2. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 19 Confidence Intervals for Proportions

3. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 3 Standard Error Both of the sampling distributions we’ve looked at are Normal. For proportions For means

4. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 4 Standard Error (cont.) When we don’t know p or σ, we’re stuck, right? Nope. We will use sample statistics to estimate these population parameters. Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

5. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 5 Standard Error (cont.) For a sample proportion, the standard error is For the sample mean, the standard error is

6. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 6 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:

7. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 7 A Confidence Interval (cont.) By the 68-95-99.7% 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…

8. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 8 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.

9. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 9 A Confidence Interval (cont.)

10. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Just Checking… A Pew Research study regarding cell phones asked questions about cell phone experience. One growing concern is unsolicited advertising in the form of text messages. Pew asked cell phone owners, “Have you ever received unsolicited text messages on your cell phone from advertisers?” and 17% reported that they had. Pew estimates a 95% confidence interval to be or between 13% and 21%. Are the following statements about people who have cell phones correct? Explain. 1. In Pew’s sample, somewhere between 13% and 21% of respondents reported that they had received unsolicited advertising text messages. 2. We can be 95% confident that 17% of U.S. cell phone owners have received unsolicited advertising text messages. 3. We are 95% confident that between 13% and 21% of all U.S. cell phone owners have received unsolicited advertising text messages. 4. We know that between 13% and 21% of all U.S. cell phone owners have received unsolicited advertising text messages. 5. 95% of all U.S. cell phone owners have received unsolicited advertising text messages. Slide 19 - 10

11. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 11 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.

12. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 12 What Does “95% Confidence” Really Mean? (cont.) The figure to the right shows that some of our confidence intervals (from 20 random samples) capture the true proportion (the green horizontal line), while others do not:

13. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 13 What Does “95% Confidence” Really Mean? (cont.) Our confidence is in the process of constructing the interval, not in any one interval itself. Thus, we expect 95% of all 95% confidence intervals to contain the true parameter that they are estimating.

14. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Polls and Margin of Error Example On January 30-31, 2007, Fox News/Opinion Dynamics polled 900 registered voters nationwide. When asked “Do you believe that global warming exists?” 82% said “Yes”. Fox reported their margin of error to be +/- 3%. It is the standard among pollsters to use a 95% confidence interval unless otherwise stated. Given that, what does Fox News mean by claiming a margin of error of +/- 3% in this context? Slide 19 - 14

15. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 15 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, making the interval wider.

16. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 16 Margin of Error: Certainty vs. Precision (cont.)

17. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 17 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.

18. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 18 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).

19. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Polls and Margin of Error Example On January 30-31, 2007, Fox News/Opinion Dynamics polled 900 registered voters nationwide with a margin of error of +/- 3%. It is a convention among pollsters to use a 95% confidence level and to report the “worst case” margin of error, based on p = 0.5 How did Fox calculate their margin of error? Slide 19 - 19

20. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 20 Critical Values The ‘2’ in (our 95% confidence interval) came from the 68-95-99.7% 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 (the number of SEs that corresponds to our confidence interval level).

21. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 21 Critical Values (cont.) Example: For a 90% confidence interval, the critical value is 1.645:

22. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Polls and Margin of Error Example On January 30-31, 2007, Fox News/Opinion Dynamics polled 900 registered voters and found that 82% of the respondents believed that global warming exists. Fox reported a 95% confidence interval with a margin of error of +/- 3%. Using the critical value of z and the standard error based on the observed proportion, what would the margin of error be for a 90% confidence interval? What’s good and bad about this change? Slide 19 - 22

23. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Just Checking… Think some more about the 95% confidence interval Fox News created for the proportion of registered voters who believe that global warming exists. 6. If Fox wanted to be 98% confident, would their confidence interval need to be wider or narrower? 7. Fox’s margin of error was about +/- 3%. If they reduced it to +/- 2%, would their level of confidence be higher or lower? 8. If Fox News had polled more people, would the interval’s margin of error had been smaller or larger? Slide 19 - 23

24. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 24 Assumptions and Conditions All statistical models make upon assumptions. Different models make different assumptions. If those assumptions are not true, the model might be inappropriate and our conclusions based on it may be wrong. You can never be sure that an assumption is true, but you can often decide whether an assumption is plausible by checking a related condition.

25. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 25 Assumptions and Conditions (cont.) Here are the assumptions and the corresponding conditions you must check before creating a confidence interval for a proportion: Independence Assumption: We first need to Think about whether the Independence Assumption is plausible. It’s not one you can check by looking at the data. Instead, we check two conditions to decide whether independence is reasonable.

26. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 26 Assumptions and Conditions (cont.) 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?  Sample Size Assumption: The sample needs to be large enough for us to be able to use the CLT. Success/Failure Condition: We must expect at least 10 “successes” and at least 10 “failures.”

27. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 27 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.

28. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 28 Choosing Your Sample Size The question of how large a sample to take is an important step in planning any study. Choose a Margin or Error (ME) and a Confidence Interval Level. The formula requires which we don’t have yet because we have not taken the sample. A good estimate for, which will yield the largest value for (and therefore for n) is 0.50. Solve the formula for n.

29. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Polls and Margin of Error Example The Fox News poll which estimated that 82% of all voters believed global warming exists had a margin of error of +/- 3%. Suppose an environmental group planning a follow-up survey of voters’ opinions on global warming wants to determine a 95% confidence interval with a margin of error of no more than +/- 2%. How large of a sample do they need? Use the Fox News estimate as the basis for your calculation. Slide 19 - 29

30. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Sample Size Example Again A credit card company is about to send out a mailing list to test the market for a new credit card. From that sample, they want to estimate the true proportion of people who will sign up for the card nationwide. A pilot study suggests that about 0.5% of the people receiving the offer will accept it. To be within a tenth of a percentage point (0.001) of the true rate with 95% confidence, how big does the test mailing have to be? Slide 19 - 30

31. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 31 What Can Go Wrong? Don’t Misstate What the Interval Means: Don’t suggest that the parameter varies. Don’t claim that other samples will agree with yours. Don’t be certain about the parameter. Don’t forget: It’s about the parameter (not the statistic). Don’t claim to know too much. Do take responsibility (for the uncertainty). Do treat the whole interval equally.

32. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 32 What Can Go Wrong? (cont.) Margin of Error Too Large to Be Useful: We can’t be exact, but how precise do we need to be? One way to make the margin of error smaller is to reduce your level of confidence. (That may not be a useful solution.) You need to think about your margin of error when you design your study. To get a narrower interval without giving up confidence, you need to have less variability. You can do this with a larger sample…

33. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 33 What Can Go Wrong? (cont.) 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.

34. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 34 What Can Go Wrong? (cont.) Violations of Assumptions: Watch out for biased samples—keep in mind what you learned in Chapter 12. Think about independence.

35. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 35 What have we learned? Finally we have learned to use a sample to say something about the world at large. This process (statistical inference) is based on our understanding of sampling models, and will be our focus for the rest of the book. In this chapter we learned how to construct a confidence interval for a population proportion. Best estimate of the true population proportion is the one we observed in the sample.

36. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 36 What have we learned? Best estimate of the true population proportion is the one we observed in the sample. Create our interval with a margin of error. Provides us with a level of confidence. Higher level of confidence, wider our interval. Larger sample size, narrower our interval. Calculate sample size for desired degree of precision and level of confidence. Check assumptions and condition.

37. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 19 - 37 What have we learned? We’ve learned to interpret a confidence interval by Telling what we believe is true in the entire population from which we took our random sample. Of course, we can’t be certain, but we can be confident.