1. Inference for Proportions

2. Preface (text notes)

3. Preface (text notes)

4. Categorical Data
Up to this point we have only talked about quantitative variables.
Many studies are about variables such as race, sex, occupation, make of a car, smoking or non- smoking, type of complaint received, etc.
Under these types of variables our data are counts or percents obtained from counts
How do we numerically summarize numerical data?

5. Big Picture p Population Parameter? Population Inference Sample
Sample Statistic

6. Cold outside? Do you think the temperature is “low” outside?
There are two possible answers to this question Yes or No (at least by our definition).
Pretend that out of a sample of 20 people in our class, 15 say yes.
So the proportion of people who walk outside and curse, “It’s cold out here” is
15 = 0.75 or 75% 20

7. Cold!
Lets assume that Stat 226 section D is a good representation of the ISU population
We are interested in knowing the opinion of the entire ISU student body
This is a new parameter that is denoted by P
Thus P represents the population proportion
Recall that we use a statistic to estimate the population parameter of interest
In this case = 0.75 is a statistic or simply is the sample statistic.

8. Sample Proportion In general:
A sample proportion from an SRS of size n is the number of “successes” over the total sample size

9. Sampling Distribution of
Recall that a sampling distribution of a statistic is the distribution or shape center and spread of that are obtained from all possible samples of the same size.

10. Sampling Distribution of
As the sample size increases, the sampling distribution of becomes approximately normal with
Mean: p (Thus is an unbiased estimator)
Standard Deviation
Where p is the true population proportion
This is denoted as

11. Standard Error
Since we don’t know what p is we can use in the standard deviation to get the standard error.
So we have

12. One Issue
Unfortunately in practice it has been shown that using the estimate to construct confidence intervals can be inaccurate
For example lets suppose that not one person noticed that the temperature outside is low (all wearing shorts?).
Then and which means that we are certain that no one thinks the temperature is low
We need to find an estimator that is still unbiased but doesn’t have this characteristic

13. Wilson Estimate Wilson estimate of p (the population proportion) is
We are essentially adding four observations and two of them are considered successes (and two failures). The Wilson estimate always keeps us away from 0 and 1 and works well in practice.
So the Wilson estimate for the proportion of people for whom the temperature outside is low:

14. Confidence Interval for p
So a Confidence Interval for p based on the Wilson estimate is
Where the standard error is
And z* is the value of the standard Normal density curve with area C between -z* and z*
The margin of error is
Same old method in a different hat (or tilda)
estimate  margin of error
Caution: Use this interval when sample size is at least n=5 and for confidence level 90%, 95%, or 99%

15. Cold! (again)
A 95% confidence interval for the proportion of ISU students who think the temperature outside is low:
So the 95% CI is (0.526, 0.889)
Interpretation:
I am 95% confident that between 52.6% and 88.9% ISU students’ first thought when they head outside is the temperature outside is low

16. Example cont. What does “95% confident” tell us?
This interval is fairly wide. Hence, the interval does not give enough information about what p is. How can we make the interval narrower?

17. Determine a Sample Size
Can we find the sample size needed to get a confidence interval for p that has a pre- determined a level of confidence and margin of error? Yes, we can!
Just solve for n in the margin of error equation

18. Determine a Sample Size
Recall:
Solving for n we get
But there is an issue!
We don’t know until after data is collected.

19. Determine a Sample Size
There are two solutions to the problem
Use a guesstimate obtained from previous studies
Use this estimate is conservative and will give the desired ME and confidence level always, but if then sample size will be larger than needed and more money will be spent than necessary.
So if you are not given a guesstimate, revert to

20. Example
Is there interest in a new product? One of your employees has suggested that your company develop a new product. You decide to take a random sample of your customers and ask whether or not there is interest in the new product. The response is on a 1 to 5 scale, with 1 indicating “definitely would not purchase”; 2, “probably would not purchase”; 3, “not sure”; 4, “probably would purchase”; 5, “definitely would purchase.” For an initial analysis, you will record the responses 1,2,and 3 as “no” and 4 and 5 as “Yes”. What sample size would you use if you wanted a 95% margin of error to be 0.1 or less?

21. Solution
We want ME (or m) = 0.1 and our level of confidence to be 95% so z*=1.96. And since three responses are no we will use
So:
And n=89 (e.g., rounded up to 93 then subtract 4 = 89)

22. Summary

23. Summary

24. Summary