This is a detailed lesson plan on Mean and Variance of the sampling distribution of the sample means using 3Is Method.

1. LESSON PLAN FOR STATISTICS & PROBABILITY
I. INFORMATION
Subject Matter: Sampling Distribution of the Sample Means from an Infinite Population
Grade Level: XII Time Allotment: 1 hour
Teacher/s: Elton John B. Embodo
Content Standard: The learner demonstrates understanding of key concepts of sampling and sampling
distributions of the sample mean.
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.
Learning Competency: The learner finds the mean and variance of the sampling distribution of the sample mean.
M11/12SP-IIId-5
Objectives: At the end of the lesson, the students must have:
a. calculated the mean, variance, and standard deviation of the sampling distribution
of the samples means.
b. explained the relevance of selecting a sample from a population in real life
scenarios.
References: The Sampling Distribution of the Sample Mean. (n.d.). https://
saylordotorg.io/text_introductory-statistics/s10-02-the-sampling-distribution-of-t.html
Instructional Materials: PowerPoint, chalk
Skills: Analysis and Collaboration
Values: Unity, cooperation, camaraderie
Method: 3Is Method
II. LEARNING EXPERIENCES
Teacher’s Activity Students’ Response
A. Introduction
1. Prayer
2. Greetings
3. Reminders
4. Checking of Attendance
5. Classroom Rules - MATH
Must come to class neat, clean, and prepared.
Actively participate in the activities and pay attention to the
discussion.
Talk appropriately and respectfully to your teacher and
classmates.
Handle the learning materials with care.
Are my rules clear to you class?
a. Review
In our previous meetings, you have learned on how to
calculate the mean and variance of a given population. Is
there anyone here who would like to describe what a mean
is?
That is correct! How about the variance or standard deviation
of a set of data? Who would like to describe?
Fantastic!
A mean refers to the average of the values from of given
set of data.
The variance or standard deviation of a given sample or
population refers to average distances of each value
from the mean of the given sample. It is also used to tell
how the spread the distribution of the data is.

2. b. motivation
You have also learned class that from the given population,
we can derive several sample means based on a certain
condition through following some indicated steps.
But how about if the given population is infinite? How do you
derive the sample means?
And how do we calculate the mean, variance, and standard
deviation of the sampling distribution of sample means from
an infinite population?
B. Interaction
To answer my questions, be with me this morning as I discuss
to you the sampling distribution of the sample means from an
infinite population. Everybody read!
Statement of the Aim
Listen attentively since you are expected to achieve these
objectives. Can somebody read?
In determining the number of samples from an infinite
population, we can repeat selecting the same values. This is
since with replacement is allowed.
For us to be guided, here are the formulas and steps in finding
the mean, variance, and standard deviation of the sampling
distribution of the sample means
Formulas
Sampling Distribution of The Sample Means from An
Infinite Population.
Objectives:
a. calculate the mean, variance, and
standard deviation of the sampling
distribution of the samples means.
b. explain the relevance of selecting a
sample from a population in real
life scenarios.
1. n
N - number of samples with replacement
(infinite population)
2. ( )
x x P x
  
 - mean of the sampling
distribution of the sample means
3. 2 2
( ) ( )
x x
P x x
 
   - variance of the
sampling distribution of the sample means

3. Steps
Giving of Examples
Example 1
Since the population is infinite, we can repeat or replace a
value with itself. So, what is the first step to follow?
What is then the number of samples using the formula n
N ?
Based on the next step, what shall we do?
Determine the number of samples using the formula n
N
.
2
3
9
n
N


There are 9 possible number of samples with size 2 from
the given population.
1. List all the possible samples and their
corresponding means.
1. Determine the number of samples using the
formula n
N .
2. List all the possible samples and their
corresponding means.
3. Construct the sampling distribution of the
means.
4. Compute the mean of the sampling
distribution of the sample means.
a. Multiple the sample mean by the
corresponding probability
b. Add the results
5. Compute the variance of the sampling
distribution of the sample means.
a. Subtract the sample means x
 from
each sample X. Label this as x
X 
 .
b. Square the difference x
X 

c. Multiply the results by the
corresponding probability. Label this
as 2
( ) ( )
x
P x x 
  .
d. Add the results
1. A population consists of three
numbers (2, 4, 6). Consider all
possible samples of size 2 which
can be drawn with replacement
from the population.

4. The next step to take is to construct a sampling distribution
of the means. The first column must be the list of distinct
means, the second column should be the frequency and the
third column is the probability of each mean.
After obtaining the sampling distribution of the sample
means, what is the next thing to do?
Very good! Since we have already obtained the mean of the
sampling distribution of the sample means, what is then the
next step to take?
Samples Means
(2,2) 2
(2,4) 3
(2,6) 4
(4,4) 4
(4,2) 3
(4,6) 5
(6,6) 6
(6,2) 4
(6,4) 5
Mean: Add the elements of the sample and divide by
their number.
Sample Mean
X
Frequency Probability
P(X)
2 1 1/9
3 2 2/9
4 3 3/9
5 2 2/9
6 1 1/9
Total 9 1
Probability: Divide the frequency of the each mean by
the total amount of frequency.
We will compute the mean of the sampling distribution
of the sample means using the formula
( )
x x P x
  

Where x is the sample mean and P(x) is its
corresponding probability.
Sample Mean
X
Probability
P(X)
X ∙ P(X)
2 1/9 0.22
3 2/9 0.67
4 3/9 1.33
5 2/9 1.11
6 1/9 0.67
Total 4
( )
x x P x
  
 = 4
Compute the variance of the sampling distribution of the
sample means using the formula which is
2 2
( ) ( )
x x
P x x
 
  

5. Collaborative Activities
Here are the mechanics for the group the activity. Everybody
read!
For next example, I am going to group you into two groups.
Here is the mechanics, everybody read!
Group Activity
Example 2
a. Subtract the sample means x
 from each
sample X. Label this as x
X 
 .
b. Square the difference x
X 

c. Multiply the results by the corresponding
probability. Label tis as 2
( ) ( )
x
P x x 
  .
d. Add the results.
X P(X) X-Mx (X-Mx)2
P(X) ∙ (X-Mx)2
2 1/9 -2 4 0.44
3 2/9 -1 1 0.22
4 3/9 0 0 0
5 2/9 1 1 0.22
6 1/9 2 4 0.44
Total 1.32
2 2
( ) ( )
x x
P x x
 
   = 1.32
Thus, the standard deviation is 2
x
 = 1.15
1. The class will be divided into two
groups.
2. Each group will be given with
different problems to be solved in
10 minutes.
3. The group which can finish
solving the problem first with
correct solutions and answers will
be declared as the winner.
4. Each group must select one
representative to explain the output
in front.
2. A population consists of four
numbers (18, 20, 22, 24). Consider
all possible samples of size 2
which can be drawn with
replacement from the population.

6. Presentation of Group Activity
Group 1 and Group 2’s expected output
1. Number of samples with replacement is 42
= 16.
Samples Means
(18,18) 18
(18,20) 19
(18,22) 20
(18,24) 21
(20,18) 19
(20,20) 20
(20,22) 21
(20,24) 22
(22,18) 20
(22,20) 21
(22,22) 22
(22,24) 23
(24,18) 21
(24,20) 22
(24,22) 23
(24,24) 24
Sample Mean
X
Frequency Probability
P(X)
18 1 1/16
19 2 2/16
20 3 3/16
21 4 4/16
22 3 3/16
23 2 2/16
24 1 1/16
Total 16 1
Sample Mean
X
Probability
P(X)
X ∙ P(X)
18 1/16 1.125
19 2/16 2.375
20 3/16 3.75
21 4/16 5.25
22 3/16 4.125
23 2/16 2.875
24 1/16 1.5
Total 21
( )
x x P x
  
 = 21
X P(X) X-Mx (X-Mx)2
P(X) ∙ (X-Mx)2
18 1/16 -3 9 0.5625
19 2/16 -2 4 0.5
20 3/16 -1 1 0.1875
21 4/16 0 0 0
22 3/16 1 1 0.1875
23 2/16 2 4 0.5
24 1/16 3 9 0.5625
Total 2.5

7. Prepared:
ELTON JOHN B. EMBODO
Teacher 1 Applicant
C. Integration
Values Integration
A while ago class, we discussed the sampling distribution of
the sample mean from an infinite population.
Class, how you ever wondered why do we take a sample from
a population when we study or analyze a set of data? Why is
there a need to use samples, rather than the population itself?
You are all correct. Taking a sample from a population is a
wise thing to do because treating or analyzing a sample is
easier, more practical, cost-effective, convenient, and
manageable.
It just like conducting research, if the population of the
respondents is quite huge, then taking a sample from it will
be helpful for the researcher to gather richer amount of
information. It will be easier, faster, and more manageable to
analyze.
2 2
( ) ( )
x x
P x x
 
  
= 2.5
Thus, the standard deviation is
2
x

= 1.58
(Students’ responses)
III. EVALUATION
Directions: In a one whole sheet of paper, calculate the mean, variance, and standard deviation of the sampling
distribution of the sample means from an infinite population indicated in the given problem.
Problem:
Consider a population consisting of 1, 2, 3, 4, & 5. Suppose samples of size 2 are drawn from this population with
replacement. What is the mean, variance, and standard deviation of the sampling distribution of sample means?
Answers:
( )
x x P x
  
 = 3
2 2
( ) ( )
x x
P x x
 
  
= 1
2
x

= 1
IV. ASSIGNMENT
Directions: Research other ways of finding the mean, variance, and standard deviation of the sampling distribution of
the sample mean from an infinite population. Compare the ways that you have found out to the one that we have
employed in the class. Present it through a PowerPoint presentation which is to be submitted next week.