1. Sampling Distribution Models
Population – all items
of interest.
Population
Parameter: p
Inference
We are ready to start putting together the pieces to come up with how to use sample statistics to make inferences, educated guesses, about population parameters. Remember our picture of the population and sample. We will not complete the process by introducing the idea of using a sample statistics as an estimate of the population parameter. But before that we need to know a little more about how sample statistics based on random samples from a population behave.
Sample – a
few items from
the population.
Sample
Statistic:
Random
selection

2. Sampling Distribution of
Shape: Approximately Normal
Center: The mean is p.
Spread: The standard deviation is

3. Sampling Distribution of
Conditions:
10% Condition: The size of the sample should be less than 10% of the size of the population.
Success/Failure Condition: np and n(1 – p) should both be greater than 10.

4. 68 – 95 – 99.7 Rule

5. Probability
If the population proportion, p, is known, we can find the probability or chance that takes on certain values using a normal model.

6. Inference
In practice the population parameter, p, is not known and we would like to use a sample to tell us something about p.
Use the sample proportion, , to make inferences about the population proportion p.

7. Example Population: All adults in the U.S.
Parameter: Proportion of all adults in the U.S. who think that abortion should be legal. Unknown!

8. Example
Sample: 1,772 randomly selected registered voters nationwide. Quinnipiac University Poll, Jan. 30 – Feb. 4, 2013.
Statistic: 992 of the 1,772 registered voters in the sample (56%) answered that abortion should be legal.

9. Rule
95% of the time the sample proportion, , will be between

10. Rule
95% of the time the sample proportion, , will be within
two standard deviations of p.

11. Standard Deviation
Because p, the population proportion is not known, the standard deviation
is also unknown.

12. Standard Error Substitute as our estimate (best guess) of p.
The standard error of is:

13. About 95% of the time the sample proportion, , will be within
two standard errors of p.

14. About 95% of the time the population proportion, p, will be within
two standard errors of .

15. Confidence Interval for p
We are 95% confident that p will fall between

16. Example

17. Interpretation
We are 95% confident that the population proportion of all adults in the U.S. who would answer that abortion should be legal is between 53.6% and 58.4%.

18. Interpretation Plausible values for the population parameter p.
95% confidence in the process that produced this interval.

19. 95% Confidence
If one were to repeatedly sample at random 1,772 adults and compute a 95% confidence interval for each sample, 95% of the intervals produced would contain, or capture, the population proportion p.

20. Simulation

22. Margin of Error Is called the Margin of Error (ME).
This is the furthest can be from p, with 95% confidence.

23. Margin of Error
What if we want to be 99.7% confident?

24. Margin of Error Confidence z* 80% 90% 95% 98% 99% 1.282 1.645
2.326
2.576

25. Another Example
Pew Research Center/USA Today Poll, Feb. 13 – 18, Asked of 1,504 randomly selected adults nationwide.
“Do you favor or oppose setting stricter emission limits on power plants in order to address climate change?”

26. Another Example n=1,504 randomly selected adults. Favor Oppose
Unsure/ Refused
52%
28%
10%

27. Another Example
90% confidence interval for p, the proportion of the population of all adults in the U.S. who favor emission limits on power plants in order to address climate change.

28. Calculation

29. What Sample Size? Conservative Formula
The sample size to be 95% confident that , the sample proportion, will be within ME of the population proportion, p.

30. Example
Suppose we want to be 95% confident that our sample proportion will be within 0.02 of the population proportion.