1. Sampling Distribution of a Sample Mean
Lecture 28
Section 8.4
Tue, Oct 31, 2006

2. Sampling Distribution of the Sample Mean
Sampling Distribution of the Sample Mean– The distribution of sample means over all possible samples of the size n from the population.

3. With or Without Replacement?
If the sample size is small in relation to the population size (< 5%), then it does not matter whether we sample with or without replacement.
The calculations are simpler if we sample with replacement.
In any case, we are not going to worry about it.

4. Example Suppose a population consists of the numbers {6, 12, 18}.
Using samples of size n = 1, 2, or 3, find the sampling distribution ofx.
Draw a tree diagram showing all possibilities.

5. The Tree Diagram (n = 1)
n = 1 6
mean = 6 12
mean = 12 18
mean = 18

6. The Sampling Distribution (n = 1)
The sampling distribution ofx is
The parameters are
 = 12
2 = 24 x
P(x) 6
1/3 12 18

7. The Sampling Distribution (n = 3)
The shape of the distribution:
density
1/3
mean 6 8 10 12 14 16 18

8. The Tree Diagram (n = 2) 6 12 18 mean 6 6 12 9 12 18 6 9 12 12 15 18 6

9. The Sampling Distribution (n = 2)
The sampling distribution ofx is
The parameters are
 = 12
2 = 12 x
P( x) 6
1/9 9
2/9 12
3/9 15 18

10. The Sampling Distribution (n = 3)
The shape of the distribution:
density
3/9
2/9
1/9
mean 6 8 10 12 14 16 18

11. The Tree Diagram (n = 3) 6 12 18 6 12 18 6 12 18 6 12 18 mean 6 6 12 810 6 12 8 6 12 10 18 12 6 10 18 12 12 18 14 6 6 8 12 10 18 12 6 12 12 10 12 12 18 14 6 12 18 12 14 18 16 6 6 10 12 12 18 14 18 6 12 12 12 14 18 16 6 14 18 12 16 18 18

12. The Sampling Distribution (n = 3)
The sampling distribution ofx is
The parameters are
 = 2
2 = 8 x
P(x) 6
1/27 8
3/27 10
6/27 12
7/27 14 16 18

13. The Sampling Distribution (n = 3)
The shape of the distribution:
density
9/27
6/27
3/27
mean 6 8 10 12 14 16 18

14. Sampling Distributions
Run the program
Central Limit Theorem for Means.exe.
Use n = 30 and population = {1, 2, 3}
Generate 100 samples.

15. 100 Samples of Size n = 30
 = 0.75
 = 0.079

16. Observations and Conclusions
Observation #1: The values ofx are clustered around .
Conclusion #1:x is probably close to .

17. Central Limit Theorem for Means.exe.
Larger Sample Size
Now we will select samples of size 30 instead of only 100 samples.
Run the program
Central Limit Theorem for Means.exe.
Pay attention to the shape of the distribution.

18. 10,000 Samples of Size n = 30
 = 0.75
 =

19. 10,000 Samples of Size n = 30

20. More Observations and Conclusions
Observation #2: The distribution ofx appears to be approximately normal.
Conclusion #2: We can use the normal distribution to calculate just how close to  we can expectx to be.

21. Central Limit Theorem for Means.exe.
Larger Sample Size
Now we will select samples of size 200 instead of size 30.
Run the program
Central Limit Theorem for Means.exe.
Pay attention to the spread (standard deviation) of the distribution.

22. 10,000 Samples of Size n = 200
 = 0.75
 =

23. Observations and Conclusions
Observation #3: As the sample size increases, the clustering is tighter.
Conclusion #3-1: Larger samples give more reliable estimates.
Conclusion #3-2: For sample sizes that are large enough, we can make very good estimates of the value of .

24. One More Observation
However, we must know the values of  and  for the distribution ofx.
That is, we have to quantify the sampling distribution ofx.

25. The Central Limit Theorem
Begin with a population that has mean  and standard deviation .
For sample size n, the sampling distribution of the sample mean is approximately normal with

26. The Central Limit Theorem
The approximation gets better and better as the sample size gets larger and larger.
That is, the sampling distribution “morphs” from the distribution of the original population to the normal distribution.
For many populations, the distribution is almost exactly normal when n  10.
For almost all populations, if n  30, then the distribution is almost exactly normal.

27. The Central Limit Theorem
Therefore, if the original population is exactly normal, then the sampling distribution of the sample mean is exactly normal for any sample size.
This is all summarized on pages 536 – 537.

28. Lottery Example
Let’s consider a simple lottery game with the following pdf for payoffs: x
P(X = x)
1000
0.0001
100
0.0010 10
0.0100 1
0.1000
0.8889

29. Lottery Example Then we find that Assume that tickets sell for $1.00.
 = 0.40.
 =
Assume that tickets sell for $1.00.
If the state sells 1,000,000 lottery tickets, what is the probability that they will make money on this game?
How much money are they likely to make?