7-1 Estimating a Population Proportion 325 3. Notation For the poll described in Exercise 1, what values do pn, qn, n, E, and p represent? If the confidence level is 95%, what is the value of a? 4. Confidence Levels Given specific sample data, such as the data given in Exercise 1, which confidence interval is wider: the 95% confidence interval or the 80% confidence interval? Why is it wider? Finding Critical Values. In Exercises 5–8, find the critical value zA,2 that corresponds to the given confidence level. 5. 90% 6. 99% 7. 99.5% 8. 98% Formats of Confidence Intervals. In Exercises 9–12, express the confidence interval using the indicated format. (The confidence intervals are based on the proportions of red, orange, yellow, and blue M&Ms in Data Set 38 “Candies” in Appendix B.) 9. Green M&Ms Express 0.116 6 p 6 0.192 in the form of pn { E. 10. Orange M&Ms Express 0.193 6 p 6 0.283 in the form of pn { E. 11. Yellow M&Ms Express the confidence interval 10.0847, 0.1532 in the form of pn - E 6 p 6 pn + E. 12. Blue M&Ms Express the confidence interval 0.255 { 0.046 in the form of pn - E 6 p 6 pn + E. Constructing and Interpreting Confidence Intervals. In Exercises 13–16, use the given sample data and confidence level. In each case, (a) find the best point estimate of the population proportion p; (b) identify the value of the margin of error E; (c) construct the confidence interval; (d) write a statement that correctly interprets the confidence interval. 13. Tennis Challenges In a recent U.S. Open tennis tournament, men playing singles matches used challenges on 240 calls made by the line judges. Among those challenges, 88 were found to be successful with the call overturned. Construct a 95% confidence interval for the proportion of successful challenges. 14. Eliquis The drug Eliquis (apixaban) is used to help prevent blood clots in certain patients. In clinical trials, among 5924 patients treated with Eliquis, 153 developed the adverse reaction of nausea (based on data from Bristol-Myers Squibb Co.). Construct a 99% confidence interval for the proportion of adverse reactions. 15. Survey Return Rate In a study of cell phone use and brain hemispheric dominance, an Internet survey was e-mailed to 5000 subjects randomly selected from an online group involved with ears. 717 surveys were returned. Construct a 90% confidence interval for the proportion of returned surveys. 16. Medical Malpractice In a study of 1228 randomly selected medical malpractice lawsuits, it was found that 856 of them were dropped or dismissed (based on data from the Physicians Insurers Association of America). Construct a 95% confidence interval for the proportion of medical malpractice lawsuits that are dropped or dismissed. Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question. 17. Births A random sample of 860 births in New York State included 426 boys. Construct a 95% confidence interval estimate of the proportion of boys in all births. It is believed that among all births, the proportion of boys is 0.512. Do these sample results provide strong evidence against that belief? 18. Mendelian Genetics One of Mendel’s famous genetics experiments yielded 580 peas, with 428 of them green and 152 yellow. a. Find a 99% confidence interval estimate of the percentage of green peas. continued
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