1. Sampling Distribution of a Sample Proportion
Lecture 28
Sections 8.1 – 8.2
Mon, Oct 15, 2007

2. Sampling Distributions
Sampling Distribution of a Statistic

3. The Sample Proportion
The letter p represents the population proportion.
The symbol p^ (“p-hat”) represents the sample proportion.
p^ is a random variable.
The sampling distribution of p^ is the probability distribution of all the possible values of p^.

4. Example
Suppose that 2/3 of all males wash their hands after using a public restroom.
Suppose that we take a sample of 1 male.
Find the sampling distribution of p^.

5. Example W N
2/3
1/3
P(W) = 2/3
P(N) = 1/3

6. Example Let x be the sample number of males who wash.
The probability distribution of x is x
P(x)
1/3 1
2/3

7. Example Let p^ be the sample proportion of males who wash. (p^ = x/n.)
The sampling distribution of p^ is p^
P(p^)
1/3 1
2/3

8. Example Now we take a sample of 2 males, sampling with replacement.
Find the sampling distribution of p^.

9. Example W N
2/3
1/3
P(WW) = 4/9
P(WN) = 2/9
P(NW) = 2/9
P(NN) = 1/9

10. Example Let x be the sample number of males who wash.
The probability distribution of x is x
P(x)
1/9 1
4/9 2

11. Example Let p^ be the sample proportion of males who wash. (p^ = x/n.)
The sampling distribution of p^ is p^
P(p^)
1/9
1/2
4/9 1

12. Samples of Size n = 3
If we sample 3 males, then the sample proportion of males who wash has the following distribution. p^
P(p^)
1/27 = .03
1/3
6/27 = .22
2/3
12/27 = .44 1
8/27 = .30

13. Samples of Size n = 4
If we sample 4 males, then the sample proportion of males who wash has the following distribution. p^
P(p^)
1/81 = .01
1/4
8/81 = .10
2/4
24/81 = .30
3/4
32/81 = .40 1
16/81 = .20

14. Samples of Size n = 5
If we sample 5 males, then the sample proportion of males who wash has the following distribution. p^
P(p^)
1/243 = .004
1/5
10/243 = .041
2/5
40/243 = .165
3/5
80/243 = .329
4/5 1
32/243 = .132

15. Our Experiment In our experiment, we had 60 samples of size 5.
Based on the sampling distribution when n = 5, we would expect the following
Value of p^
0.0
0.2
0.4
0.6
0.8
1.0
Actual 1 3 7 18 21 10
Predicted
2.5
9.9
19.8
7.9

16. The pdf when n = 1 1

17. The pdf when n = 2
1/2 1

18. The pdf when n = 3
1/3
2/3 1

19. The pdf when n = 4
1/4
2/4
3/4 1

20. The pdf when n = 5
1/5
2/5
3/5
4/5 1

21. The pdf when n = 10
2/10
4/10
6/10
8/10 1

22. Observations and Conclusions
Observation: The values of p^ are clustered around p.
Conclusion: p^ is close to p most of the time.

23. Observations and Conclusions
Observation: As the sample size increases, the clustering becomes tighter.
Conclusion: Larger samples give better estimates.
Conclusion: We can make the estimates of p as good as we want, provided we make the sample size large enough.

24. Observations and Conclusions
Observation: The distribution of p^ appears to be approximately normal.
Conclusion: We can use the normal distribution to calculate just how close to p we can expect p^ to be.

25. One More Observation
However, we must know the values of  and  for the distribution of p^.
That is, we have to quantify the sampling distribution of p^.

26. The Central Limit Theorem for Proportions
It turns out that the sampling distribution of p^ is approximately normal with the following parameters.

27. The Central Limit Theorem for Proportions
The approximation to the normal distribution is excellent if

28. Example
If we gather a sample of 100 males, how likely is it that between 60 and 70 of them, inclusive, wash their hands after using a public restroom?
This is the same as asking the likelihood that 0.60  p^  0.70.

29. Example Use p = 0.66. Check that
np = 100(0.66) = 66 > 5,
n(1 – p) = 100(0.34) = 34 > 5.
Then p^ has a normal distribution with

30. Example So P(0.60  p^  0.70) = normalcdf(.60,.70,.66,.04737)=

31. Why Surveys Work
Suppose that we are trying to estimate the proportion of the male population who wash their hands after using a public restroom.
Suppose the true proportion is 66%.
If we survey a random sample of 1000 people, how likely is it that our error will be no greater than 5%?

32. Why Surveys Work
Now we have

33. Why Surveys Work
Now find the probability that p^ is between 0.61 and 0.71:
normalcdf(.61, .71, .66, ) =
It is virtually certain that our estimate will be within 5% of 66%.

34. Why Surveys Work
What if we had decided to save money and surveyed only 100 people?
If it is important to be within 5% of the correct value, is it worth it to survey 1000 people instead of only 100 people?

35. Quality Control
A company will accept a shipment of components if there is no strong evidence that more than 5% of them are defective.
H0: 5% of the parts are defective.
H1: More than 5% of the parts are defective.

36. Quality Control
They will take a random sample of 100 parts and test them. If no more than 10 of them are defective, they will accept the shipment.
What is ?
What is ?