Free solutions & answers for Introductory Statistics Chapter 10 - (Page 1) [step by step] (2024)

In parts of the eastern United States, whitctail deer are a major nuisance tofarmers and homeowners, frequently damaging crops, gardens, and landscaping. Aconsumer organization arranges a test of two of the leading deer repellents\(\mathrm{A}\) and \(\mathrm{B}\) on the market. Fifty-six unfenced gardens inareas having high concentrations of deer are used for the test. Twenty-ninegardens are chosen at random to receive repellent \(\mathrm{A}\), and the other27 receive repellent \(\mathrm{B}\). For each of the 56 gardens, the timeelapsed between application of the repellent and the appearance in the gardenof the first deer is recorded. For repellent \(\mathrm{A}\), the mean time is101 hours. For repellent \(B\), the mean time is 92 hours. Assume that the twopopulations of elapsed times have normal distributions with populationstandard deviations of 15 and 10 hours, respectively. a. Let \(\mu_{1}\) and \(\mu_{2}\) be the population means of elapsed times forthe two repellents, respectively. Find the point estimate of\(\mu_{1}-\mu_{2}\). b. Find a \(97 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) c. Test at a \(2 \%\) significance level whether the mean elapsed times forrepellents \(A\) and \(B\) are different. Use both approaches, the critical-valueand \(p\) -value, to perform this test.

The U.S. Department of Labor collects data on unemployment insurance paymentsmade to unemployed people in different states. Suppose that during 2011 arandom sample of 1000 unemployed people in Florida received an average weeklyunemployment benefit of \(\$ 219.65\), while a random sample of 900 unemployedpeople in Mississippi received an average weekly unemployment benefit of \(\$191.47\). Assume that the population standard deviations of 2011 weeklyunemployment benefits paid to all unemployed workers in Florida andMississippi were \(\$ 35.15\) and \(\$ 28.22\), respectively. (Note: A 2011 studyby DailyFinance.com (http://www.dailyfinance.com/2011/05/12/unemployment-benefits-best-worst-states/) rated Mississippi and Florida as the two worststates for unemployment benefits.) a. Let \(\mu_{1}\) and \(\mu_{2}\) be the means of weekly unemployment benefitspaid to all unemployed workers during 2011 in Florida and Mississippi,respectively. What is the point estimate of \(\mu_{1}-\mu_{2} ?\) b. Construct a \(96 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). c. Using a \(2 \%\) significance level, can you conclude that the means of allweekly unemployment benefits paid to all unemployed workers during 2011 inFlorida and Mississippi are different? IIse both the \(p\) -value and thecritical-value annroaches to make this.te-

A local college cafeteria has a self-service soft ice cream machine. Thecafeteria provides bowls that can hold up to 16 ounces of ice cream. The foodservice manager is interested in comparing the average amount of ice creamdispensed by male students to the average amount dispensed by female students.A measurement device was placed on the ice cream machine to determine theamounts dispensed. Random samples of 85 male and 78 female students who gotice cream were selected. The sample averages were \(7.23\) and \(6.49\) ounces forthe male and female students, respectively. Assume that the populationstandard deviations are \(1.22\) and \(1.17\) ounces, respectively. a. Let \(\mu_{1}\) and \(\mu_{2}\) be the population means of ice cream amountsdispensed by all male and all female students at this college, respectively.What is the point estimate of \(\mu_{1}-\mu_{2} ?\) b. Construct a \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). c. Using a \(1 \%\) significance level, can you conclude that the average amountof ice cream dispensed by all male college students is larger than the averageamount dispensed by all female collegs students? Use both approaches to makethis test.

Employees of a large corporation are concerned about the declining quality ofmedical services provided by their group health insurance. A random sample of100 office visits by employees of this corporation to primary care physiciansduring 2004 found that the doctors spent an average of 19 minutes with each patient. This year a random sample of 108 such visits showed that doctorsspent an average of \(15.5\) minutes with each patient. Assume that the standarddeviations for the two populations are \(2.7\) and \(2.1\) minutes, respectively. a. Construct a \(95 \%\) confidence interval for the difference between the twopopulation means for these two years. b. Using a \(2.5 \%\) level of significance, can you conclude that the mean timespent by doctors with each patient is lower for this year than for \(2004 ?\) c. What would your decision be in part \(\mathrm{b}\) if the probability ofmaking a Type I error were zero? Explain.

A car magazine is comparing the total repair costs incurred during the firstthree years on two sports cars, the T-999 and the XPY. Random samples of 45T-999s and 51 XPY's are taken. All 96 cars are 3 years old and have similarmileages. The mean of repair costs for the 45 T-999 cars is \(\$ 3300\) for thefirst 3years. For the 51 XPY cars, this mean is \(\$ 3850\). Assume that thestandard deviations for the two popuLations are \(\$ 800\) and \(\$ 1000\),respectively. a. Construct a \(99 \%\) confidence interval for the difference between the twopopulation means. b. Using a \(1 \%\) significance level, can you conclude that such mean repaircosts are different for these two types of cars? c. What would your decision be in part \(b\) if the probability of making a TypeI error were zero? Explain.

The management at New Century Bank claims that the mean waiting time for allcustomers at its branches is less than that at the Public Bank, which is itsmain competitor. A business consulting firm took a sample of 200 customersfrom the New Century Bank and found that they waited an average of \(4.5\)minutes before being served. Another sample of 300 customers taken from thePublic Bank showed that these customers waited an average of \(4.75\) minutesbefore being served. Assume that the standard deviations for the twopopulations are \(1.2\) and \(1.5\) minutes, respectively. a. Make a \(97 \%\) confidence interval for the difference between thepopulation means. b. Test at a 2.5\% significance level whether the claim of the management ofthe New Century Bank is truc. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value,would you reject the null hypothesis if \(\alpha=01 ?\) What if \(\alpha=05\) ?