1. Sampling Distribution of a Sample Mean
Lecture 28
Section 8.4
Tue, Oct 16, 2007

2. Sampling Distribution of the Sample Mean

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. Experiment – Tenure of Senators
State
Years AL 10 CT 26 IL 30 MD 20 NE 6 NC 2 SC 4 VA 28 DE 8 MA 45 ND 15 WA AK 39 34 IA 22 NV SD 14 5 FL MI OH WV 48 AZ 12 KS NH TN GA 11 MN OK 13 WI 18 AR KY NJ 1 TX HI 7 MS OR WY CA 44 LA NM 24 UT ID 16 MO PA CO ME NY VT IN MT RI 32 29

5. Experiment – Tenure of Senators
State
Years 1 10 14 26 27 30 40 20 53 6 66 2 79 4 92 28 15 8 41 45 54 67 80 93 3 39 16 34 29 22 42 55 68 81 94 5 17 43 56 69 82 95 48 12 18 31 44 57 70 83 96 19 32 11 58 71 13 84 97 7 33 46 59 72 85 98 21 47 60 73 86 99 9 35 61 24 74 87
100 23 36 49 62 75 88 37 50 63 76 89 25 38 51 64 77 90 52 65 78 91

6. 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.

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

8. 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

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

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

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

12. 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

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

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

15. 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

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

17. 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

18. 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

19. The Sampling Distribution (n = 4)
The sampling distribution ofx is
The parameters are
 = 12
2 = 6 x
P(x) 6
1/81
7.5
4/18 9
10/81
10.5
16/81 12
19/81
13.5 15
16.5
4/81 18

20. The Sampling Distribution (n = 4)
The shape of the distribution:
density
20/81
16/81
12/81
8/81
4/18
mean 6 8 10 12 14 16 18

21. The Sampling Distribution (n = 4)
The shape of the distribution:
density
Normal curve
20/81
16/81
12/81
8/81
4/18
mean 6 8 10 12 14 16 18

22. The Sampling Distribution (n = 5)
The sampling distribution ofx is
The parameters are
 = 12
2 = 4.8 x
P(x) 6
1/243
0.004
7.2
5/243
0.021
8.4
15/243
0.062
9.6
30/243
0.123
10.8
45/243
0.185 12
51/243
0.210
13.2
14.4
15.6
16.8 18

23. The Sampling Distribution (n = 5)
The shape of the distribution:
density
50/243
40/243
30/243
20/243
10/243
mean 6 8 10 12 14 16 18

24. The Sampling Distribution (n = 5)
The shape of the distribution:
density
Normal curve
50/243
40/243
30/243
20/243
10/243
mean 6 8 10 12 14 16 18

25. Observations and Conclusions
Observation: The values ofx are clustered around .
Conclusion:x is likely to be close to .

26. Observations and Conclusions
Observation: As the sample size increases, the clustering is tighter.
Conclusion: Larger samples give more reliable estimates.
Conclusion: We can make our estimates as good as we like if we use large enough samples.

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

28. 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.

29. 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

30. 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 original distribution to the normal distribution.

31. The Central Limit Theorem
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.

32. The Central Limit Theorem
Also, 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.

33. Bag A vs. Bag B There are two bags, Bag A and Bag B.
Each bag contains 20,000 vouchers with values from $10 to $60.
Their distributions are shown on the following slide.

34. Bag A vs. Bag B Bag A Bag B = 1000 vouchers 10 20 30 40 50 60

35. Bag A vs. Bag B
Use the TI-83 to compute the mean and standard of each population (Bag A and Bag B).

36. Bag A vs. Bag B The Bag A population: The Bag B population:  = 23.5
 = 14.24
The Bag B population:
 = 46.5

37. Bag A vs. Bag B The hypotheses:
H0: The bag is Bag A.
H1: The bag is Bag B.
Suppose that we sample 100 vouchers (one at a time, with replacement).
Decision rule: Reject H0 if the average of the 100 vouchers is more than 35.

38. Bag A vs. Bag B Find the sampling distribution ofx if H0 is true.

39. The Two Sampling Distributions15 20 25 30 35 40 45 50 55 H1 15 20 25 30 35 40 45 50 55

40. The Two Sampling Distributions15 20 25 30 35 40 45 50 55
N(46.5, 1.424) H1 15 20 25 30 35 40 45 50 55

41. The Two Sampling Distributions15 20 25 30 35 40 45 50 55 H1 15 20 25 30 35 40 45 50 55

42. Bag A vs. Bag B
What is ?
What is ?
How reliable is this test?

43. Example
Suppose a brand of light bulb has a mean life of 750 hours with a standard deviation of 120 hours.
What is the probability that 36 of these light bulbs would last a total of at least hours?