. The activity is
good only for 10-15 minutes. Consider the given
problem. Answer what is being asked in the
situation )
1.The Graduate Record Exam (GRE) is a
standardized test required to be admitted to many
graduate schools in the United States. A high score
in the GRE makes admission more likely. According
to the Educational Testing Service, the mean score
for takers of GRE who do not have training courses
is 555 with a standard deviation of 139. Brain
Philippines (BP) offers expensive GRE training
courses, claiming their graduates score better than
those who have not taken any training courses. To
test the company’s claim, a statistician randomly
selected 30 graduates of BP and asked their GRE
scores.
a. Formulate the appropriate null and
alternative hypotheses.
b. Identify situations when Type I and Type II
errors are committed and state their
possible consequences.
c. Suppose the decision rule is “Reject Ho if
the mean score of the sampled BP
graduates is greater than 590; otherwise,
fail to reject Ho.” Compute for the level of
significance for this test. Also, find the risk
of concluding that the BP graduates did not
score better than 555 when in fact the mean
score is 600.
ANSWER:
Ho: Graduates of BP courses did not score
better than 555 while Ha: Graduates of BP courses
did score better than 555.
Type I error is committed when we declare
that the company’s claim is true where in fact BP
graduates do not perform better than 555 and a
possible consequence is that the tuition fee paid for
181
the training is wasted. On the other hand, Type II is
committed when we declare that the BP’s claim is
false when in fact BP graduates do score better
than 555 and a possible consequence is that
opportunity to score better than 555 is lost
The probability of Type I error is the same
as the level of significance denoted by α.
𝛼 = 𝑃[𝑐𝑜𝑚𝑚𝑖𝑡𝑡𝑖𝑛𝑔 𝑎 𝑇𝑦𝑝𝑒 𝐼 𝑒𝑟𝑟𝑜𝑟]
= 𝑃 [𝑅𝑒𝑗𝑒𝑐𝑡 𝐻𝑜I𝐻𝑜 𝑖𝑠 𝑇𝑟𝑢𝑒]
𝑋 − 555
590 − 555
]
≥
139
139
√30
√30
= 𝑃[𝑍 ≥ 1.38]
= 1 − 𝑃[𝑍 < 1.38] = 1 − 0.9162 = 0.0838
= 𝑃 [𝑋 ≥ 590 I 𝜇 = 555] = 𝑃 [
On the other hand, the risk of concluding that the BP
graduates did not score better than 555 when in fact
their mean score is 600 is the probability of committing
Type II and such risk or probability is computed as
follows:
𝛽 = 𝑃[𝑐𝑜𝑚𝑚𝑖𝑡𝑡𝑖𝑛𝑔 𝑎 𝑇𝑦𝑝𝑒 𝐼𝐼 𝑒𝑟𝑟𝑜𝑟]
= 𝑃[𝐴𝑐𝑐𝑒𝑝𝑡 𝐻𝑜I𝐻𝑜 𝑖𝑠 𝐹𝑎𝑙𝑠𝑒]
= 𝑃[𝐴𝑐𝑐𝑒𝑝𝑡 𝐻𝑜I𝐻𝑎 𝑖𝑠 𝑇𝑟𝑢𝑒]
𝑋 − 600 590 − 600
]
<
139
139
√30
√30
= 𝑃[𝑍 < −0.80]
= 1 − 𝑃[𝑍 < 0.80] = 1 − 0.7881 = 0.2119
= 𝑃[𝑋 < 590 I 𝜇 = 600] = 𝑃 [
G. Making
generalizations and
abstractions about the
lesson
H. Evaluating learning
We can determine the probability of making a
correct decision in accepting or rejecting a true null
hypothesis by calculating the probabilities of
committing a Type I and Type II error. It is the
probability of making a decision based on the
uncertainty of the true state of nature of the
hypothesis being tested. We may use a numerical
example and a formula to illustrate the computation
of the probabilities of committing Type I and Type
II errors.
Consider the given problem and ask them to do
what is being asked for.
Consider a manufacturing process that is
known to produce bulbs that have life lengths
with a standard deviation of 75 days. A
potential customer will purchase bulbs from
the company that manufactures the bulbs if
she is convinced that the average life of the
bulbs is 1550 days.
a. Formulate the appropriate null and
alternative hypotheses.
182
b. Identify situations when Type I and Type II
errors are committed and state their
possible consequences.
c. Suppose the decision rule is “Reject Ho if
a random sample of 50 bulbs has a life
less than 1532 days; otherwise, fail to
reject Ho.” Compute for the level of
significance for this test. Also, find the risk
of concluding that the average is greater
than 1550 days when in fact their mean
score is 1500.
ANSWER:
a.
Ho: null hypothesis, that the average life of
bulbs is (at least) 1550 days against the
alternative hypothesis, that the average is less
than 1550
b.
Type I error is committed when we declare
that the average life is less than 1550 days where
in fact the average life is 1550 days or more. On
the other hand, Type II is committed when we
declare that the average is at least 1550 days,
when in fact, it is less than 1550 days.
c.
The probability of Type I error is the same
as level of significance denoted by α.
= 𝑃 [𝑐𝑜𝑚𝑚𝑖𝑡𝑡𝑖𝑛𝑔 𝑎 𝑇𝑦𝑝𝑒 𝐼 𝑒𝑟𝑟𝑜𝑟 ]
= 𝑃 [𝑅𝑒𝑗𝑒𝑐𝑡 𝐻𝑜I𝐻𝑜 𝑖𝑠 𝑇𝑟𝑢𝑒]
= 𝑃[𝑋 ≤ 1532 I 𝜇 = 1550]
𝑋 − 1550
1532 − 1550
]
≤
75
75
√50
√50
= 𝑃[𝑍 ≥ 1.38]
= 1 − 𝑃[𝑍 ≤ 1 − 1.70] ≈ 0.04
= 𝑃[
V.REMARKS
VI. REFLECTION
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require additional
activities for remediation
C. Did the remedial lessons
work? No. of learners
who caught up with the
lesson
D. No. of learners who
continue to require
remediation
183
E.
Which of my teaching
strategies worked well?
Why did these work?
F.
What difficulties did I
encounter which my
principal or supervisor can
help me solve?
G. What innovation or
localized materials did I
use/discover which I
wish to share with other
teachers?
184
School
Teacher
Time and Date
OBJECTIVE
S
A. Content
Standard
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
I.
B. Performance
Standard
C. Learning
Competencie
s/ Objectives
Write The LC
code for each
II.
III.
CONTENT
The learner demonstrates understanding of key concepts of
tests of hypotheses on the population mean.
The learner is able to perform appropriate tests of
hypotheses involving the population mean to make
inferences in real-life problems in different disciplines.
The learner identifies the parameter to be tested given a
real-life problem.
M11/12SP-IVa-3.
Tests of Hypothesis
LEARNING
RESOURCE
S
A. References
1. Teacher’s
Guide pages
2. Learner’s
Material
pages
3. Textbook
pages
4. Additional
Materials from
Learning
Resource
(LR) portal
B. Other
Learning
Resources
IV. PROCEDURE
A. Reviewing
previous lesson
or presenting the
new lesson
B. Establishing a
purpose of the
lesson
Commission on Higher Education & Philippine Normal
University (2016). Teaching Guide for Senior High School:
Statistics and Probability.
https://www.cliffsnotes.com/studyguides/statistics/sampling/populations
Laptop, projector, powerpoint presentation
Post the words PARAMETER and STATISTIC on the
board. Ask the students to give the definition of the two
words by writing them in a meta strips and post them in
line with the given words.
With the given problems below, identify the parameter to be
tested.
1. The father of a senior high school student lists down the
expenses he will incur when he sends his daughter to the
university. At the university he wants his daughter to study,
he hears that the average tuition fee is at least Php 20,000
per semester.
185
Ans.:The parameter to be tested is the average
tuition fee or the true population mean of the tuition fee.
2. The principal of an elementary school believes that this
year, there would be more students from the school who
would pass the National Achievement Test (NAT), so that
the proportion of students who passed the NAT is greater
than the proportion obtained in previous year, which is 0.75.
What will be the appropriate null and alternative hypotheses
to test this belief?
Ans.:The parameter to be tested is the
proportion of students of the school who passed the
NAT this year.
C. Presenting
examples/
instances of the
new lesson
D. Discussing
new concepts
and practicing
new skills #1
A researcher wants to estimate the average height of
women aged 20 years or older. From a simple random
sample of 45 women, the researcher obtains a sample
mean height of 63.9 inches.
Which of the statements below is the
parameter of the given situation?
a. Average height of all women aged 20 years or older.
b. Average height of 63.9 inches from the sample of 45
women.
Discuss the answer of the activity, pointing out the
difference between parameter and statistic.
Present another problem. Let the students identify
the parameter.
A school administrator wants to estimate the mean
score on the verbal portion of the SAT for students whose
first language is not English. From a simple random sample
of 20 students whose first language is not English, the
administrator obtains a sample mean SAT verbal score of
458.
Ans.:The parameter is the mean verbal
SAT score for students whose first language is not
English.
QUIZ BEE
E. Developing
Mastery(Leads
to Formative
Assessment)
Group the class into 5.
Identify the parameter of each of the problems to be
presented.
1. A nutritionist wants to estimate the mean amount of
sodium consumed by children under the age of 10.
From a random sample of 75 children under the age
of 10, the nutritionist obtains a sample mean of 2993
milligrams of sodium consumed.
ans.:The mean amount of sodium
consumed by children under the age of ten.
2. Nexium is a drug that can be used to reduce the acid
produced by the body and heal damage to the
esophagus. A researcher wants to estimate the
proportion of patients taking Nexium that are healed
within 8 weeks. A random sample of 224 patients
186
suffering from acid reflux disease is obtained, and
213 of those patients were healed after 8 weeks.
ans.:The proportion of patients healed
by Nexium in 8 weeks.
3. A researcher wants to estimate the average farm
size in Sorsogon. From a simple random sample of
40 farms, the researcher obtains a sample mean
farm size of 731 acres.
ans.:The average farm size in Sorsogon.
4. An energy official wants to estimate the average oil
output per well in the Philippines. From a random
sample of 50 wells throughout the Philippines, the
official obtains a sample mean of 10.7 barrels per
day.
ans.:The average oil output per well in
the Philippines.
5. An education official wants to estimate the
proportion of adults aged 18 or older who had read
at least one book during the previous year. A
random sample of 1006 adults aged 18 or older is
obtained, and 835 of those adults had read at least
one book during the previous year.
ans.:The proportion of adults 18 or
older who read a book in the previous year.
F. Making
generalizations
and abstraction
about the lesson
G. Finding
practical
applications of
concepts and
skills in daily
living
How to identify the parameter to be tested given a real life
problem?
How important is having knowledge on identifying the
parameter to be tested of a real life problem?
With the following problems,identify the parameter to be
tested.
H. Evaluating
Learning
187
I. Additional
activities for
application or
remediation
V. REMARKS
VI. REFLECTION
A. No. of learners
who earned 80%
in the evaluation
B. No. of learners
who require
additional
activities for
remediation who
scored below
80%
C. Did the remedial
lesson work? No.
of learners who
have coped up
with the lesson
D. No. of learners
who continue to
require
remediation
E. Which of my
teachings
strategies
worked well?
Why did these
work?
F. What difficulties
did I encounter
which my
principal/
supervisor can
help me solve?
G. What innovation
or localized
materials did I
use/ discover
which I wish to
share with the
other teachers?
188
SCHOOL
GRADE LEVEL
LEARNING
AREA
QUARTER
TEACHER
TIME& DATE
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
Write The LC code
for each
II.
11
STATISTICS AND
PROBABILITY
CONTENT
The learner demonstrates understanding of key concepts of
tests of hypotheses on the population mean.
The learner is able to perform appropriate tests of
hypotheses involving the population mean to make
inferences in real-life problems in different disciplines.
The learner formulates the appropriate null and alternative
hypotheses on a population mean.
M11/12SP-IVb-1.
Tests of Hypothesis
III.
LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
Commission on Higher Education & Philippine Normal
University (2016). Teaching Guide for Senior High School:
Statistics and Probability. pp.374-375
2. Learner’s Material
pages
3. Textbook pages
4. Additional
Materials from
Learning Resource
(LR) portal
https://stattrek.com/hypothesis-test/mean.aspx
https://www.ck12.org/statistics/null-and-alternativehypotheses/lesson/Null-and-Alternative-Hypotheses-ADVPST/
cfcc.edu/faculty/cmoore/0801-HypothesisTests.pdf
https:// m358k-ps-six-hypothesis-tests.pdf
B. Other Learning
Resources
Manila paper,
worksheet
IV.
pentel
pen,
powerpoint
presentation,
PROCEDURE
A. Reviewing
previous lesson or
presenting the new
lesson
Arrange the steps of hypothesis testing procedures below:
a. Make a decision whether to reject or fail to reject the null
hypothesis.
b. Using a simple random of observation, compute the
value of the test statistic.
c. Formulate the null and alternative hypotheses.
d. State the conclusion.
e. Identify the test statistic to use. With the given level of
significance and the distribution of the test statistics, state
the decision rule and specify the rejection region.
189
With the given problems below, formulate the null and
alternative hypotheses.
1. The father of a senior high school student lists down the
expenses he will incur when he sends his daughter to
the university. At the university he wants his daughter to
study, he hears that the average tuition fee is at least
Php 20,000 per semester. He wants to do a test of
hypothesis.
B. Establishing a
purpose of the
lesson
C. Presenting
examples/
instances of the
new lesson
D. Discussing new
concepts and
practicing new
skills #1
E. Developing
Mastery(Leads to
Formative
Assessment)
2. The principal of an elementary school believes that this
year, there would be more students from the school who
would pass the National Achievement Test (NAT), so
that the proportion of students who passed the NAT is
greater than the proportion obtained in previous year,
which is 0.75. What will be the appropriate null and
alternative hypotheses to test this belief?
Differentiate the two problems. Use the first problem to
discuss the concept.
Discuss the result of the activity, pointing out the following:
 A statistical hypothesis is a statement about a
parameter and deals with evaluating the value of the
parameter.
 The null and alternative hypotheses should be
complementary and non-overlapping.
 Generally, the null hypothesis is a statement of equality
or includes the equality condition as in the case of ‘at
least’
( greater than or equal) or ‘atmost’ (less than or equal).
Present the situation below and ask the students to formulate
the appropriate null and alternative hypotheses.
“The average daily number of text messages that a
Grade 11 student sends is equal to 100.”
Group Activity
The class will be divided into three groups. Each group will
be given a worksheet.
Rubric for Assessing the Activity
190
F. Making
generalizations
and abstraction
about the lesson
G. Finding practical
applications of
concepts and skills
in daily living
How to formulate null and alternative hypotheses?
Cite some applications of the concepts learned in daily life.
With the following problems, formulate the appropriate null
and alternative hypotheses.
H. Evaluating
Learning
I.
Additional activities
for application or
remediation
V.
REMARKS
191
VI.
REFLECTION
A. No. of learners
who earned 80%
in the evaluation
B. No. of learners
who require
additional activities
for remediation
who scored below
80%
C. Did the remedial
lesson work? No.
of learners who
have coped up
with the lesson
D. No. of learners
who continue to
require
remediation
E. Which of my
teachings
strategies worked
well? Why did
these work?
F. What difficulties
did I encounter
which my principal/
supervisor can
help me solve?
G. What innovation or
localized materials
did I use/ discover
which I wish to
share with the
other teachers?
192
School
Teacher
Time and Date
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
I.OBJECTIVES
A. Content Standard
The learner demonstrates understanding of key
concepts of tests of hypotheses on the population mean
and population proportion.
B. Performance
Standard
The learner is able to perform appropriate tests of
hypotheses involving the population mean and
population proportion to make inferences in real-life
problems in different disciplines.
C. Learning
Competency/Obje
ctive:
The learner identifies the appropriate form of the teststatistic when: (a) the population variance is assumed
to be known (b) the population variance is assumed to
be unknown; and (c) the Central Limit Theorem is to be
used.
II. CONTENT
III.LEARNING
RESOURCES
A. References
1. Teacher’s guide
pages
2. Learner’s
material pages
3. Textbook pages
4. Additional
materials from
learning
resource
(LR)
portal
B. Other
Learning
Resource
IV. PROCEDURE
M11/12SP-IVb-2
Appropriate Form of the Test-Statistic
374-398
Statistics and Probability pp.246-265
A. Reviewing
previous lesson or Reviewing the steps of hypothesis testing procedure:
presenting the new 1. Formulate the null and alternative hypotheses.
lesson
2. Identify the test statistic to use. With the given level
of significance and the distribution of the test statistics,
state the decision rule and specify the rejection region.
3. Using a simple random sample of observation,
compute the value of the test statistic.
4. Make a decision whether to reject or fail to reject
(accept) Ho.
5. State the conclusion.
193
B. Establishing a
purpose for the
lesson:
How does test statistics play a very important role in the
decision on Hypothesis Testing?
The test statistic is a standardized expression of the
point estimator of the parameter identified in the
hypothesis. Also, the distribution of the test statistic is
also needed to be specified.
C. Presenting
examples/instance TEST STATISTIC – a value used to determine the
s of the new lesson probability needed in decision-making and a value
determined by a computational formula that is
compared with confidence coefficient (like 1.96 and
2.58)
Problem #1
The father of a senior high school student lists down
the expenses he will incur when he sends his daughter
to the university. At the university where he wants his
daughter to study, he hears that the average tuition fee
is at least Php20,000 per semester. He wants to do a
test of hypothesis.
In this problem, the parameter of interest is the average
tuition fee or the true population mean of the tuition fee.
In symbol, this parameter is denoted as µ. As applied
to the problem, the appropriate null and alternative
hypotheses are:
Ho: The average tuition fee in the targeted university is
at least Php20,000. In symbols, Ho: µ ≥ Php20,000.
Ha: The average tuition fee in the targeted university is
less than Php20,000. In symbols, Ha: µ < Php20,000.
 In the problem, the parameter is the population
mean. To identify the test statistics, which is part
of the second step, certain assumptions have to
be made.
D. Discussing new
concepts and
practicing new skill
#1
With the assumption of known population variance (σ2)
and the variable of interest is measured at least in the
interval scale and follows the normal distribution, the
appropriate test statistic, denoted as Zc is computed as
Zc =
𝑋−𝜇𝑜
𝜎/√𝑛
where X is the sample mean computed from a simple
random sample of n observations; µ0 is the
hypothesized value of the parameter; and σ is the
population standard deviation. The test statistic follows
the standard normal distribution which means the
194
tabular value in the Z-table will be used as critical or
tabular value.
With the assumption of unknown population variance
(σ2) and the variable of interest is measured at least in
the interval scale and follows the normal distribution,
the appropriate test statistic, denoted as tC is computed
as
tc =
𝑋−𝜇𝑜
𝑠/√𝑛
where X and s, are the sample mean and sample
standard deviation, respectively, computed from a
simple random sample of n observations; and µ0 is the
hypothesized value of the parameter. The test statistic
follows the Student’s t-distribution with n-1 degrees of
freedom which means the tabular value in the Student’s
t-table will be used as critical or tabular value.
E. Discussing
concepts
practicing
skills #2
new
and
new
 Present the table to the class.
195
F. Developing
GROUP ACTIVITY
mastery (Leads to The class will be divided into 4 groups. Each group will
Formative
be given a corresponding problem for them to work on
Assessment 3)
and materials such as bond paper, pentel pen and
manila paper.
Direction: Identify the appropriate test statistic to
be used on the following problem:
GROUP 1: A researcher used a developed problem
solving test to randomly select 50 Grade VI pupils. In
this sample, X=80 and s=10. The mean µ and the
standard deviation of the population used in the
standardization of the test were 75 and 15, respectively.
196
ANSWER: Test Statistic Zc =
𝑋−𝜇𝑜
𝜎/√𝑛
GROUP 2: The owner of a factory that sells a
particular bottled fruit juice claims that the average
capacity of their product is 250 ml. To test the claim, a
consumer group gets a sample of 100 such bottles,
calculates the capacity of each bottle, and then finds
the mean capacity to be 248 ml. The standard deviation
s is 5 ml.
ANSWER: Test Statistic tc =
𝑋−𝜇𝑜
𝑠/√𝑛
Group 3: The Graduate Record Exam (GRE) is a
standardized test required to be admitted to many
graduate schools in the United States. A high score in
the GRE makes admission more likely. According to
the Educational Testing Service, the mean score for
takers of GRE who do not have training courses is 555
with a standard deviation of 139. Brain Philippines (BP)
offers expensive GRE training courses, claiming their
graduates score better than those who have not taken
any training courses. To test the company’s claim, a
statistician randomly selected 30 graduates of BP and
asked their GRE scores.
ANSWER: Test Statistic Zc =
𝑋−𝜇𝑜
𝜎/√𝑛
Group 4: A brand of powdered milk is advertised as
having a net weight of 250 grams. A curious consumer
obtained the net weight of 10 randomly selected cans.
The values obtained are: 256, 248, 242, 245, 246, 248,
250, 255, 243 and 249 grams. Is there reason to believe
that the average net weight of the powdered milk cans
is less than 250 grams at 10% level of significance?
Assume the net weight is normally distributed with
unknown population variance.
ANSWER: Test Statistic tc =
G. Making
generalizations
and abstractions
about the lesson
H. Evaluating
Learning
𝑋−𝜇𝑜
𝑠/√𝑛
The decision that we make depends on the computed
test statistic. The formula for computing the test
statistic depends on the sample size.
Direction: Identify the appropriate test statistic to be
used on the following problem:
1. The principal of an elementary school believes
that this year, there would be more students from the
school who would pass the National Achievement Test
(NAT), so that the proportion of students who passed
the NAT is greater than the proportion obtained in
197
previous year, which is 0.75. What will be the
appropriate null and alternative hypotheses to test this
belief?
ANSWER: Test Statistic Zc =
𝑋−𝜇𝑜
𝜎/√𝑛
2. The minimum wage earners of the National
Capital Region are believed to be receiving less than
Php500 per day. The CEO of a large supermarket
chain in the region is claiming to be paying its
contractual higher than the minimum daily wage rate of
Php500. To check on this claim, a labour union leader
obtained a random sample of 144 contractual
employees from this supermarket chain. The survey of
their daily wage earnings resulted to an average wage
of Php510 per day with standard deviation of Php100.
The daily wage of the region is assumed to follow a
distribution with an unknown population variance.
Perform a test of hypothesis at 5% level of significance
to help the labour union leader make an
empiricalbased conclusion on the CEO’s claim.
ANSWER: Test Statistic Zc =
I.
Additional activities
for application or
Remediation
V. REMARKS
VI. REFLECTION
A. No. of learners who
earned 80% on the
formative
assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E.
Which
of
my
teaching strategies
worked well? Why
did it work?
F. What difficulties did I
encounter which my
198
𝑋−𝜇𝑜
𝜎/√𝑛
principal
or
supervisor can help
me solve?
G. What innovation or
localized material/s did I
use/discover which I wish
to share with other
teachers?
199
School
Teacher
Time and Date
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
I.OBJECTIVES
A. Content Standard
The learner demonstrates understanding of key
concepts of tests of hypotheses on the population
mean and population proportion.
B. Performance
Standard
The learner is able to perform appropriate tests of
hypotheses involving the population mean and
population proportion to make inferences in real-life
problems in different disciplines.
C. Learning
Competency/Objecti
ve:
The learner identifies the appropriate rejection region
for a given level of significance when: (a) the
population variance is assumed to be known (b) the
population variance is assumed to be unknown; and
(c) the Central Limit Theorem is to be used.
M11/12SP-IVc-1
II. CONTENT
Appropriate Rejection Region
III.LEARNING
RESOURCES
A. References
1. Teacher’s
guide
pages
2. Learner’s material
pages
3. Textbook pages
4. Additional
materials
from
learning resource
(LR) portal
B. Other
Learning
Resource
IV. PROCEDURE
374-398
Statistics and Probability pp.246-265
A. Reviewing previous
lesson or presenting Review the appropriate test statistics to be used
the new lesson
when:
(a) the population variance is assumed to be known
(b) the population variance is assumed to be
unknown; and
(c) the Central Limit Theorem is to be used
B. Establishing a
purpose for the
lesson:
Ask the learners what they will do in case the variable
of interest cannot be assumed to follow a normal
distribution. Is there a way to test the hypotheses?
200
 The answer to this question is: Yes, there is
a way to do it but they must be assured that
the sample size is large enough to invoke the
Central Limit Theorem they learned under the
lesson on sampling distribution of the sample
mean. Let us say that for the given problem,
a random sample of size 36 is sufficient for us
to invoke the theorem.
C. Presenting
examples/instances
of the new lesson
The father of a senior high school student lists
down the expenses he will incur when he sends
his daughter to the university. At the university
where he wants his daughter to study, he hears
that the average tuition fee is at least Php20,000
per semester. He wants to do a test of hypothesis.
For the problem, the first is the appropriate decision
rule. Suppose the level of significance (α) is set at
0.05, then the decision rule for the problem could be
stated as ‘Reject Ho if Zc < -Z0.05 = -1.645. Otherwise,
we fail to reject Ho.” Note that this test procedure is
referred to as “one-tail Z-test for the population mean
when the population variance is known’ and the
rejection region is illustrated as follows:
The father of a senior high school student lists
down the expenses he will incur when he sends
his daughter to the university where he wants her
to study. He hypothesizes that the average tuition
fee is at least Php20,000 per semester. He knows
the variable of interest, which is the tuition fee, is
measured at least in the interval scale or
specifically in the ratio scale. He assumes that the
variable of interest follows the normal distribution
but both population mean and variance are
unknown. The father asks, at random, 25
students of the university about their tuition fee
per semester. He is able to get an average of
Php20,050 with a standard deviation of Php500.
201
For the problem, the first is the appropriate decision
rule. Suppose the level of significance (α) is set at
0.05, then the decision rule for the problem can be
stated as “Reject Ho if the tc < -tα,24 = -2.064.
Otherwise, we fail to reject Ho.” Note that this test
procedure is referred to as “one-tail t-test for the
population mean” and the rejection region is
illustrated as follows:
D. Discussing new
concepts and
practicing new skill
#1
Present the table to the students while highlighting
the appropriate rejection region.
202
203
E. Developing mastery
(Leads to Formative GROUP ACTIVITY
Assessment 3)
The class will be divided into 3 groups. Each group
will be given a corresponding problem for them to
work on and materials such as bond paper, pentel
pen and manila paper.
Direction: Identify the appropriate rejection
region to be used on the following problem:
Group 1: The Graduate Record Exam (GRE) is a
standardized test required to be admitted to many
graduate schools in the United States. A high score
in the GRE makes admission more likely. According
to the Educational Testing Service, the mean score
for takers of GRE who do not have training courses
is 555 with a standard deviation of 139. Brain
Philippines (BP) offers expensive GRE training
courses, claiming their graduates score better than
those who have not taken any training courses. To
test the company’s claim, a statistician randomly
selected 30 graduates of BP and asked their GRE
scores.
ANSWER:
Group 2: The minimum wage earners of the National
Capital Region are believed to be receiving less than
Php500 per day. The CEO of a large supermarket
chain in the region is claiming to be paying its
contractual higher than the minimum daily wage rate
of Php500. To check on this claim, a labour union
leader obtained a random sample of 144 contractual
employees from this supermarket chain. The survey
of their daily wage earnings resulted to an average
wage of Php510 per day with standard deviation of
Php100. The daily wage of the region is assumed to
follow a distribution with an unknown population
variance. Perform a test of hypothesis at 5% level of
204
significance to help the labour union leader make an
empiricalbased conclusion on the CEO’s claim.
ANSWER:
Group 3: A brand of powdered milk is advertised as
having a net weight of 250 grams. A curious
consumer obtained the net weight of 10 randomly
selected cans. The values obtained are: 256, 248,
242, 245, 246, 248, 250, 255, 243 and 249 grams. Is
there reason to believe that the average net weight
of the powdered milk cans is less than 250 grams at
10% level of significance? Assume the net weight is
normally distributed with unknown population
variance.
ANSWER:
F. Making
Ask a student to give generalization on how to
generalizations and identify the appropriate rejection region given a
abstractions
about particular population variance.
the lesson
G. Evaluating Learning
Direction: Identify the appropriate rejection
region to be used on the following problem:
1. A researcher used a developed problem
solving test to randomly select 50 Grade VI
pupils. In this sample, X=80 and s=10. The
mean µ and the standard deviation of the
population used in the standardization of the
test were 75 and 15, respectively.
ANSWER:
205
H. Additional activities
for application or
Remediation
V. REMARKS
VI. REFLECTION
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did
the
remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies
worked
well? Why did it work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized material/s did I
use/discover which I wish to
share with other teachers?
206
SCHOOL
GRADE LEVEL
LEARNING
AREA
QUARTER
TEACHER
TIME& DATE
I.
OBJECTIVES
A. Content Standard
B. Performance Standard
C. Learning Competencies/
Objectives
Write The LC code for each
II.
11
STATISTICS AND
PROBABILITY
CONTENT
The learner demonstrates understanding of key
concepts of tests of hypotheses on the population
mean
The learner is able to perform appropriate tests of
hypotheses involving the population mean to make
inferences in real-life problems in different
disciplines.
M11/12SP-IVd-1. The learner computes for the
test-statistics value (population mean).
M11/12SP-IVd-2. Draws conclusion about the
population mean based on the test-statistic value
and the rejection region.
Calculating Test-Statistic Value given that the
population Variance is known
III.
LEARNING
RESOURCES
A. References
1. Teacher’s Guide pages
Commission on Higher Education & Philippine
Normal University (2016). Teaching Guide for
Senior High School: Statistics and Probability.
pp.372-384
2. Learner’s Material pages
3. Textbook pages
4. Additional Materials from
Learning Resource (LR)
portal
B. Other Learning Resources
IV.
Hypothesis testing.WWW.ck12.org
PROCEDURE
A. Reviewing previous lesson
or presenting the new
lesson
Complete the table below about the form of
statistics to be used, the decision rule and the
rejection region given the hypothesis and some
assumptions about the distribution. Call some
students to write their answers on the board.
207
Consider the situations below.
B. Establishing a purpose of
the lesson
The father of a senior high school student lists
down the expenses he will incur when he sends
his daughter to a private college. At the school
where he wants his daughter to study, he hears
that the average tuition fee is at least Php20,000
per semester. He wants to know if this claim is
true so he does a test of hypothesis. Suppose
from a simple random sample of 16 students, a
sample mean of Php19,750 was obtained.
Further, the variable of interest, which is the
tuition fee in the college, is said to be normally
distributed with an assumed population
variance equal to Php160,000.
The students will be asked of the following
questions. (Solicit students’ responses)
1. What hypotheses can be formulated based
from the given situation?
2. What form of test-statistics is appropriate in
the given problem?
Emphasize that the lesson will focus on
calculating the test-statistic value (about the
population mean) and drawing conclusion about
the population mean based on the test-statistic
value and rejection region given that the
population variance is known.
C. Presenting examples/
instances of the new lesson
D. Discussing new concepts
and practicing new skills #1
Present to students the hypothesis testing
procedures:
a. Formulate the null and alternative hypotheses.
b. Identify the test-statistic to use. With the given
level of significance and the distribution of the
test statistics, state the decision rule and
specify the rejection region.
c. Using a simple random sample of observation
given in each problem, compute the value of
the test statistic by applying the formula of test
statistics identified in part b.
d. Make a decision whether to reject or fail to
reject Ho.
e. State the conclusion.
Using the situation about the tuition fee situation,
calculate the test-statistic value following the
hypothesis
testing
procedures.
Students
participation may be encouraged during the
discussion.
208
c. Hypotheses:
Ho: The average tuition fee in the targeted
university is at least
Php20,000. In symbols, Ho: μ ≥
Php20,000.
Ha: The average tuition fee in the targeted
university is less than
Php20,000. In symbols, Ha: μ <
Php20,000.
d. Test-statistics to use:
With the assumption of known population
variance (σ2) and the variable of interest is
measured at least in the interval scale and
follows the normal distribution, the appropriate
test statistic, denoted as ZC is computed as
Where :
is the sample mean computed
from a simple random sample
of n observations;
μ0 is the hypothesized value of the
parameter; and
σ is the population standard
deviation.
Decision rule and Rejection Region: Suppose the
level of significance (α) is set at 0.05, then the
decision rule for the problem could be stated as
Reject Ho if ZC < -Z0.05 = -1.645. Otherwise, we fail
to reject Ho.”
e. Compute for the value of the test statistic
Given:
= 19, 750
μ0 = 20, 000
σ2 = 160,000
σ = 400
Solution:
-2.50
209
d. Making Decision
Since the computed Z statistic value of -2.50
is less than the critical value of -1.645 at 0.05
level of significance, therefore we reject the null
hypothesis.
e. Conclusion
With the rejection of the null hypothesis, the
father can then say that the average tuition fee
in the university where he wants his daughter to
study is less than Php20,000.
The class will be divided into 4 groups. Provide each
group with the materials needed in accomplishing
their tasks such as Hand-outs, permanent markers,
calculator, manila paper.
E. Developing Mastery(Leads
to Formative Assessment)
Each group will be assigned a problem to work
with. For each problem, they will perform the
following Test of Hypothesis procedures in order to
calculate the test statistic value and be able to
formulate a conclusion based from the result. (Use
0.05 level of significance to evaluate the
hypothesis)
a.
Formulate the null and alternative
hypotheses.
b.
Identify the test statistic to use. With the
given level of significance and the distribution of
the test statistics, state the decision rule and
specify the rejection region.
c.
Using a simple random sample of
observation given in each problem, compute the
value of the test statistic by applying the formula
of test statistics identified in part b.
d.
Make a decision
e.
State the conclusion.
Tasks:
Group 1 and 2. The school nurse thinks the
average height of 7th graders has increased. The
average height of a 7th grader five years ago was
145 cm with a standard deviation of 20 cm. She
takes a random sample of 200 students and finds
that the average height of her sample is 147 cm.
Are 7th graders now taller than they were before?
Group 3 and 4. Mang Ruben is trying out a
planting technique that he hopes will increase the
yield on his cacao trees. The average weight of
yield on each tree is 80 kg with a standard deviation
of 30 kg. This year, after trying his new planting
technique, he takes a random sample 35 trees and
finds the average weight of yield is to be 98 kg. He
wonders whether or not this is a statistically
significant increase.
210
Each group shall present their outputs in front. They
will also compare their solution from the output of
other groups.
Check the solution of each group, and clarify some
errors in their solutions.
Summarize the lesson by emphasizing to students
that with the assumption of known population
variance (σ2) and the variable of interest is
measured at least in the interval scale and follows
the normal distribution, the appropriate test statistic,
denoted as ZC is computed as
F. Making generalizations and
abstraction about the lesson
Where :
is the sample mean computed
from a simple random sample of n observations;
μ0 is the hypothesized value of the
parameter; and
σ
is the population standard
deviation.
Emphasize to students that decisions are made
about the hypothesis based on the result of the
calculated statistical-value (ZC) at a given level of
significance. This decision is used to formulate the
conclusion.
G. Finding practical
applications of concepts and
skills in daily living
Quiz: Given the problem below, State the null
hypothesis, calculate the statistical value and
make a conclusion out of the resulted z-value.
H. Evaluating Learning
Consider a manufacturing process that is known to
produce bulbs that have life lengths with a standard
deviation of 75 days. A potential customer will
purchase bulbs from the company that
manufactures the bulbs if she is convinced that the
average life of the bulbs is at least 1550 days. To
test this, a random sample of 50 bulbs were tested
and found to have an average life of 1532 days.
Answer:
Ho: The average life of the bulb is at least
1550 days. In
symbols, μ ≥ 1550.
Ha: The average life of the bulb is less than
1550 days. In
symbols, μ < 1550.
211
Suppose the level of significance (α) is set at
0.05, then the decision rule for the problem could be
stated as
Reject Ho if ZC <-Z0.05 = -1.645. Otherwise, we fail
to reject Ho.”
-1.697
Since the computed Z statistic value of -1.697 is
less than the critical value of
-1.645 at 0.05 level of significance, therefore
reject the null hypothesis.
Because of the rejection of the null hypothesis,
the potential costumer can
conclude that the
average life of the bulb is less than 1550 days. The
consequences of this may lead the potential
customer to not purchase the bulbs from that
company.
I.
Additional activities for
application or remediation
V.
REMARKS
VI.
REFLECTION
A. No. of learners who earned
80% in the evaluation
B. No. of learners who require
additional activities for
remediation who scored
below 80%
C. Did the remedial lesson
work? No. of learners who
have coped up with the
lesson
D. No. of learners who
continue to require
remediation
E. Which of my teachings
strategies worked well?
Why did these work?
F. What difficulties did I
encounter which my
principal/ supervisor can
help me solve?
212
G. What innovation or localized
materials did I use/ discover
which I wish to share with
the other teachers?
213
SCHOOL
GRADE LEVEL
LEARNING
AREA
QUARTER
TEACHER
TIME& DATE
I.
OBJECTIVES
A. Content Standard
B. Performance Standard
C. Learning Competencies/
Objectives
Write The LC code for each
II.
11
STATISTICS AND
PROBABILITY
CONTENT
The learner demonstrates understanding of key
concepts of tests of hypotheses on the population
mean
The learner is able to perform appropriate tests of
hypotheses involving the population mean to make
inferences in real-life problems in different
disciplines.
M11/12SP-IVd-1. The learner computes for the teststatistics value (population mean).
M11/12SP-IVd-2. Draws conclusion about the
population mean based on the test-statistic value
and the rejection region.
Calculating Test-Statistic Value given that the
population Variance is UNKNOWN
III.
LEARNING
RESOURCES
A. References
1. Teacher’s Guide pages
2. Learner’s Material pages
3. Textbook pages
4. Additional Materials from
Learning Resource (LR)
portal
B. Other Learning Resources
IV.
PROCEDURE
A. Reviewing previous lesson
or presenting the new
lesson
Commission on Higher Education & Philippine
Normal University (2016). Teaching Guide for
Senior High School: Statistics and Probability.
pp.372-384
https://courses.edx.org/c4x/UTAustinX/UT.7.01x/asset/Chapter_12.pdf
Check-point: Ask the students to identify from the
following the terms which they are familiar with. Ask
them to define those terms based from their
understanding.
f. Normal Distribution
g. T-Distribution
h. Critical value
i. Rejection region
j. Degrees of Freedom
214
Consider the situations below.
The high school MAPEH Teacher asked if the
athletes in their school are doing as well academically
as the other student athletes. We know from a previous
study that the average Final Rating of the student
athletes is 87. After an initiative to help improve the
academic ratings of student athletes, the MAPEH
teacher randomly samples 20 athletes and finds that
the average Final Ratings of the sample is 88.8. with a
sample standard deviation of 15. Is there a significant
improvement? Use a 0.05 significance level.
B. Establishing a purpose of
the lesson
The students will be asked of the following
questions. (Solicit students’ responses)
3. What hypotheses can be formulated based
from the given situation?
4. What form of test-statistics is appropriate in
the given problem?
Emphasize that the lesson will focus on
calculating the test-statistic value (about the
population mean) and drawing conclusion about the
population mean based on the test-statistic value
and rejection region given that the population
variance is unknown.
C. Presenting examples/
instances of the new lesson
Present to students the hypothesis testing
procedures:
f. Formulate the null and alternative hypotheses.
g. Identify the test-statistic to use. With the given
level of significance and the distribution of the
test statistics, state the decision rule and
specify the rejection region.
h. Using a simple random sample of observation
given in each problem, compute the value of
the test statistic by applying the formula of test
statistics identified in part b.
i. Make a decision whether to reject or fail to
reject Ho.
j. State the conclusion.
D. Discussing new concepts
and practicing new skills #1
Using the situation about the tuition fee situation,
calculate the test-statistic value following the
hypothesis
testing
procedures.
Students
participation may be encouraged during the
discussion.
a. Hypotheses:
Ho: The average Final rating of the student
athletes is equal to 87. In symbols, Ho: μ =
87.
215
Ha: The average Final rating of the student
athletes is not equal to 87. In symbols, Ho: μ
≠ 87.
b. Test-statistics to use:
With the assumption of unknown population
variance (σ2), the appropriate test statistic, is
the T-test denoted as TC is computed as
TC
Where : TC is the test statistic and has n-1
degrees of freedom
is the sample mean computed
from a simple random
sample of n observations;
μ0 is the hypothesized value of the
parameter; and
S is the sample standard
deviation.
Decision rule and Rejection
Region: Suppose the level of
significance (α) is set at 0.05, then
the decision rule for the problem is:
Reject Ho if |tC|> ± 2.093 0.05/2, 19.
Otherwise, we fail to reject Ho.
c. Compute for the value of the test
statistic
Given:
= 88.8
μ0 = 87
s = 15
n = 20
Solution: Tc
Tc
=
Tc = 0.537
d. Making Decision
Since the computed T statistic value of
0.537 is less than the critical value of ±
2.093 at 0.05 level of significance, therefore
we fail to reject the null hypothesis.
e. Conclusion
Therefore, the average final rating of the
sample of student athletes is not significantly
different from the average final rating of
student athletes. Therefore, we can conclude
that the difference between the sample mean
216
and the hypothesized value is not sufficient to
attribute it to anything other than sampling
error. Thus, the MAPEH teacher can conclude
that the mean academic performance of the
athletes does not differ from the mean
performance of other athletes.
The class will be divided into 4 groups. Provide
each group with the materials needed in
accomplishing their tasks such as Hand-outs,
permanent markers, calculator, manila paper.
E. Developing Mastery(Leads
to Formative Assessment)
Each group will be assigned a problem to work
with. For each problem, they will perform the
following Test of Hypothesis procedures in order to
calculate the test statistic value and be able to
formulate a conclusion based from the result. (Use
0.05 level of significance to evaluate the
hypothesis)
f. Formulate the null and alternative
hypotheses.
g. Identify the test statistic to use. With the
given level of significance and the distribution of
the test statistics, state the decision rule and
specify the rejection region.
h. Using a simple random sample of
observation given in each problem, compute
the value of the test statistic by applying the
formula of test statistics identified in part b.
i. Make a decision
j. State the conclusion.
Tasks:
Group 1 and 2. “Duracell manufactures batteries
that the CEO claims will last an average of 300
hours under normal use. A researcher randomly
selected 20 batteries from the production line and
tested these batteries. The tested batteries had a
mean life span of 270 hours with a standard
deviation of 50 hours. Do we have enough evidence
to suggest that the claim of an average lifetime of
300 hours is false?”
Group 3 and 4. “The father of a senior high school
student lists down the expenses he will incur when
he sends his daughter to the university where he
wants her to study. He hypothesizes that the
average tuition fee is at least Php20,000 per
semester. He knows the variable of interest, which
is the tuition fee, is measured at least in the interval
scale or specifically in the ratio scale. He assumes
that the variable of interest follows the normal
217
distribution but both population mean and variance
are unknown. The father asks, at random, 25
students of the university about their tuition fee per
semester. He is able to get an average of
Php20,050 with a standard deviation of Php500.”
Each group shall present their outputs in front. They
will also compare their solution from the output of
other groups.
Check the solution of each group, and clarify some
errors in their solutions.
Summarize the lesson by emphasizing to students
that with the assumption of unknown population
variance (σ2) and the variable of interest is
measured at least in the interval scale and follows
the normal distribution, the appropriate test statistic,
denoted as TC is computed as
F. Making generalizations and
abstraction about the lesson
Tc
Where :
TC is the test statistic and has n-1
degrees of freedom
is the sample mean computed
from a simple random sample of n observations;
μ0 is the hypothesized value of the
parameter; and
S
is the sample standard
deviation.
Emphasize to students that decisions are made
about the hypothesis based on the result of the
calculated statistical-value (TC) at a given level of
significance. This decision is used to formulate the
conclusion.
G. Finding practical
applications of concepts
and skills in daily living
Quiz: Given the problem below, State the null
hypothesis, calculate the statistical value and make
a conclusion out of the resulted T-value.
H. Evaluating Learning
A brand of powdered milk is advertised as having a
net weight of 250 grams. A curious consumer
obtained the net weight of 10 randomly selected
cans. The values obtained are: 256, 248, 242, 245,
246, 248, 250, 255, 243 and 249 grams. Is there
reason to believe that the average net weight of the
powdered milk cans is less than 250 grams at 10%
level of significance? Assume the net weight is
218
normally distributed with unknown population
variance.
Answer:
Ho: The average net weight of the
powdered milk cans is equal to 250 grams. In
symbols, μ = 250
Ha: The average net weight of the
powdered milk cans is
equal to 250 grams. In symbols, μ< 250
With 10% level of significance, the decision
rule is “Reject the null hypothesis (Ho) if the t C
< -t0.10,9 = -2.998.”
With the computed test statistic equal to -1.23,
the null hypothesis is not rejected.
We can then say that the advertised average
net weight of the powdered milk is indeed true
or μ = 250 grams.
I.
V.
Additional activities for
application or remediation
REMARKS
VI.
REFLECTION
A. No. of learners who earned
80% in the evaluation
B. No. of learners who require
additional activities for
remediation who scored
below 80%
C. Did the remedial lesson
work? No. of learners who
have coped up with the
lesson
D. No. of learners who
continue to require
remediation
E. Which of my teachings
strategies worked well?
Why did these work?
219
F. What difficulties did I
encounter which my
principal/ supervisor can
help me solve?
G. What innovation or localized
materials did I use/ discover
which I wish to share with
the other teachers?
220
Solution to Problems
A. Group Task
Duracell manufactures batteries that the CEO claims will last an average of 300 hours
under normal use. A researcher randomly selected 20 batteries from the production
line and tested these batteries. The tested batteries had a mean life span of 270 hours
with a standard deviation of 50 hours. Do we have enough evidence to suggest that
the claim of an average lifetime of 300 hours is false?
a. Formulate the null and alternative hypotheses.
Ho: μ = 300.
Ha: μ ≠ 300.
b. Identify the test statistic to use. With the given level of significance and the
distribution of the test statistics, state the decision rule and specify the rejection region.
With the assumption of unknown population variance (σ2), the appropriate test statistic,
is the T-test denoted as TC is computed as
TC
Suppose the level of significance (α) is set at 0.05, then the decision rule for the
problem is:
Reject Ho if |tC|> ± 2.093 0.05/2, 19. Otherwise, we fail to reject Ho.
c. Using a simple random sample of observation given in each problem, compute the
value of the test statistic by applying the formula of test statistics identified in part b.
TC
= -2.68
d. Make a decision
Since our calculated t-test value is outside of our t-critical value –it lies in the critical region –
we reject the Null Hypothesis.
e. State the conclusion.
The average battery life of the sample is significantly different from the average battery life
claim by the CEO.
2. “The father of a senior high school student lists down the expenses he will incur
when he sends his daughter to the university where he wants her to study. He
hypothesizes that the average tuition fee is at least Php20,000 per semester. He
221
knows the variable of interest, which is the tuition fee, is measured at least in the
interval scale or specifically in the ratio scale. He assumes that the variable of
interest follows the normal distribution but both population mean and variance are
unknown. The father asks, at random, 25 students of the university about their
tuition fee per semester. He is able to get an average of Php20,050 with a standard
deviation of Php500.”
f. Formulate the null and alternative hypotheses.
Ho: μ ≥ 20, 000.
Ha: μ < 20, 000.
g. Identify the test statistic to use. With the given level of significance and the
distribution of the test statistics, state the decision rule and specify the rejection region.
With the assumption of unknown population variance (σ2), the appropriate test statistic,
is the T-test denoted as TC is computed as
TC
Suppose the level of significance (α) is set at 0.05, then the decision rule for the
problem is:
Reject Ho if tC < -1.7110.05, 24. Otherwise, we fail to reject Ho.
h. Using a simple random sample of observation given in each problem, compute the
value of the test statistic by applying the formula of test statistics identified in part b.
TC
= 0.5
i. Make a decision
Since our calculated t-test value is outside of our t-critical value, we fail to reject the Null
Hypothesis.
j. State the conclusion.
With the acceptance of the null hypothesis, the father can say that the average tuition fee at
the university where he wanted his daughter to study is at least Php20,000.
222
Solution to Problems
A. Group Task
1. The school nurse thinks the average height of 7th graders have increased. The
average height of a 7th grader five years ago was 145 cm with a standard deviation of
20 cm. She takes a random sample of 200 students and finds that the average height
of her sample is 147 cm. Are 7th graders now taller than they were before?
k. Formulate the null and alternative hypotheses.
Ho: The average height of 7th graders is less than or equal to 145 cm. In symbols,
μ ≤ 145.
Ha: The average height of 7th graders is more than 145 cm. In symbols,
μ > 145.
l. Identify the test statistic to use. With the given level of significance and the
distribution of the test statistics, state the decision rule and specify the rejection
region.
With the assumption of known population variance (σ 2) and the variable of
interest is measured at least in the interval scale and follows the normal distribution,
the appropriate test statistic, denoted as ZC is computed as
Suppose the level of significance (α) is set at 0.05, then the decision rule for
the problem could be stated as
Reject Ho if ZC > Z0.05 = 1.645. Otherwise, we fail to reject Ho.”
m. Using a simple random sample of observation given in each problem, compute the
value of the test statistic by applying the formula of test statistics identified in part b.
Given:
μ0
σ
n
= 147
= 145
= 20
= 200
Solution:
1.414
n. Make a decision
Since the computed Z statistic value of 1.414 is less than the critical value of
1.645 at 0.05 level of significance, therefore we fail to reject the null hypothesis.
o. State the conclusion.
Because of not rejecting the null hypothesis, the school nurse can conclude
that the average height of the 7th graders did not increase.
2. Mang Ruben is trying out a planting technique that he hopes will increase the yield
on his cacao trees. The average weight of yield on each tree is 80 kg with a standard
deviation of 30 kg. This year, after trying his new planting technique, he takes a random
sample of 35 trees and finds the average weight of yield is to be 98 kg. He wonders
whether or not this is a statistically significant increase.
a. Formulate the null and alternative hypotheses.
Ho: The average weight of yield on each tree is less than or equal to 145 kg. In
symbols, μ ≤ 80.
Ha: The average weight of yield on each tree is more than 80 kg. In
symbols, μ > 80.
223
b. Identify the test statistic to use. With the given level of significance and the
distribution of the test statistics, state the decision rule and specify the rejection
region.
With the assumption of known population variance (σ 2) and the variable of
interest is measured at least in the interval scale and follows the normal distribution,
the appropriate test statistic, denoted as ZC is computed as
Suppose the level of significance (α) is set at 0.05, then the decision rule for
the problem could be stated as
Reject Ho if ZC > Z0.05 = 1.645. Otherwise, we fail to reject Ho.”
c. Using a simple random sample of observation given in each problem, compute the
value of the test statistic by applying the formula of test statistics identified in part b.
Given:
μ0
σ
n
= 98
= 80
= 30
= 35
Solution:
3.550
d. Make a decision
Since the computed Z statistic value of 3.550 is more than the critical value of
1.645 at 0.05 level of significance, therefore reject the null hypothesis.
e. State the conclusion.
Because of rejecting the null hypothesis, Mang Ruben can conclude that with
the application of the new farming technique brought a significant increase on the yield
of his cacao trees. Meaning, this new farming technique is effective to increase the
yield of cacao trees.
224
School
Teacher
Time and Date
I.
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
OBJECTIVES
A. Content Standard
B. Performance Standard
C. Learning
Competencies/
Objectives
Write The LC code for
each
II.
CONTENT
III.
LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s Material
pages
3. Textbook pages
4. Additional Materials
from Learning
Resource (LR) portal
B. Other Learning
Resources
IV.
PROCEDURE
A. Reviewing previous
lesson or presenting
the new lesson
The learners demonstrate understanding of key
concepts of tests of hypotheses on the population
mean and population proportion.
The learner is able to perform appropriate tests of
hypotheses involving the population mean and
population proportion to make inferences in reallife problems in different discipline.
The learner solves problems involving tests of
hypothesis on the population mean
M11/12SP-IVe-1
TEST ON POPULATION MEAN
Statistics and Probability, pages 374-384
Start the lesson by reviewing the steps of
hypothesis testing procedure:
1. Formulate the null and alternative
hypotheses
2. Identify the test statistic to use. With the
given level of significance and the
distribution of the test statistics, state
the decision rule and specify the
rejection region.
3. Using a simple random sample of
observation, compute the value of the
test statistic.
4. Make decision whether to reject or fail
to reject the H0.
5. State the conclusion.
225
B. Establishing a purpose
of the lesson
C. Presenting examples/
instances of the new
lesson
D. Discussing new
concepts and
practicing new skills #1
Post the problem on the board/ project it using the
projector.
The father of a senior high school student
lists down the expenses he will incur when
he sends his daughter to the university
where he wants her to study. He
hypothesized that the average tuition/
miscellaneous fee is at least Php20, 000
per semester. He knows the variable of
interest which is the tuition/ miscellaneous
fee, is measured at least in the interval
scale or especially in the ratio scale. He
assumes that the variable of interest
follows the normal distribution but booth
population mean and variance are
unknown. The father asks, at random, 25
students of the university about their tuition/
miscellaneous fee per semester. He is able
to get an average of Php20, 050 with
standard deviation of Php500.
Let the student analyze the problem.
The class will be group with 5 members.
Each member will have a piece of colored paper
to be used in the activity. Each students will follow
the path, wherein they cannot pass until their
answer is correct ( the teacher/ selected students
will checked). The first group who finishes the task
will be having an additional 5 points and the rest
will have 4, 3, 2, and 1 depending on the time they
finish the task.
Each group will discuss first the situation and plan
some strategies to perform the activity in the
fastest way.
E. Discussing new
concepts and
practicing new skills #2
F. Developing
Mastery(Leads to
Formative
Assessment)
STEP 1
Let the students identify the null hypothesis
(𝐻0 ) and the alternative hypothesis (𝐻𝑎 ).
H0: the average tuition fee in the targeted
university is at least Php20, 000.
: 𝐻0 : 𝜇 ≥ 20, 000 𝑝𝑒𝑠𝑜𝑠
Ha: the average tuition fee in the targeted
university is less than Php20, 000.
:𝐻𝑎 : 𝜇 < 20, 000 𝑝𝑒𝑠𝑜𝑠
226
STEP 2
With the assumption of unknown
population variance (𝜎 2 ) and the variable
of interest is measured at least in the
interval scale and follows the normal
distribution.
The appropriate test statistic, denoted as 𝑡𝑐
𝑥̅ −𝜇
is computed as 𝑡𝑐 = 𝑠 0 , where 𝑥̅ and 𝑠
√𝑛
are the sample mean and standard
deviation, respectively, computed from a
simple random sample of 𝑛 observation;
and 𝜇0 is the hypothesized value of the
parameter.
The decision rule cam be one of the
following possibilities:
1. Reject 𝐻0 if 𝑡𝑐 < −𝑡𝑎 , 𝑛 − 1 .
Otherwise, we fail to reject 𝐻0
2. Reject 𝐻0 if 𝑡𝑐 < 𝑡𝑎 , 𝑛 − 1. Otherwise,
we fail to reject 𝐻0
3. Reject 𝐻0 if |𝑡𝑐 | < 𝑡𝑎/2 , 𝑛 − 1. Otherwise,
we fail to reject 𝐻0
Suppose the level of the significance (𝛼) is
set at 0.05, then the decision rule of the
problem can be stated as
“Reject 𝐻0 if the𝑡𝑐 < −𝑡𝛼,24 = −2.064”.
Otherwise, we fail to reject𝐻0 .
rejected region
G. Making generalizations
and abstraction about
the lesson
−𝑡𝛼,𝑛−1 = −2.064
STEP 3
𝑥̅ − 𝜇0
20, 050 − 20, 000
𝑡𝑐 =
=
= 0.50
𝑠
500
√𝑛
√25
STEP 4
Fail to reject the null hypothesis
STEP 5
We can conclude that the father can say that the
average tuition fee at the university where he
wanted his daughter to study is at least Php20,
000
The students will summarize the lesson today by
identifying the null hypothesis, alternative
hypothesis, assumptions, appropriate statistic and
the decision rule and rejected region.
227
How do you solve problems involving hypothesis
testing on the population mean?
H. Finding practical
applications of
concepts and skills in
daily living
I.
Evaluating Learning
Worksheet and answer sheet will be distributed
by the teacher and the students will be paired
for this activity.
A researcher used s developed problem
solving test to randomly selected grade 11
students. In this sample, 𝑥̅ = 80 and 𝑠 = 10.
The 𝜇 and the standard deviation of the
population used in the standardization of the
test were 75 and 15 respectively. Use the 95%
confidence level to answer the following
questions:
 Does the sample mean differ significantly
from the population mean?
J. Additional activities for
application or
remediation
V.
REMARKS
VI.
REFLECTION
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
228
C. Did the remedial
lesson work? No. of
learners who have
coped up with the
lesson
D. No. of learners who
continue to require
remediation
E. Which of my teachings
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
principal/ supervisor
can help me solve?
G. What innovation or
localized materials did
I use/ discover which I
wish to share with the
other teachers?
229
School
Teacher
Time and Date
I.
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
Write The LC code
for each
II.
CONTENT
III.
LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s Material
pages
3. Textbook pages
The learners demonstrate understanding of key
concepts of tests of hypotheses on the population
mean and population proportion.
The learner is able to perform appropriate tests of
hypotheses involving the population mean and
population proportion to make inferences in reallife problems in different discipline.
The learner formulates the appropriate null and
alternative hypotheses on a population proportion
M11/12SP-IVe-2
TEST ON POPULATION PROPORTION
Statistics and Probability, pages 385-389
Statistics and Probability for Senior High School,
Jimczyville Publications, pages 169-176
4. Additional Materials
from Learning
Resource (LR) portal
B. Other Learning
Resources
IV.
Teacher’s Guide
Laptop
activity Sheets
Power Point presentation
Projector
Strips of Colored Paper
PROCEDURE
A. Reviewing previous
lesson or presenting
the new lesson
Activity #1
THERE’S MY PATH!
(attached)
Students will look for the right path in a maze.
Every word that they pass through will be put in a
strip of paper and post it in front.
Answer:
Formulate the null and alternative hypothesis on
the population proportion.
Review of Terms:
Types of HypothesisPopulation
Proportion
230
B. Establishing a
purpose of the
lesson
Post the problem on the board/ project it using the
projector.
The principal of Rawis National High School
believes that this year there would be more
students from the school who pass the National
Achievement Test (NAT), so that the proportion of
the students who passed the NAT is greater than
the proportion obtained in previous year, which is
0.75.
Questions:
1. What can you say about the given
situation?
2. How it is differ from problems in previous
lesson about hypothesis testing on
population mean?
3. What would be the appropriate null and
alternative hypothesis to test this
belief?(students answer will be noted)
The teacher will introduce the concept of
population proportion.
After discussing the concept of population
proportion, present the null and alternative
hypothesis of the given problem.
C. Presenting
examples/ instances
of the new lesson
𝐻0 : The proportion of students of the school
who pass the NAT this year is equal to 0.75. In
symbol, 𝑃 = 0.75
𝐻𝑎 : The proportion of students of the school
who pass the NAT this year is greater than
0.75. In symbol, 𝑃 > 0.75
The teacher will post the situation to the students.
And the teacher will guide the students in finding
null hypothesis and alternative hypothesis
A random sample of 750 students is selected,
of whom 92 are left- handed. Use this sample
data to test the claim that 10% of the students
are left- handed.
Discussing new
concepts and practicing
new skills #1
𝐻0 : 𝑃 = 0.10
𝐻𝑎 : 𝑃 ≠ 0.10
An independent research group is interested to
show that the percentage of babies delivered
through Caesarian Section is decreasing. For the
past years, 20% of the babies were delivered
through Caesarian Section. The research group
randomly inspects the medical records of 144
births and finds that 25 of the births were by
Caesarian Section.
231
𝐻0 : The proportion of births that were
delivered by Caesarian Section is not
decreasing, that is, it is still at least equal to
0.20. In symbol, 𝑃 ≥ 0.20
𝐻𝑎 : the proportion of births that were delivered
by Caesarian Section is decreasing, that is, it
is less than 0.20. in symbol, 𝑃 < 0.20
D. Discussing new
concepts and
practicing new skills
#2
The class will be group into 4. The teacher will
post problems on the board. In a few minutes, let
the students talk about themselves the null and
alternative hypothesis of the given situation. And
after a while, the teacher will call one student
coming from a group to answer the null hypothesis
and alternative hypothesis.
E. Developing
Mastery(Leads to
Formative
Assessment)
A machine is known to produce 20% defective
products, and is therefore sent for repair. After
the machine is repaired, 400 products produced
by the machine are chosen at random and 64 of
them found to be defective. Do the data provide
enough evidence that the proportion of defective
products produced by the machine has been
reduced as a result of the repair?
𝐻0 :. 𝑃 = 0.20 (no change; the repair did not help)
𝐻0 :. 𝑃 < 0.20 (the repair was effective at reducing
the proportion of defective parts)
Polls on certain topics are conducted routinely in
order to monitor changes in the public’s opinions
over time. One such topic is the death penalty. In
2003 a pool estimated that 64% of adults support
the death penalty for a person convicted to
murders. Do the results of this pool provide
evidence that the proportion of adults who
support the death penalty for convicted
murderers change between 2003 and the later
pool?
232
F. Making
generalizations and
abstraction about the
lesson
G. Finding practical
applications of
concepts and skills in
daily living
H. Evaluating Learning
I.
Additional activities
for application or
remediation
V.
REMARKS
VI.
REFLECTION
How do we formulate the null hypothesis and
alternative hypothesis of the population proportion
of every problem?
Students will cite at least three situations when the
test of proportion is applicable.
Worksheet and answer sheet will be distributed
by the teacher and the students will be paired
for this activity.
 Mr. Alba asserts that fewer than 5% of the
bulbs that he sells are defective. Suppose
300 bulbs are randomly selected, each is
tested and 10 defective bulbs are found.
What is appropriate null and alternative
hypothesis can be formulated?
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
learners who have
coped up with the
lesson
D. No. of learners who
continue to require
remediation
E. Which of my
teachings strategies
worked well? Why
did these work?
F. What difficulties did I
encounter which my
233
principal/ supervisor
can help me solve?
G. What innovation or
localized materials
did I use/ discover
which I wish to share
with the other
teachers?
234
235
School
Teacher
Time and Date
I. OBJECTIVES
A. Content Standard
B. Performance
Standards
C. Learning
Competency
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key
concepts of test of hypothesis on the population mean
and population proportion.
The learner is able to perform appropriate tests of
hypotheses involving the population mean and the
population proportion to make inferences in real – life
problems in different disciplines.
M11/12 SP – IVe-4
Identifies the appropriate rejection region for a given
level of significance when the central limit theorem is
to be used.
Performance Indicators:
1. Illustrate graphically the rejection region and
acceptance region for a given level of
significance or confidence level.
2. Given a 2 – value with a level of
significance/confidence level, indicate if it is in
the rejection region or acceptance region.
Testing Hypothesis Involving Population Proportion
II. CONTENT
III. LEARNING
RESOURCES
A. References
1. Teacher’s
Guide
2. Learners’
Materials
3. Reference
Books
4. Additional
Materials for
Learning
B. Other Learning
Statistics and Probability by Belecina, et.al
Resources
IV. Instructional Procedure
1. Activity
Activity 1 (Teacher’s Activity)
Review by showing the following tables:
Table 1. Four Possible Outcome in Decision Making
Decision About H0
Do not
reject H0
Reject
(or accept
H0)
Type I
Correct
H0 is true
error
Decision
Reality
Correct
Type II
H0 is falsr
Decision
error
Table 2. Types of Errors
236
Error
Type
Proba Correc Type
Proba
in
bility
t
bility
decisi
Decisi
on
on
Reject I
α
Accept A
1–α
a true
a true
H0
H0
Accept II
β
Reject B
1–β
a false
a false
H0
H0
Teacher’s Input (Exposition)
In hypothesis testing, decisions are made.
Errors are likely to be committed. The best that can be
done is to control the probability with which an error
occurs. The types of errors and some details are
shown in tables 1 and 2. The most frequently used
probability values for α and β are 0.05 (5%) and 0.01
(1%).
Below are the graphical representations of a two
– tailed (non – directional) test and one – tailed
(directional) tests. The shaded parts are the rejection
regions while the unshaded are the acceptance
regions.
Two – tailed (non – directional):
both
Critical
Value
𝛼
2
Critical
the probability is found on
Value
tails of𝛼the distribution.
2
1α
M
One – tailed, left tail: the
probability is found at the left
tail of the distribution.
One – tailed, right tail: the
probability is found at the right
tail of the distribution.
Activity 2 (Students’ Activity)
(Let the class work in small groups of three members
and give/distribute metacards where either one of
these two problems are written. Let them work for 5
minutes.
1. Identify the given data and answer the
questions that follow.
237

2. Analysis
For a 95% confidence level or level of
significance, what is the value of α?
 What are the critical values for a one –
tailed test?
 What are the critical values for a two –
tailed test?
 Show the data and the critical values (in
a normal curve) graphically.
2. Identify the given data and answer the
questions that follow.
 For a 99% confidence level or level of
significance, what is the value of α?
 What are the critical values for a one –
tailed test?
 What are the critical values for a two –
tailed test?
 Show the data and the critical values
graphically.
(Call on volunteers to present groups’ outputs and
answers)
(Teacher processes the outputs and answers given or
shown by the volunteers. Teacher may add inputs and
correct misconception, if there are.
Teaching Notes:
For a 95% confidence level or level of significance, two
0.95
– tailed
= 0.4750, in the normal curve, this area
2
corresponds to Z = 1.96. Hence, critical values for 95%
confidence level or level of significance are ± 1.96.
Graphical representation is shown below.
For a 95% confidence interval or level of significance,
one – tailed, 5% from the extreme left or right is
bounded by the critical value. To compute for it, 0.50 –
0.05 = 0.4500, this area corresponds to a z – value
which is 1.65. Hence, the critical values are ± 1.65.
Below is the graphical representation.
238
Right –
tailed
Left – tailed
For a 99% confidence level or level of significance, two
.99
– tailed,
= 0.4950 and this area corresponds to two
2
2
z – values, 257 and 2.58. in this case, the larger the
value is considered.
Two – tailed
For a 99% confidence level or level of significance,
one – tailed, 1% from the extreme left or right is
bounded by the critical value. To compute, 0.50 – 0.01
= 0.4950 and this area corresponds to a z – value
which is 2.33. hence, the critical values are ± 2.33.
239
Left – tailed
Right –
tailed
3. Abstraction
(Teacher leads the class to formulate generalization )
Students’ Activity:
Based from the previous activity and
discussions, gather important information and fill the
table with missing data.
DIRECTION: Fill the table below with
appropriate/correct data.
Confidence Level
Two – Tailed
One – Tailed
-Zα/2 =
-z =
95% ( 1 – α)
Zα/2 =
z=
-Zα/2 =
-z =
99% (1 – α)
Zα/2 =
z=
Provision for practice:
Given: Z = 2, 95% confidence level/level of
significance, two tailed.
Do the following:
 Draw the normal curve
 Locate the z- value
 Indicate if the z – value is in the rejection or
acceptance region.
Expected answer:
z = 2, 95% confidence level, two – tailed (non –
directional) critical values: ± 1.96
240
4. Application
Home Task (For
Reflection)
(Pen – and – paper activity)
For each of the given set of data, do the following:
1. Draw the normal curve.
2. Locate the z – value.
3. Indicate if the z – value is in the rejection or
acceptance region.
a. z = 2, 99%, two – tailed
b. z = 2, 95%, one – tailed, right
c. z = -2.65, 95%, two – tailed
d. z = 1.86, 99%, one – tailed
e. z = -4.1, 95%, two – tailed
f. z = 1.39, 99%, one – tailed
What decision can you associate with the z – value if it
is 1.69 at 95% confidence level, two tailed?
241
School
Teacher
Time and Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standards
C. Learning
Competency
II.
III.
CONTENT
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide
2. Learners’
Materials
3. Reference
Books
4. Additional
Materials for
Learning
B. Other Learning
Resources
IV.
Instructional
Procedure
Grade Level
Learning Area
Quarter
11
MATHEMATICS
3rd
The learner demonstrates understanding of key concepts of
test of hypothesis on the population mean and population
proportion.
The learner is able to perform appropriate tests of
hypotheses involving the population mean and the population
proportion to make inferences in real – life problems in
different disciplines.
M11/12 SP – IVf –g-1
Solves problems involving test of hypotheses on the
population proportion.
Performance Indicators:
3. Conduct tests involving population proportion;
4. Interpret test of proportions.
Testing Hypothesis Involving Population Proportion
Statistics and Probability by Belecina et. al.
Statistics and Probability by Arciaga et. al.
Review/ Motivation
Ask students to give reactions on these statements.
242
1. Activity
1. When the evidence is not enough, do not reject the
null hypothesis.
2. When the evidence is sufficient to reject the null
hypothesis, a significant difference exists.
(Teacher solicits reactions from the students and discusses
with them that the two statements are true.)
Teacher’s exposition and unlocking of difficulties.
There are certain situations where inferences are to be made
using only the proportions or percentages of population.
These inferences are made in the context of probability- P.
The formula for the test statistic z for proportions is
̂
𝑝−𝑝
𝑧 = ̂̂ 0
∝𝑝
Where
𝑥
𝑝̂ =
𝑛
𝑝_𝑜 = ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑧𝑒𝑑 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 , 𝑝
∝ 𝑝̂ =
𝑝𝑞
√ 𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑖𝑛𝑔 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑜𝑛 𝑝̂ 𝑜𝑟
𝑝𝑜 𝑞𝑜
𝑛
∝ 𝑝̂ = √
𝑖𝑓 𝑝𝑜 𝑖𝑠 𝑢𝑠𝑒𝑑.
𝑞𝑜= 1 − 𝑝𝑜
So
𝑝̂−𝑝
𝑧 = 𝑝 𝑞𝑜
√ 𝑜𝑛 𝑜
 For a one-tailed test:
𝐻𝑜: 𝑝 = 𝑝𝑜
𝐻1: 𝑃>𝑃𝑂 & the rejection region is 𝑧 > 𝑧∝
or
𝐻1:𝑃< 𝑃𝑂 & 𝑡ℎ𝑒 𝑟𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛 𝑖𝑠 𝑧 < −𝑧𝛼
 For a two – tailed test
𝐻𝑜 : 𝑝 = 𝑝𝑂
𝐻1 : 𝑝 ≠ 𝑝𝑜
The rejection region is 𝑧 < −𝑧𝛼 𝑜𝑟 𝑧 > 𝑧𝛼
2
2
Recalling the previous table for critical values,
Confidence
Significance Critical Values
level
Level
One -Tailed
Two-Tailed
. 05 (5%)
. 95%
±1.65
±1.96
. 01(1%)
. 99%
±2.33
±2.58
Students’ activity 1(student ’s task)
In small groups of 3 members, let the students solve for z if:
𝑛 = 94
15
𝑝̂ =
94
𝑝𝑜 = 15 %
(Call on a student volunteer to show solution on the board)
Expected solution:
𝑝̂ − 𝑝𝑜
𝑧=
𝑝 𝑞
√ 𝑜 0
𝑛
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15
− 15
𝑧 = 94
√(. 15)(. 85)
94
𝑧 ≈ 0.26
𝑞𝑜 = 1 − 𝑝𝑜
= 1- .15
= .85
Student’s Activity 2(Divided Problem Solving)
Post the problem below :
A whitening soap as product of factory A is claimed as 60%
effective. A research was conducted to 100 adults and 70 of
them found the soap effective. Can this fact be used to
conclude that its product is more effective over the other
whitening soap being sold in the leading store? Conduct a
hypothesis test using the 0.05 level of significance.
Divide the class into groups of 5 members & let each group
work on solving the given problem using the guide questions/
directives below:
1. Is CLT applicable? Why?
2. What is the population parameter?
3. What are the given data?
4. Formulate the hypotheses.
5. Select the test statistic.
6. Compute using the test statistic.
7. State the decision rule.
8. Interpret the result.
2. Analysis
(Call on a representative of each group to present the
solution, one at a time.)
Expected solution:
1. Is CLT applicable?
Why?
2. What is the
population
parameter?
3. What are the given
data?
Yes, because 𝑛 > 30, n=
100
Population proportion p.
4. Formulate
Hypotheses
𝐻0 : 𝑝̂ =𝑝0
244
𝑝̂ = 70%= .70
n= 100
𝑝𝑜 = 60%=. 60
𝑞𝑜 = 1 − 𝑝0
𝑞𝑜 = .40
𝐻1 : 𝑝̂ > 𝑝0
5. Select the test
statistic
Test Statistic: Z -test
𝑝̂ − 𝑝𝑜
𝑧=
𝑝 𝑞
√ 𝑜 𝑜
𝑛
𝑝̂ − 𝑝0
6. Compute using the
𝑧=
test statistic
√𝑝𝑜 𝑞𝑜
. 70 − .60
𝑧=
√(. 60)(. 40)
100
= 2.041241452
≈ 2.04
7. State the decision
Z= 2.04 at 𝛼 = .05, the
rule
critical values are ± 1.65,
right- tailed 2.04> 1.65
Z value lies in the rejection
area
Decision rule : reject the 𝐻𝑜
8. Interpret the result
There is a significant
difference between the two
parameters compared.
The whitening soap as a
product of factory A is more
effective than the other
whitening soap being sold
in the leading stores.
Teacher’s exposition is essential in this part to guide and
clarify salient concepts especially on the statement of the
decision rule & interpretation parts.
3. Abstraction
4. Application
(The teacher lead the class to formulate generalization )
Student’s activity
Based from the previous activity and discussion, what are the
suggested steps in solving problems involving test of
hypotheses on the population or proportion?
Expected answers:
1. Determine the given data.
2. Formulate the hypotheses. Let the test is directional
or non directional .
3. Select the test Statistic
4. Compute for using the test statistic.
5. State the decision rule.
6. Interpret the results.
Let the students work on the following problem in pairs,
The supplier of a truckload of fruits asserts that less than 5%
of these are with defects, 20 prone these assertion, 250 fruits
were randomly selected, tested and were found to be with
245
defects is this sufficient to conclude that less than 5% of the
fruits are with defects? Use 0.01 as significance level.
(Call on 2 to 3 volunteers to present solution )
Expected answers:
Given data:
P= 0.05(5%)= 𝑝𝑜 ( given a null hypothesis)
n= 200- CLT is applicable
10
𝑝̂ =
= 0.04
250
𝛼 = 0.01; 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒: 𝑙𝑒𝑓𝑡 ∶ −1.65
𝑞𝑜 = 1 − 𝑝𝑜
= 1-0.05
=.95
Hypotheses:
𝐻𝑂 = 𝑃 = 𝑃𝑂 = 0.05
𝐻1 : 𝑝 < 0.05 (𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 − 𝑙𝑒𝑓𝑡 − 𝑡𝑎𝑖𝑙𝑒𝑑)
Test statistic: z statistic
𝑝̂−𝑝
𝑧 = 𝑝 𝑞𝑜
√ 𝑜𝑛 𝑂
=
0.04−0.05
(0.05)(0.95)
250
√
= -0.7254762501
≈-0.73
Decision Rule
Z= -0.73
Home Task (For
Reflection)
246