3. Sample size and the Central Limit Theorem

You are a senior in college writing your sociology honors thesis. You are having trouble finding time to exercise. You wonder whether your peers are doing a better job of scheduling it in, so you decide to take a poll to see how many hours the students in your honors seminar exercise each week. The following table lists the actual hours of exercise done weekly by each of the 12 students in your seminar. The average of this population of numbers is = 6.

Weekly Hours of Exercise

Student	Hours of Exercise	Sample 1	Sample 2	Sample 3	Sample 4
1	5				5
2	8			8	8
3	6	6			6
4	10	10		10
5	8	8		8	8
6	4		4	4
7	12	12			12
8	1		1	1
9	1
10	7		7	7	7
11	6			6	6
12	4		4	4	4

1. Suppose you decide to take a small sample of 4 of your 12 peers. In the column of the foregoing table labeled "Sample 1" are the values for hours of exercise done weekly by each of the 4 students in this sample.

You can calculate the average of a sample of numbers by adding together the different values collected and dividing them by the number of values (in this case, 4). Feel free to use a calculator. What is the average of this first sample of 4 students?

3

4

10

9

2. You decide to take a second sample of 4 of your 12 peers. In the column of the table labeled "Sample 2" are the values for this second sample of 4 students' hours of exercise done weekly. What is the average of the second sample of 4 students?

9

10

4

3

3. Now, suppose you decide on a larger sample of 8 of the 12 students in your seminar. In the column of the table labeled "Sample 3" are the values for hours of exercise done weekly by each of the 8 students in the sample. What is the average of this first sample of 8 students?

4

6

5

7

4. You decide to take a second sample of 8. In the column of the table labeled "Sample 4" are the values for this second sample of 8 students' hours of exercise done weekly. What is the average of this second sample of 8 students?

7

6

5

4

5. The averages of the samples provide statistics that can be used to estimate the mean of the population. The averages from the larger samples are accurate estimates of the population average than the smaller samples. The size of the standard error should be for the larger sample sizes than for the smaller sample sizes.

These findings are related to the Central Limit Theorem. Even if the variable that is sampled normally distributed throughout the population, as long as the sample size is , you can assume that the sampling distribution is normal in shape with a mean equal to the mean and a standard deviation equal to .