6-2 Basic Skills and Concepts
Transcription
6-2 Basic Skills and Concepts
4025_CH06_p264-331 01/02/04 9:13 AM Page 280 280 CHAPTER 6 Estimates and Sample Sizes 6-2 Basic Skills and Concepts Finding Critical Values. In Exercises 1–4, find the critical value za>2 that corresponds to the given confidence level. 1. 99% 2. 90% 3. 98% 4. 92% 5. Express the confidence interval 0.220 , p , 0.280 in the form of p̂ E. 6. Express the confidence interval 0.456 , p , 0.496 in the form of p̂ E. 7. Express the confidence interval (0.604, 0.704) in the form of p̂ E. 8. Express the confidence interval 0.742 0.030 in the form of p̂ 2 E , p , p̂ 1 E. Interpreting Confidence Interval Limits. In Exercises 9–12, use the given confidence interval limits to find the point estimate p̂ and the margin of error E. 9. (0.444, 0.484) 11. 0.632 , p , 0.678 10. 0.278 , p , 0.338 12. 0.887 , p , 0.927 Finding Margin of Error. In Exercises 13–16, assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. 13. n 5 800, x 5 200, 95% confidence 14. n 5 1200, x 5 400, 99% confidence 15. 99% confidence; the sample size is 1000, of which 45% are successes. 16. 95% confidence; the sample size is 500, of which 80% are successes. Constructing Confidence Intervals. In Exercises 17–20, use the sample data and confidence level to construct the confidence interval estimate of the population proportion p. 17. n 5 400, x 5 300, 95% confidence 18. n 5 1200, x 5 200, 99% confidence 19. n 5 1655, x 5 176, 98% confidence 20. n 5 2001, x 5 1776, 90% confidence Determining Sample Size. In Exercises 21–24, use the given data to find the minimum sample size required to estimate a population proportion or percentage. 21. Margin of error: 0.060; confidence level: 99%; p̂ and q̂ unknown 22. Margin of error: 0.038; confidence level: 95%; p̂ and q̂ unknown 23. Margin of error: five percentage points; confidence level: 95%; from a prior study, p̂ is estimated by the decimal equivalent of 18.5%. 24. Margin of error: three percentage points; confidence level: 90%; from a prior study, p̂ is estimated by the decimal equivalent of 8%. 25. Interpreting Calculator Display The Insurance Institute of America wants to estimate the percentage of drivers aged 18–20 who drive a car while impaired because of alcohol consumption. In a large study, 42,772 males aged 18–20 were surveyed, and 5.1% of them said that they drove in the last month while being impaired from alcohol Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 4025_CH06_p264-331 01/02/04 9:13 AM Page 281 6-2 Estimating a Population Proportion (based on data from “Prevalence of Alcohol-Impaired Driving,” by Liu, Siegel, et al., Journal of the American Medical Association, Vol. 277, No. 2). Using the sample data and a 95% confidence level, the TI-83 Plus calculator display is as shown. a. Write a statement that interprets the confidence interval. b. Based on the preceding result, does alcohol-impaired driving appear to be a problem for males aged 18–20? (All states now prohibit the sale of alcohol to persons under the age of 21.) c. When setting insurance rates for male drivers aged 18–24, what percentage of alcohol-impaired driving would you use if you are working for the insurance company and you want to be conservative by using the likely worst case scenario? 26. Interpreting Calculator Display In 1920 only 35% of U.S. households had telephones, but that rate is now much higher. A recent survey of 4276 randomly selected households showed that 4019 of them had telephones (based on data from the U.S. Census Bureau). Using those survey results and a 99% confidence level, the TI-83 Plus calculator display is as shown. a. Write a statement that interprets the confidence interval. b. Based on the preceding result, should pollsters be concerned about results from surveys conducted by telephone? TI-83 Plus TI-83 Plus 27. Internet Shopping In a Gallup poll, 1025 randomly selected adults were surveyed and 29% of them said that they used the Internet for shopping at least a few times a year. a. Find the point estimate of the percentage of adults who use the Internet for shopping. b. Find a 99% confidence interval estimate of the percentage of adults who use the Internet for shopping. c. If a traditional retail store wants to estimate the percentage of adult Internet shoppers in order to determine the maximum impact of Internet shoppers on its sales, what percentage of Internet shoppers should be used? 28. Death Penalty Survey In a Gallup poll, 491 randomly selected adults were asked whether they are in favor of the death penalty for a person convicted of murder, and 65% of them said that they were in favor. a. Find the point estimate of the percentage of adults who are in favor of this death penalty. b. Find a 95% confidence interval estimate of the percentage of adults who are in favor of this death penalty. c. Can we safely conclude that the majority of adults are in favor of this death penalty? Explain. 29. Mendelian Genetics When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 428 green peas and 152 yellow peas. a. Find a 95% confidence interval estimate of the percentage of yellow peas. b. Based on his theory of genetics, Mendel expected that 25% of the offspring peas would be yellow. Given that the percentage of offspring yellow peas is not 25%, do the results contradict Mendel’s theory? Why or why not? 30. Misleading Survey Responses In a survey of 1002 people, 701 said that they voted in a recent presidential election (based on data from ICR Research Group). Voting records show that 61% of eligible voters actually did vote. a. Find a 99% confidence interval estimate of the proportion of people who say that they voted. b. Are the survey results consistent with the actual voter turnout of 61%? Why or why not? Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 281 4025_CH06_p264-331 01/02/04 9:13 AM Page 282 282 CHAPTER 6 Estimates and Sample Sizes 31. Drug Testing The drug Ziac is used to treat hypertension. In a clinical test, 3.2% of 221 Ziac users experienced dizziness (based on data from Lederle Laboratories). a. Construct a 99% confidence interval estimate of the percentage of all Ziac users who experience dizziness. b. In the same clinical test, people in the placebo group didn’t take Ziac, but 1.8% of them reported dizziness. Based on the result from part (a), what can we conclude about dizziness as an adverse reaction to Ziac? 32. Smoking and College Education The tobacco industry closely monitors all surveys that involve smoking. One survey showed that among 785 randomly selected subjects who completed four years of college, 18.3% smoke (based on data from the American Medical Association). a. Construct the 98% confidence interval for the true percentage of smokers among all people who completed four years of college. b. Based on the result from part (a), does the smoking rate for those with four years of college appear to be substantially different than the 27% rate for the general population? 33. Sample Size for Internet Purchases Many states are carefully considering steps that would help them collect sales taxes on items purchased through the Internet. How many randomly selected sales transactions must be surveyed to determine the percentage that transpired over the Internet? Assume that we want to be 99% confident that the sample percentage is within two percentage points of the true population percentage for all sales transactions. 34. Sample Size for Left-Handed Golfers As a manufacturer of golf equipment, the Spalding Corporation wants to estimate the proportion of golfers who are left-handed. (The company can use this information in planning for the number of right-handed and left-handed sets of golf clubs to make.) How many golfers must be surveyed if we want 99% confidence that the sample proportion has a margin of error of 0.025? a. Assume that there is no available information that could be used as an estimate of p̂. b. Assume that we have an estimate of p̂ found from a previous study that suggests that 15% of golfers are left-handed (based on a USA Today report). c. Assume that instead of using randomly selected golfers, the sample data are obtained by asking TV viewers of the golfing channel to call an “800” phone number to report whether they are left-handed or right-handed. How are the results affected? 35. Sample Size for Motor Vehicle Ownership You have been hired by the Ford Motor Company to do market research, and you must estimate the percentage of households in which a vehicle is owned. How many households must you survey if you want to be 94% confident that your sample percentage has a margin of error of three percentage points? a. Assume that a previous study suggested that vehicles are owned in 86% of households. b. Assume that there is no available information that can be used to estimate the percentage of households in which a vehicle is owned. c. Assume that instead of using randomly selected households, the sample data are obtained by asking readers of the Washington Post newspaper to mail in a survey form. How are the results affected? Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 4025_CH06_p264-331 01/02/04 9:13 AM Page 283 6-2 Estimating a Population Proportion 36. Sample Size for Weapons on Campus Concerned about campus safety, college officials want to estimate the percentage of students who carry a gun, knife, or other such weapon. How many randomly selected students must be surveyed in order to be 95% confident that the sample percentage has a margin of error of three percentage points? a. Assume that another study indicated that 7% of college students carry weapons (based on a study by Cornell University). b. Assume that there is no available information that can be used to estimate the percentage of college students carrying weapons. 37. Color Blindness In a study of perception, 80 men are tested and 7 are found to have red > green color blindness (based on data from USA Today). a. Construct a 90% confidence interval estimate of the proportion of all men with this type of color blindness. b. What sample size would be needed to estimate the proportion of male red > green color blindness if we wanted 96% confidence that the sample proportion is in error by no more than 0.03? Use the sample proportion as a known estimate. c. Women have a 0.25% rate of red > green color blindness. Based on the result from part (a), can we safely conclude that women have a lower rate of red > green color blindness than men? 38. TV Ratings The CBS television show 60 Minutes has been successful for many years. That show recently had a share of 20, meaning that among the TV sets in use, 20% were tuned to 60 Minutes (based on data from Nielsen Media Research). Assume that this is based on a sample size of 4000 (typical for Nielsen surveys). a. Construct a 97% confidence interval estimate of the proportion of all sets in use that were tuned to 60 Minutes at the time of the broadcast. b. What sample size would be required to estimate the percentage of sets tuned to 60 Minutes if we wanted 99% confidence that the sample percentage is in error by no more than one-half of one percentage point? (Assume that we have no estimate of the proportion.) c. At the time of this particular 60 Minutes broadcast, ABC ran “Exposed: Pro Wrestling,” and that show received a share of 11. Based on the result from part (a), can we conclude that 60 Minutes had a greater proportion of viewers? Did professional wrestling really need to be exposed? d. How is the confidence interval in part (a) affected if, instead of randomly selected subjects, the survey data are based on 4000 television viewers volunteering to call an “800” number to register their responses? 39. Cell Phones and Cancer A study of 420,000 Danish cell phone users found that 135 of them developed cancer of the brain or nervous system. Prior to this study of cell phone use, the rate of such cancer was found to be 0.0340% for those not using cell phones. The data are from the Journal of the National Cancer Institute. a. Use the sample data to construct a 95% confidence interval estimate of the percentage of cell phone users who develop cancer of the brain or nervous system. b. Do cell phone users appear to have a rate of cancer of the brain or nervous system that is different from the rate of such cancer among those not using cell phones? Why or why not? 40. Pilot Fatalities Researchers studied crashes of general aviation (noncommercial and nonmilitary) airplanes and found that pilots died in 5.2% of 8411 crash landings (based on data from “Risk Factors for Pilot Fatalities in General Aviation Airplane Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 283 4025_CH06_p264-331 01/02/04 9:13 AM Page 284 284 CHAPTER 6 Estimates and Sample Sizes Crash Landings,” by Rostykus, Cummings, and Mueller, Journal of the American Medical Association, Vol. 280, No. 11). a. Construct a 95% confidence interval estimate of the percentage of pilot deaths in all general aviation crashes. b. Among crashes with an explosion or fire on the ground, the pilot fatality rate is estimated by the 95% confidence interval of (15.5%, 26.9%). Is this result substantially different from the result from part (a)? What can you conclude about an explosion or fire as a risk factor? c. In planning for the allocation of federal funds to help with medical examinations of deceased pilots, what single percentage should be used? (We want to be reasonably sure that we have enough resources for the worst case scenario.) 41. Wearing Hunter Orange A study of hunting injuries and the wearing of “hunter” orange clothing showed that among 123 hunters injured when mistaken for game, 6 were wearing orange. Among 1115 randomly selected hunters, 811 reported that they routinely wear orange. The data are from the Centers for Disease Control. a. Construct a 95% confidence interval estimate of the percentage of injured hunters who are wearing orange. b. Construct a 95% confidence interval estimate of the percentage of hunters who routinely wear orange. c. Do these results indicate that a hunter who wears orange is less likely to be injured because of being mistaken for game? Why or why not? 42. Appearance Counts A Sales and Marketing Management survey included 651 sales managers, and 94% of them said that being a sloppy dresser can make a sales representative’s job more difficult. For that same group, 75% said that being an unstylish dresser can make a sales representative’s job more difficult. a. Construct a 90% confidence interval estimate of the percentage of sales managers who say that being a sloppy dresser can make a sales representative’s job more difficult. b. Construct a 90% confidence interval estimate of the percentage of sales managers who say that being an unstylish dresser can make a sales representative’s job more difficult. c. Given that sample proportions naturally vary, can we conclude that when sales managers state reasons for a sales representative’s job becoming more difficult, the percentage is higher for sloppy dressing than for unstylish dressing? Why or why not? 43. Red M&M Candies Refer to Data Set 19 in Appendix B and find the sample proportion of M&Ms that are red. Use that result to construct a 95% confidence interval estimate of the population percentage of M&Ms that are red. Is the result consistent with the 20% rate that is reported by the candy maker Mars? 44. Alcohol and Tobacco Use in Children’s Movies Refer to Data Set 7 in Appendix B. a. Construct a 95% confidence interval estimate of the percentage of animated children’s movies showing any tobacco use. b. Construct a 95% confidence interval estimate of the percentage of animated children’s movies showing any alcohol use. c. Compare the preceding results. Does either tobacco or alcohol appear in a greater percentage of animated children’s movies? d. In using the results from parts (a) and (b) as measures of the depiction of unhealthy habits, what important characteristic of the data is not included? Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 4025_CH06_p264-331 01/02/04 9:13 AM Page 285 6-3 Estimating a Population Mean: S Known 6-2 Beyond the Basics 45. Probing for Precision An example of this section used the photo-cop survey data with n 5 829 and p̂ 5 0.51 to construct the 95% confidence interval of 0.476 , p , 0.544. However, p̂ cannot be exactly 0.51 because 51% of 829 people is 422.79 people, which is not possible. The sample statistic of 51% has been rounded to the nearest whole number. Find the minimum and maximum values of x for which x > 829 is rounded to 0.51, then construct the confidence intervals corresponding to those two values of x. Do the results differ substantially from the confidence interval of 0.476 , p , 0.544 that was found using 0.51? 46. Using Finite Population Correction Factor This section presented Formulas 6-2 and 6-3, which are used for determining sample size. In both cases we assumed that the population is infinite or very large and that we are sampling with replacement. When we have a relatively small population with size N and sample without replacement, we modify E to include the finite population correction factor shown here, and we can solve for n to obtain the result given here. Use this result to repeat part (b) of Exercise 38, assuming that we limit our population to a town with 10,000 television sets in use. E 5 za>2 p̂q̂ N 2 n Å n ÅN 2 1 n5 Np̂q̂3za>2 4 2 p̂q̂3za>2 4 2 1 sN 2 1dE 2 47. One-Sided Confidence Interval A one-sided confidence interval for p can be expressed as p , p̂ 1 E or p . p̂ 2 E, where the margin of error E is modified by replacing za>2 with za. If Air America wants to report an on-time performance of at least x percent with 95% confidence, construct the appropriate one-sided confidence interval and then find the percent in question. Assume that a simple random sample of 750 flights results in 630 that are on time. 48. Confidence Interval from Small Sample Special tables are available for finding confidence intervals for proportions involving small numbers of cases, where the normal distribution approximation cannot be used. For example, given x 5 3 successes among n 5 8 trials, the 95% confidence interval found in Standard Probability and Statistics Tables and Formulae (CRC Press) is 0.085 , p , 0.755. Find the confidence interval that would result if you were to use the normal distribution incorrectly as an approximation to the binomial distribution. Are the results reasonably close? Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 285