Can be used in teaching Statistics and Probability

1. Finding the
Mean and
Variance of the
Sampling
Distribution of
the Sample
Means

2. Activity 1
ACTIVITY 1: Consider a population
consisting of 1, 2, 3, 4, and 5. Suppose
samples of size 2 are drawn from this
population. Describe the sampling
distribution of the sample means.

3. Activity 1
1. Compute the mean of the population.
2. Compute the variance of the
population.
𝜎2 =
𝑋
2
𝑁
-
𝑋
𝑁
2

4. Activity 1
3. Determine the number of possible
samples of size n = 2.
4. List all possible samples and their
corresponding means.
5. Construct the sampling distribution of
the sample means.

5. Activity 1
6. Compute the mean of the sampling
distribution of the sample means.
7. Compute the variance of the sampling
distribution of the sample means.

6. Activity 1
Sample Means from a Finite Population
ACTIVITY 2: Consider a population
consisting of 1, 2, 3, 4, and 5. Suppose
samples of size 3 are drawn from this
population. Describe the sampling
distribution of the sample means.

7. Activity 1
ACTIVITY 1 ACTIVITY 2
Population
(N = 5)
Sampling
Distribution
of the
Sample
Means
(n = 2)
Population
(N = 5)
Sampling
Distribution
of the
Sample
Means
(n = 3)
Mean 3.00 3.00 3.00 3.00
Variance 2.00 0.75 2.00 0.33
Standard
Deviation
1.41 0.87 1.41 0.57

8. Activity 1
Properties of the Sampling
Distribution of Sample Means
•Mean of the sampling
distribution of the sample means
is equal to the population mean.

9. Activity 1
Variance: 𝝈𝑿
𝟐
=
𝝈𝟐
𝒏
.
𝑵 −𝒏
𝑵 −𝟏
; for finite
population
Variance: 𝝈𝑿
𝟐
=
𝝈𝟐
𝒏
; for infinite
population

10. Activity 1
Finite population – consists of a finite
or fixed number of elements,
measurements, or observations.
Infinite Population – contains,
hypothetically at least, infinitely
elements.

11. Activity 1
Finite population correction factor
𝑵 − 𝒏
𝑵 − 𝟏
Standard error of the mean = standard
deviation of the sampling distribution of
the sample means

12. Activity 1
Standard error of the mean = measures
the degree of accuracy of the sample
mean as an estimate of the population
mean.

13. Activity 1
The Central Limit Theorem
If random samples of size n are drawn
from a population, then as n becomes
larger, the sampling distribution of the
mean approaches the normal distribution,
regardless of the shape of the population
distribution.

14. Activity 1
Describing the Sampling Distribution
of the Sample Means from an Infinite
Population

15. Activity 1
1. A population has a mean of 60 and a
standard deviation of 5. A random sample
of 16 measurements is drawn from this
population. Describe the sampling
distribution of the sample means by
computing its mean and standard
deviation.

16. Activity 1
2. The heights of male college students are
normally distributed with a mean of 68 inches
and a standard deviation of 3 inches. If 80
samples consisting of 25 students each are
drawn from the population, what would be the
expected mean and SD of the resulting
sampling distribution of means?