2. Confidence Intervals for Proportions
Chapter 19
Confidence Intervals for Proportions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

3. A Confidence Interval
Recall that the sampling distribution model of is centered at p, with standard deviation
Since we don’t know p, we can’t find the true standard deviation of the sampling distribution model, so we need to find the standard error:

4. A Confidence Interval (cont.)
By the % Rule, we know
about 68% of all samples will have ’s within 1 SE of p
about 95% of all samples will have ’s within 2 SEs of p
about 99.7% of all samples will have ’s within 3 SEs of p
We can look at this from ’s point of view…

5. A Confidence Interval (cont.)
Consider the 95% level:
There’s a 95% chance that p is no more than 2 SEs away from .
So, if we reach out 2 SEs, we are 95% sure that p will be in that interval. In other words, if we reach out 2 SEs in either direction of , we can be 95% confident that this interval contains the true proportion.
This is called a 95% confidence interval.

6. A Confidence Interval (cont.)

7. Margin of Error: Certainty vs. Precision (cont.)

8. Margin of Error: Certainty vs. Precision (cont.)
The choice of confidence level is somewhat arbitrary, but keep in mind this tension between certainty and precision when selecting your confidence level.
The most commonly chosen confidence levels are 90%, 95%, and 99% (but any percentage can be used).

9. Margin of Error: Certainty vs. Precision (cont.)
To be more confident, we wind up being less precise.
We need more values in our confidence interval to be more certain.
Because of this, every confidence interval is a balance between certainty and precision.
The tension between certainty and precision is always there.
Fortunately, in most cases we can be both sufficiently certain and sufficiently precise to make useful statements.

10. Margin of Error: Certainty vs. Precision (cont.)

11. Formula for Confidence Interval
Define terms!!

12. Critical Values (cont.)
Example: For a 90% confidence interval, the critical value is 1.645:
What about 95%? 99%?

13. Assumptions and Conditions
Randomization Condition: Were the data sampled at random or generated from a properly randomized experiment? Proper randomization can help ensure independence.
10% Condition: Is the sample size no more than 10% of the population?
Success/Failure Condition: We must expect at least 10 “successes” and at least 10 “failures.”
Plausible Independence Condition: Is there any reason to believe that the data values somehow affect each other? This condition depends on your knowledge of the situation—you can’t check it with data

14. Example 1: Local Newspaper
The Houston Chronicle polls a random sample of 330 voters, finding 144 who say they will vote “yes” on the upcoming school budget. Create a 95 % confidence interval for actual sentiment of all voters.
Plausible Independence Condition
Random Sampling Condition
10% Condition
Success Failure Condition
Interpretation!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

15. Activity – Coin Flipping
Let’s work together to create and graph confidence intervals for each of your samples.
Follow directions on activity sheet.

16. What Does “95% Confidence” Really Mean? (cont.)
The figure to the right shows that some of our confidence intervals capture the true proportion (the green horizontal line), while others do not:

17. Example 2: New Medicine
An experiment finds that 27% of 53 subjects report improvement after using a new medicine.
Create a 95% confidence interval for the actual cure rate.
Make it wider – 90% confidence interval.
What are the advantages and the disadvantages?

18. Optional – Hershey’s Kisses
Toss (gently) a Hershey’s kiss 60 times. Record the number of times that it lands base down.
Calculate your sample proportion for landing base down.
Create a 90% confidence interval for landing base down. (Make sure that you check conditions!!!)
Interpret what this confidence interval means.

19. Choosing Your Sample Size!!
In general, the sample size needed to produce a confidence interval with a given margin of error at a given confidence level is:
where z* is the critical value for your confidence level.
To be safe, round up the sample size you obtain.

20. Example 3: More Medicine
Remember the example about the experiment with a certain medicine? (Example 2) What sample size would we need in a follow-up study if we want a margin of error of 5% with 98% confidence?