Schools Division Office of Camarines Sur
San Jose, Pili, Camarines Sur
LEARNING ACTIVITY SHEET No. 18
(Solving Problems Involving Sampling Distribution of Sample Means)
Week 7 - 8
Name of Student:_______________________________________________________
Learning Area: STATISTICS AND PROBABILITY Grade Level:_______________
Section: ______________________________________ Date:____________________
I. Introductory Concept
The central limit theorem states that the sampling distribution of the
sample means approaches the normal distribution as the sample size get larger – no
matter what the shape of the population distribution is. In this activity sheet, you are
going to answer questions/problems about sample means in the same manner that
a normal distribution can be used to answer questions about individual values.
II. Learning Skills from the MELCs
After going through this activity sheet, you are going to solve problems
involving sampling distribution of the sample means. (M11/12SP– III – e – f -1)
III. Activities
The central limit theorem states that as the sample size n increases
without limit, the shape of the distribution of the sample means taken with
replacement from a population with mean 𝜇 and standard deviation 𝜎 will
approach a normal distribution. As previously shown in LAS 17, this distribution
If the sample means are randomly selected with𝜎replacement, the
will have a mean 𝜇𝑥 = 𝜇 and standard deviation 𝜎𝑥 = 𝑛.
For this reason, we can use the central limit theorem to answer
questions about sample means in the same manner that a normal distribution
can be used to answer questions about individual values. The only difference
is that a new formula must be used for the z – values, it is
𝑧=
𝑋− 𝜇
𝜎
𝑛
Where: 𝑋 − 𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛
𝜇 − 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛
𝜎 − 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑛 − 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒
1
It is important to remember two things when you use the central limit
theorem:
1. When the variable is normally distributed, the distribution of
the sample means will be normally distributed, for any
sample size n.
2. When the distribution of the original variable is not normal, a
sample size of 30 or more is needed to use a normal
distribution to approximate the distribution of the sample
means. The larger the sample, the better the approximation
will be.
For you to better understand the use of central limit theorem in solving
problems involving sampling distribution of sample means, I will show you examples.
Example 1:
Fresh Cola uses a filling machine to fill the plastic bottles with soda. The
contents of every bottle vary according to a normal distribution with =
and
=
. 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/facts in the
problem.
2. Identify what is
asked for.
3. Identify the
formula to be
used.
4. Solve the
problem.
Solution
=
=
=
=
The probability that the mean samples will be less than 250 ml.
(In symbols,
)
We are dealing with the data about sample means. So, we will
use the formula
.
=
=
=
=
We shall find the
under the normal curve.
2
=−
=
−
by getting the area
To find the area of the z – value we are going to look up the
corresponding area in Annex A.
5. State the final
answer.
Thus,
=
−
=
The probability that 10 randomly selected bottles will have a
mean less than 250 ml is 0.0008 or 0.08%.
Example 2:
A. C. Neilsen reported that the children between the ages 2 and 5
watch an average of 25 hours of television per week. Assume that the variable is
normally distributed, and the standard deviation is 3 hours. If 20 children between
the ages 2 and 4 are randomly selected, find the probability that the mean of the
hours they watch television is greater than 26.3 hours.
I.
II.
III.
Given:
=
=
=
=
Required:
Solution:
=
=
=
We shall find
the normal curve.
=
3
by getting the area under
From appendix A, the area to the left of 1.94 is 0.9738. But
we are looking for the area to the right of 1.94. So, to
obtain the area to the right of 1.93, all we need to do is to
subtract 0.9738 from 1. Thus, we have
=
IV.
=
−
=
.
Final Answer: Therefore, the probability of obtaining a
sample mean larger than 26.3 is 2.62%.
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ACTIVITY 1: DUCK EGGS
Directions: Answer the given problem by filling in the blank parts of the
solution.
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.
3. Identify the
formula to be
used.
4. Solve the
problem.
Solution
=
=
=
=
(1)
(2)
The probability that the average weight of the 12 ducks
selected at random will be more than 75 grams. In symbols,
.
We are dealing with the data involving sample means. So, we
will use the formula
=
.
=
=
(for 3 and 4 substitute the values of
=
We shall find the
normal curve.
5. State the final
answer.
.(5)
by getting the area under the
=
______) (6)
=__________ (7)
So, the probability that 12 randomly selected duck eggs will
have a mean greater than 75 grams is ____________. (8)
5
)
ACTIVITY 2: LIFE EXPECTANCY
Directions: Solve the given problem by using the steps you learned from this
learning activity sheet. Write your answers on the space provided for each step.
Problem:
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. 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.
Solution
Steps
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.
6
ACTIVITY 3: PREGNANCY
what
Directions: Solve the given problem by using the steps you learned from this
learning activity sheet.
Problem:
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. What is the probability that the sample mean length
of pregnancy lasts for less than 250 days?
Solution
Steps
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.
IV. Rubric for Scoring
ACTIVITY 1
1 point for every correct
answer
ACTIVITY 2
2 points for every correct
step.
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ACTIVITY 3
2 points for every correct
step.
V. Answer Key
1.
2.
3.
4.
5.
6.
7.
8.
ACTIVITY 1
=
2. =
75
70
=
1.73
0.0418 or 4.18%
0.0418 or 4.18%
ACTIVITY 2
I.
=
ACTIVITY 3
=
=
I.
=
=
=
=
=
II.
III.
=
IV.
=
II.
III.
.
IV.
=
=−
=
=−
−
=
=
V.
.
=
The probability that
the mean life
expectancy will be
less than 50 years is
0.01%
=
V.
−
The probability that
the sample mean
length of
pregnancy will be
less than 250 days is
0.03%
VI. References
Bluman, A. G. (2009). Elementary Statistics: A Step by Step Approach (7th
ed.). New York: McGraw - Hill. (pp. 334)
Department of Education. Prototype Contextualized Daily Lesson Plan in
Grade 11/12 Statatistics and Probability. (pp. 85 -97)
DEVELOPMENT TEAM OF THE LEARNING ACTIVITY SHEET
WRITER
:
MANAGEMENT TEAM/ REVIEWERS:
Editor
Lay-out Artist
Illustrator
Validators
:
:
:
:
JOHN EMMANUEL R. IBE – Magarao NHS
JHOMAR B. JARAVATA- Bula NHS
SONIA V. MORAL- Colacling National HS
JUMAR R. VELASCO- RSMOHS
JHOMAR B. JARAVATA- Bula NHS
PATERNO P. MAPULA, JR. – Bolo Norte HS
DAVID V. ORTIOLA-Laganac HS
ROGEL JOHN O. NAVAL-Sta. Cruz NHS
FROILAN R. DOBLON – San Fernando NHS
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