Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 19- 1

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 19 Confidence Intervals for Proportions

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Introduction to Confidence Intervals Motivating Example A sample of 81 KSU students finds that 27 attend a KSU sporting event once a semester. = x/n = 27/81 =.33. Based on a sample of only 81 KSU students, do we expect the true proportion (p) to equal.33 exactly?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Introduction to Confidence Intervals We want to use to estimate p. HOWEVER, we know that it is very unlikely that equals p exactly due to sampling error. SO, we construct an interval within which we expect p falls within based on.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide A confidence interval for a parameter is an interval of numbers within which we expect the true value of the population parameter to be contained. The endpoints of the interval are computed based on sample information. Let’s say we want to estimate the true average GPA on campus. Some confidence intervals for the true average are: (2.95,3.05) (2.9,3.1) (2.0,4.0) (2.9999,3.0001)

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Margin of Error: Certainty vs. Precision (cont.)

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide A Confidence Interval (cont.)

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Margin of Error: Certainty vs. Precision (cont.)

Copyright © 2006 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 © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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).

Copyright © 2006 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 © 2006 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 © 2006 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 © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide At 99% confidence, z*=2.575 At 98% confidence, z*=2.33 At 95% confidence, z*=1.96 At 90% confidence, z*=1.645

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide A sample of 81 KSU students finds that 27 attend a KSU sporting event once a semester. Find a 95% confidence interval for the true population proportion of KSU students that attend a KSU sporting event once a semester. A random sample of 140 KSU students finds that 113 of those students polled avoid classes that start before 9:30 AM. Construct a 99% confidence interval for the true population proportion of students who avoid classes that start before 9:30 AM.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Alex collects a sample of 175 KSU students from the Library. This sample yields the confidence interval (77%,91%) for the true proportion of students who currently have the HOPE scholarship. Do you have faith in Alex’s estimate? Explain.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What’s wrong in the following use of statistics? A daily newspaper ran a survey by asking readers to call in their response to this question: “Do you support the development of atomic weapons that could kill millions of innocent people?” It was reported that 20 readers responded and 87% said “no” while 13% said “yes.”

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chipper Jones hit.325 during the regular season. During a playoff game, Chipper comes up to bat with the bases loaded and the announcers point out that Chipper had 20 hits in 49 at bats with the bases loaded during the regular season. The announcers infer that Chipper is better at the plate when the bases are loaded than in general. At a 99% level of confidence, are the announcers correct or is this just an example of chance variation?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chipper Jones had 20 hits in 49 at bats with the bases loaded during the regular season last year. Construct a 99% level of confidence interval for Chipper’s performance for any given year.