1. Sampling and Sampling Distributions
Chapter 8.1 – 8.3

2. Review of Terminology
The population is the entire collection of all individuals or objects of interest
The sample is the portion of the population that is selected for study.
Inferential Statistics is the process of using sample information to draw inferences or conclusions about the population.
population
Sample

3. Sampling Distribution Activity
Consider a population of 5 commuters who are all neighbors, living in a cul-de-sac.
What is N?
5, the size of the population
Each commuter was asked how many miles he/she commutes to work each day.
Find the population mean of the set of data:
µ = 68
Find the population standard deviation:
σ=30.28 C1 C2 C3 C4 C5 50 84 38
120 48

4. Sampling Distribution Activity
Suppose it is very difficult to catch all five commuters at the same time to get the information needed for the study.
Therefore, let us look at a sample of two commuters (n=2), and find the mean to estimate µ.
The first sample we can look at includes C1 and C2.
What is the mean of this sample? C1 C2 C3 C4 C5 50 84 38
120 48

5. Sampling Distribution Activity
If we list all the samples of size n=2, there will be 10 possible samples.
For all 10 possible samples and calculate the mean, , for each sample.
Commuters
Data Values
C1, C2
50, 84
C1, C3
50, 38
C1, C4
50, 120
C1, C5
50, 48
C2, C3
84, 38
C2, C4
84, 120
C2, C5
84, 48
C3, C4
38, 120
C3, C5
38, 48
C4, C5
120, 48

6. Sampling Distribution Activity
The data set of all the sample means in column 3 is called a sampling distribution of the means.
Commuters
Data Values
Mean,
C1, C2
50, 84 67
C1, C3
50, 38 44
C1, C4
50, 120 85
C1, C5
50, 48 49
C2, C3
84, 38 61
C2, C4
84, 120
102
C2, C5
84, 48 66
C3, C4
38, 120 79
C3, C5
38, 48 43
C4, C5
120, 48 84

7. Are all sample means created equal?
There are some sample means that are closer to the true population mean than others.
For example, the sample mean that is closest to the population mean µ = 68 is 67
And the sample mean that is furthest from the population mean µ = 68 is
102
The sampling error is the difference between the value of a sample mean, , and the population mean, µ.
Symbolically,
Sampling error of the mean = µ

8. Calculating the Sample Error
For the sample means 67 and 102, calculate the sample error.
67:
Sample error = 67 – 68
= -1
102:
Sample error = 102 – 68
= 34

9. Sampling Distribution Activity
Look back at column 3 of our table, the sampling distribution of the means.
Calculate the mean of this data set – we are calculating the mean of the sampling distribution of the means:
Mean of the sampling distribution of the means is 68.
The same as the population mean … AMAZING!
The sampling distribution of the means has special notation:
“mu sub x-bar”
always equals the population mean,
Mean, 67 44 85 49 61
102 66 79 43 84

10. Sampling Distribution Activity
From column 3, the sampling distribution of the means:
Calculate the standard deviation of this data set – we are calculating the standard deviation of the sampling distribution of the means:
18.54
The standard deviation of the sampling distribution of the means has special notation:
“sigma sub x-bar”
The standard deviation is a measure of how spread out the sample means are from µ.
Mean, 67 44 85 49 61
102 66 79 43 84

11. The Mean and Standard Deviation of the Sampling Distribution of the Mean
The standard deviation of the sampling distribution of the means , is also known as the standard error.
will always be smaller than the population standard deviation,
Most times, the sampling distribution of the means will have more than 10 values (as in our first example) and it becomes impossible to put the data in a calculator to find the standard error.
We use a formula to find the standard error in this case.

12. The Mean and Standard Deviation of the Sampling Distribution of the Mean
If we know the population standard deviation, σ, and the sample size, n, we can use the following formula to find the standard deviation of the sampling distribution of the mean or the standard error of the mean
The larger the standard error of the mean is, the more dispersed the samples means are from the population mean.
The smaller the standard error, the closer the sample means are to the population mean.

13. Interpretation of the Standard Error of the Mean
There are some sample means that are closer to the true population mean than others.
Recall that different sample means have different sampling errors.
The distance from µ=68 is the sampling error for each sample mean.
The average sampling error takes into consideration all sampling errors.
This is what we call the standard error.
It is a measure of how much the sample means deviate from the population mean for the sampling distribution. 43 44 49 61 66 67 79 85
102 84
µ=68

14. The Mean and Standard Deviation of the Sampling Distribution of the Mean
Let’s see if the formula for standard error will produce the same standard error that we got when we put the sampling distribution in the calculator (18.54).
Remember, the population standard deviation for the five commuters was σ=30.28
And sample size for commuters, n = 2.
Why is there a discrepancy in our answers?

15. Example 8.3
According to a study of TV viewing habits, the average number of hours a teenager watches MTV per week is µ=19 hours with a standard deviation of σ=3.8 hours. If a sample of 64 teenagers is randomly selected from the population, then determine the mean and standard error of the mean of the sampling distribution of the mean.
The mean of the sampling distribution of the mean is always equal to the population mean:
So, hours.
The standard error of the mean of the sampling distribution of the mean is found with the formula:
hours

16. Example 8.4
The registrar at a large University states that the mean grade point average of all the students is µ=2.95 with a population standard deviation of σ=0.20.
(a) Determine the mean, , and standard error, , of the sampling distribution if the sampling distribution mean consists of all possible sample means from samples of size 25.
(b) Determine the mean, , and standard error, , of the sampling distribution if the sampling distribution mean consists of all possible sample means from samples of size 100.
(c) What effect did increasing the sample size have on the mean and standard error of the sampling distribution?
In which sampling distribution would you have a better chance of selecting a sample mean which is closer to µ=2.95? Why?

17. Example 8.4
µ=2.95 of σ=0.20.
(a) Determine the mean, , and standard error, , of the sampling distribution if the sampling distribution mean consists of all possible sample means from samples of size 25.
Since , the mean of the sampling distribution
The standard error of the sampling distribution

18. Example 8.4
µ=2.95 of σ=0.20.
(b) Determine the mean, , and standard error, , of the sampling distribution if the sampling distribution mean consists of all possible sample means from samples of size 100.
Since , the mean of the sampling distribution
This is the same value regardless of the sample size.
The standard error of the sampling distribution
Note that the standard error of the mean is affected by the sample size.

19. Example 8.4
(c) What effect did increasing the sample size have on the mean and standard error of the sampling distribution?
Increasing the sample size had no effect on the mean of the sampling distribution,
Increasing the sample size from 25 to 100 decreased the standard error from 0.04 to 0.02.
In general, as the sample size increases, the standard error of the sampling distribution will decrease.

20. Example 8.4
(d) In which sampling distribution would you have a better chance of selecting a sample mean which is closer to µ=2.95? Why?
Remember the significance of a small standard deviation vs. a large standard deviation…
A smaller standard deviation indicates that the data values of the distribution are more closely clustered about the mean.
A larger standard deviation indicates that the data values of the distribution are more dispersed (spread out) from the mean.
So in which case would you have a better chance of selecting a sample mean closer to µ=2.95?
The sampling distribution of the mean for all possible samples of size 100 would give a better chance because it has the smaller standard error.

21. Section Exercises # 18
Suppose a random sample of size n=64 is selected from a population with mean, µ, and standard deviation, σ.
For each of the following values of µ andσ, find the value of µx andσx.
(a) µ=20, σ=2
(b) µ=90, σ=5
(c) µ=150, σ=10
(d) µ=200, σ=50

22. Section Exercises # 18 (a) µ=20, σ=2 (b) µ=90, σ=5 (c) µ=150, σ=10
(d) µ=200, σ=50