This document discusses sampling distributions of sample means. It contains two activities that ask the reader to describe the sampling distribution when taking samples of different sizes from a population of numbers from 1 to 5. The document explains that the mean of the sampling distribution is equal to the population mean, and provides the formula for calculating the variance of the sampling distribution. It also defines key terms like finite population, infinite population, and standard error of the mean.

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?