Statistics and
Probability
The Probabilities and Percentiles
Under the Normal Curve
SENIOR
HIGH
SCHOOL
Module
9
Quarter 1
Statistics and Probability
Quarter 1 – Module 9: The Probabilities and Percentiles Under the Normal Curve
First Edition, 2020
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Statistics and
Probability
SENIOR
HIGH
SCHOOL
Module
9
Quarter 1
The Probabilities and Percentiles
Under the Normal Curve
Introductory Message
For the facilitator:
Welcome to the Statistics and Probability (Senior High School) on The Probabilities
and Percentiles Under the Normal Curve! This module was collaboratively designed,
developed and reviewed by educators from Schools Division Office of Pasig City
headed by its Officer-In-Charge Schools Division Superintendent, Ma. Evalou
Concepcion A. Agustin in partnership with the Local Government of Pasig through
its mayor, Honorable Victor Ma. Regis N. Sotto. The writers utilized the standards
set by the K to 12 Curriculum using the Most Essential Learning Competencies
(MELC) while overcoming their personal, social, and economic constraints in
schooling.
This learning material hopes to engage the learners into guided and independent
learning activities at their own pace and time. Further, this also aims to help
learners acquire the needed 21st century skills especially the 5 Cs namely:
Communication, Collaboration, Creativity, Critical Thinking and Character while
taking into consideration their needs and circumstances.
In addition to the material in the main text, you will also see this box in the body of
the module:
Notes to the Teacher
This contains helpful tips or strategies
that will help you in guiding the learners.
As a facilitator you are expected to orient the learners on how to use this module.
You also need to keep track of the learners' progress while allowing them to
manage their own learning. Moreover, you are expected to encourage and assist the
learners as they do the tasks included in the module.
For the learner:
Welcome to Statistics and Probability Module on The Probabilities and Percentiles
Under the Normal Curve! The hand is one of the most symbolized part of the
human body. It is often used to depict skill, action and purpose. Through our
hands we may learn, create and accomplish. Hence, the hand in this learning
resource signifies that you as a learner is capable and empowered to successfully
achieve the relevant competencies and skills at your own pace and time. Your
academic success lies in your own hands!
This module was designed to provide you with fun and meaningful opportunities
for guided and independent learning at your own pace and time. You will be
enabled to process the contents of the learning material while being an active
learner.
This module has the following parts and corresponding icons:
Expectation - These are what you will be able to know after
completing the lessons in the module
Pre-test - This will measure your prior knowledge and the
concepts to be mastered throughout the lesson.
Recap - This section will measure what learnings and skills
that you understand from the previous lesson.
Lesson- This section will discuss the topic for this module.
Activities - This is a set of activities you will perform.
Wrap Up- This section
applications of the lessons.
summarizes
the
concepts
and
Valuing-this part will check the integration of values in the
learning competency.
Post-test - This will measure how much you have learned from
the entire module.
EXPECTATION
Learning Objective:
At the end of the learning episode, you are expected to:
1. compute probabilities and percentiles using the standard normal
table.
PRETEST
Directio
n: Choose the letter that corresponds to the correct answer.
1. Which of the following is the probability of the area less
than z = 0?
a. 0%
b. 50%
c. 75%
d. 100%
2. What is the probability of the area below z = -1.25?
a. 0.1056
b. 0.1038
c. 0.1025
d. 0.
3. Find the probability of the area which is at least z = 1.
a. 1
b. 0.8413
c. 0.1587
d. 0
4. Compute the probability of the area in between z = -3 and z = 3.
a. 0.6826
b. 0.9544
c. 0.9974
d. 1
5. Which of the following represents the above 75th percentile of the
distribution?
a. to the left of z=-0.675
b. to the right of z = -0.675
c. below z= 0.675
d. above z =0.675
RECAP
In the previous lesson, you have learned on how to use the zTable or what we call the Table of Areas under the Normal Curve, finding the
z-score values of the random variables with the use of the formula z =
the population data and z =
𝑥−𝑥̅
𝑆
𝑥−𝑢
𝜎
for
for the sample data. The raw score for the
random variable X was also converted using the formula X = µ + z(σ) for the
population data and X = 𝑥̅ + z(s) for the sample data. This raw score X is
above the
LESSON
mean if z is positive and it is below the mean when z is
negative.
The probability, or proportion, or the percentage associate with
the specific sets of measurement values. The value of probability is a
number from 0 to 1. All probabilities associated with the standard normal
random variables can be shown as areas under the standard normal curve.
In finding the probabilities under the normal curve, we will use the z-Table
which is also known as Table of Areas under the Normal Curve and
probability notation in equating the desired probability of an area.
The Probability Notations Under the Normal Curve
The following mathematical notations for a random variable are
used in various solutions concerning the normal curve.
P ( z < a ) denotes the probability that the z-score is less than a
P ( z > a ) denotes the probability that the z-score is greater than a
P ( a < z < b ) denotes the probability that the z-score is between a and b
where: a and b are z-score values.
Note: It is important to correctly interpret the phrases such as:
P(z<a)
P(z>a)
P(a<z<b)
less than z
greater than z
z is between a and
b
at most z
not more than z
below z
to the left of z
at least z
more than z
above z
to the right of z
Example 1: Find the probability of the area below z = 0.50.
Step 1: Draw a normal curve.
Step 2: Locate the z-score value.
Step 3: Draw a line through z = 0.50
Step 4: Shade the region to the left of z = 0.50
Step 5: Consult the z-Table and find the area that corresponds to z = 0.50
Step 6: Examine the graph and use probability notation P(z < a)
P (z < 0.50) = 0.6915 or
P (z < 0.50) = 69.15%
-3
P(z<0.50) =
0.6915 or
69.15%
µ
-1
0 0.501
-2
2
3
Z
Thus, the probability of the area below z = 0.50 is 0.6915 or 69.15%.
Example 2: Find the area that is at least z = -2.
Step
Step
Step
Step
Step
1:
2:
3:
4:
5:
Draw a normal curve.
Locate the z-score value.
Draw a line through z = -2.00
Shade the region to the right of z =-2.00.
Consult the z-Table and find the area that corresponds to z =-2.00
and it is 0.0228
Step 6: Examine the graph and use probability notation P(z > a) = 1 – P(z < a)
P (z > -2.00) = 1 – P(z<-2.00)
= 1 – 0.0228
P(z >-2) = 0.9772 or 97.72%
P(z>-2)
= 0.9772 or 97.72%
µ
-3
-2
-1
0
1
2
3
Z
Thus, the probability of the area that is at least z = -2 is 0.9772 or
97.72%.
Example 3: Find the area between z = -1.5 and z = 2.
Step 1: Draw a normal curve.
Step 2: Locate the z-score value.
Step 3: Draw a line through z = -1.5 and z = 2.
Step 4: Shade the region between z = -1.5 and z = 2.
Step 5: Consult the z-Table and find the area that corresponds to z =2 and
P(-1.5<z<2) = P(z<2) – P(z<-1.5)
= 0.9772 – 0.0668
= 0.9104
P(-1.5<z<2) = 0.9104 or 91.04%
P(-1.5<z<2)
= 0.9104 or
91.04%
-3
-2
-1.5
-1
µ
0
1
2
3
Z
Thus, the probability of the area between z = -1.5 and z = 2 is
0.9104 or 91.04%.
Example 4: Find the 85th percentile of a normal distribution.
Step
Step
Step
Step
1.03
1: Draw the appropriate normal curve.
2: Express the given percentage as probability.
3: Refer to the z-Table. Locate the area 0.8500.
4: Find z by interpolation. Since 0.8500 is between two z-values
and 1.04, find its average as follows
1.03+1.04
2
Step 5: Locate z = 1.035 under the curve in Step 1.
=
2.07
2
= 1.035
Step 6: Draw a line and shade the region to the left of z = 1.035.
Step 7: Describe the shaded region.
The shaded region under the normal curve is the 85th percentile
of the distribution.
85 %
-3
-2
-1
µ
0
1.035
1
2
3
Z
The 85th percentile of the distribution is the shaded region to the
left of z = 1.035 as shown in the normal curve.
Example 5: In a job fair sponsored by the 3 big companies, 2500 applicants
applied for a job.Their mean age was found to be 35 with a standard
deviation of 5 years.
a. Draw a normal curve distribution showing the z-scores and the raw
scores.
b. What is the probability of the job applicants who are below 33 years old?
c. How many applicants have ages between 28 and 44 years?
d. Find the percentage of the applicants who are above 39 years old.
Answer in a.
-3
20
-2
25
-1
30
µ
0
35
1
40
2
45
3
50
Z
X
Answer in b.
Step 1: Find z, when X = 33, µ = 35, σ = 5
Use the formula :
z=
𝑋−µ
𝜎
Substituting to the formula:
z=
33−35
z=
−2
5
5
z = -0.40
Step 2: Refer to the z-Table. Locate the area -0.40 and corresponds to
0.3446.
Step 3: Draw a line and shade the region to the left of z=-0.40.
0.3446 or
34.46%
-3
-2
-1
20
25
30
µ
0.4 0
33 35
1
2
3
Z
40
45
50
X
Thus, the probability of the job applicants who are below 33 years
old is 0.3446 or 34.46%.
Answer in c.
Step 1: Find z, using the formula : z =
𝑋−µ
𝜎
Substituting to the formula: when X=28, µ=35, σ=5 ; for X=44, µ=35,
σ=5
z=
28−35
z =−
z=
5
7
44−35
9
z =5
5
z = -1.40
5
z = 1.80
Step 2: Refer to the z-Table. Locate the area z=-1.4 and z=1.8, these
correspond
to 0.0808 and 0.9641.
Step 3: Examine the graph and use probability notation
P (a<z<b) = P(z<b) - P(z<a)
P(-1.4<z<1.8) = P(z<1.8) – P(z<-1.4)
= 0.9641 – 0.0808
88.33%
-3
20
-2
25
-1.4-1
28 30
µ
0
35
Step 4: Find the 88.33% of 2500.
1
40
1.82
4445
3
50
Z
X
(0.8833)(2500)=2,202.25
(Note: Round – up 2,202.25 is 2,203.)
Thus, the number of applicants ages between 28 and 44 years is 2,203.
Answer in d.
𝑋−µ
Step 1: Find z, when X=39, µ=35, σ=5. Use the formula : z =
Substituting to the formula: z =
𝜎
39−35
4
z =5
5
z = 0.80
Step 2: Refer to the z-Table. Locate the area of z=0.80 and it corresponds
to 0.7881.
Step 3: Draw a line and shade the region to the right of z=0.80
.
Step 4: Examine the graph and use probability notation
P(z > a) = 1 – P(z < a)
P (z > 0.80) = 1 – P(z<0.80)
= 1 – 0.7881
= 0.2119
P(z >0.80) = 21.19%
21.19%
-3
20
-2
25
-1
30
µ
0
35
0.81
3940
2
45
3
50
Z
X
Thus, the percentage of the applicants who are above 39 years old is
21.19%.
ACTIVITIES
ACTIVITY 1: PRACTICE
Direction: Answer the following and illustrate each under the normal curve:
1.
2.
3.
4.
Compute the probability area to the left of z = -1.25.
Compute the probability area above z = 1.
Find the probability area between z = -0.25 and z = 1.5.
Find the 90th percentile of a normal curve.
5. Compute the upper 5% of the normal curve.
ACTIVITY 2: Keep Practicing
Direction: In Barangay Mapagkalingap, there are 500 families who are
members of SAP. Their
mean of family members is 6 with a standard
deviation of 2.
1. What is the probability of the families with less than 7 members?
2. How many families have 4 to 11 members?
3.
Find the percentage of families with more than 9
WRAP–UP
members?
In finding the probabilities and percentiles under the normal
curve, use the z-Table which is also known as Table of Areas under the
Normal Curve and the following probability notations:
denotes the probability that the z-score is less than a
P(z<a)
P(z>a)
denotes the probability that the z-score is greater than a
P ( a < z < b ) denotes the probability that the z-score is between a and b
where: a and b are z-score values.
The following phrases for each probability notation may also help:
P(z<a)
P(z>a)
P(a<z<b)
less than z
greater than z
z is between a and
b
at most z
not more than z
below z
to the left of z
at least z
more than z
above z
to the right of z
VALUING
In our lesson, we find the probabilities and percentiles using the
standard normal table. As we begin to live in reality, yes, we are giving rate
or percentage for every accomplishment we have achieved in life despite of
the fact that everything else falls into places. What’s most important is the
feeling of gratification specially when we sacrifice ourselves for other people.
What sacrifices and helps can you extend to them in time of pandemic like
this Covid 19?
POSTTEST
I. Compute for the probabilities/percentiles of the following areas using
the standard normal table, then illustrate under the normal curve.
1. the area less than z = 0
2. the area above z = 1.75
3. the area in between z = -3 and z = -1.03
4. the 75th percentile of the distribution
5. above 25th percentile of the distribution
II. Given the 600 points in a loyalty card of a customer with the mean
µ = 150 and standard deviation σ = 25.
1. What is the probability for less than 125 points of the costumer?
2. How many points are in between 100 and 200?
3. What is the percentage of above 220 points of the costumer?
Rene R. Belecina, Elisa S. Bacay, and Efren B. Mateo, Statistics and
REFERENCES
Activity 2
1. 0.6915 or 69.15%
2. 418.
Post Test
3. 6.68%
I. 1. 0.50 or 50%
2. 0.0401 or 4.01%
3. 0.1502 or 15.02%
4. below z=0.675
5. above z=0.675
II.1. 0.1587 or 15.87%
2. 0.9544 or 95.44%
3. 0.0026 or 0.26%
Activity 1
1. 0.1056 or 10.56%
PRE-TEST
1.
2.
3.
4.
5.
2. 0.1587 or 15.87%
3. 0.5319 or 53.19%
4. below z = 1.285
5. above z = 1.645
b
a
c
c
b
KEY TO CORRECTION
Probability, REX Book Store: Philippine, 2016
Zita VJ Albacea, Ph.D., Marl John V. Ayaay, Isidro P. David, Ph.D., and
Imelda E. de Mesa, Statistics and Probability, Commission on
Higher Education: 2016