Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Are you here? Slide Yes, and I’m ready to learn 2. Yes, and I need a nap 3. No

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 19 Confidence Intervals for Proportions

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exciting Statistics about ISU Students 71% of sexually active students use condoms American College Health Association n=706 Slide 1- 3

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 4 A Confidence Interval 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…

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 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.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 6 A Confidence Interval (cont.)

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A catalog sales company promises to deliver orders placed on the Internet within 3 days. Follow-up calls to a few randomly selected customers show that a 95% CI for the proportion of all orders that arrive on time is 81% ± 4% Slide Between 77% and 85% of all orders arrive on time. 2. One can be 95% confident that the true population percentage of orders place on the Innerrette that arrive within 3 days is between 77% and 85% 3. One can be 95% confident that all random samples of customers will show that 81% of orders arrive on time 4. 95% of all random samples of customers will show that between 77% and 85% of orders arrive on time.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 8 What Does “95% Confidence” Really Mean? Each confidence interval uses a sample statistic to estimate a population parameter. But, since samples vary, the statistics we use, and thus the confidence intervals we construct, vary as well.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 9 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:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A catalog sales company promises to deliver orders placed on the Internet within 3 days. Follow-up calls to a few randomly selected customers show that a 95% CI for the proportion of all orders that arrive on time is 81% ± 4% Slide Between 77% and 85% of all orders arrive on time. 2. One can be 95% confident that the true population percentage of orders place on the Innerrette that arrive within 3 days is between 77% and 85% 3. One can be 95% confident that all random samples of customers will show that 81% of orders arrive on time 4. 95% of all random samples of customers will show that between 77% and 85% of orders arrive on time.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What Does “95% Confidence” Really Mean? (cont.) Thus, we expect 95% of all 95% confidence intervals to contain the true parameter that they are estimating. Our confidence is in the process of constructing the interval, not in any one interval itself.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Certainty vs. Precision

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Certainty vs. Precision 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).

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Critical Values The ‘2’ in (our 95% confidence interval) came from the % Rule. Using a table or technology, we find that a more exact value for our 95% confidence interval is 1.96 instead of 2. We call 1.96 the critical value and denote it z*. For any confidence level, we can find the corresponding critical value.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide One-Proportion z-Interval When the conditions are met, we are ready to find the confidence interval for the population proportion, p. The confidence interval is where The critical value, z*, depends on the particular confidence level, C, that you specify.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Critical Values (cont.) Example: For a 90% confidence interval, the critical value is 1.645:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Z* 80%  z*= %  z*= %  z*= %  z*= %  z*=2.576 Slide 1- 17

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley HW – Problem 18 Often, on surveys there are two ways of asking the same question. 1) Do you believe the death penalty is fair or unfairly applied? 2) Do you believe the death penalty is unfair or fairly applied? Slide 1- 18

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley HW – Problem 18 Survey 1) n=597 2) n=597 For the second phrasing, 45% said the death penalty is fairly applied. Construct a 95% confidence interval. Slide 1- 19

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Suppose 47% of the respondents in survey #1 said the death penalty was fairly applied. Does this fall within your confidence interval? Slide Yes, it falls within my CI 2. No, it does not fall within my CI

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Margin of Error: Certainty vs. Precision We can claim, with 95% confidence, that the interval contains the true population proportion. The extent of the interval on either side of is called the margin of error (ME). In general, confidence intervals have the form estimate ± ME. The more confident we want to be, the larger our ME needs to be.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Margin of Error - Problem Suppose found that 56% of ISU student drink every weekend. We want to create a 95% confidence interval, but we also want to be as precise as possible. How many people should we sample? How large should our margin of error be? Slide 1- 22

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley How many people should we sample to get a ME of 1%? Slide , Between 1,000 and 4, Between 4,000 and 8, Between 8,000 and 16,000

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Margin of Error: Certainty vs. Precision The more confident we want to be, the larger our ME needs to be. Or To be more precise (i.e. have a smaller ME), we need a larger sample size.