1. Sampling Distribution of a Sample Proportion
Lecture 24
Sections 8.1 – 8.2
Wed, Feb 27, 2008

2. Preview of the Central Limit Theorem
We looked at the distribution of the average of 1, 2, and 3 uniform random variables U(0, 1).
We saw that the shapes of their distributions was moving towards the shape of the normal distribution.

3. Preview of the Central Limit Theorem2 1 1

4. Preview of the Central Limit Theorem2 1 1

5. Preview of the Central Limit Theorem2 1 1

6. Preview of the Central Limit Theorem
Some observations:
Each distribution is centered at the same place, ½.
The distributions are being “drawn in” towards the center.
That means that their standard deviation is decreasing.
Can we quantify this?

7. Preview of the Central Limit Theorem2
= ½
2 = 1/12 1 1

8. Preview of the Central Limit Theorem2
= ½
2 = 1/24 1 1

9. Preview of the Central Limit Theorem2
= ½
2 = 1/36 1 1

10. Preview of the Central Limit Theorem2
Area = 0.200 1 1

11. Preview of the Central Limit Theorem2
Area = 0.36 1 1

12. Preview of the Central Limit Theorem2
Area = 0.432 1 1

13. Preview of the Central Limit Theorem
What would the area be for the average of 12 values?
X12 is N(0.5, 1/12).
Can we figure out what it would be for the average of 1000 values?

14. Preview of the Central Limit Theorem
This tells us that a mean based on three observations is much more likely to be close to the population mean than is a mean based on only one or two observations.

15. Parameters and Statistics
THE PURPOSE OF A SAMPLE STATISTIC IS TO ESTIMATE A POPULATION PARAMETER.
A sample mean is used to estimate the population mean.
A sample proportion is used to estimate the population proportion.

16. Parameters and Statistics
Sample statistics are variable.
Population parameters are fixed.
In fact, a statistic is a random variable.
Therefore, it has a probability distribution.

17. Some Questions
We hope that the sample proportion is close to the population proportion.
How close can we expect it to be?
Would it be worth it to collect a larger sample?
If the sample were larger, we would expect the sample proportion to be closer to the population proportion.
How much closer? And how much larger?

18. Experiment
A recent study showed that males wash their hands about 66% of the time after using the restroom.

19. Experiment
Person
Washes 1
Yes 11 No 21 31 41 2 12 22 32 42 3 13 23 33 43 4 14 24 34 44 5 15 25 35 45 6 16 26 36 46 7 17 27 37 47 8 18 28 38 48 9 19 29 39 49 10 20 30 40 50