13:49N all 91%Homework 3.pdfHomework 3.pdf ▼Find he minumum requirea sampre size ror esumaung ine average return on rear estateinvestments to within 0.5% per year with 95% confidence. The standard deviation of returns isbelieved to be 2% per year.Problem 8:Consider the use of metal detectors in airports to test people for concealed weapons. In essence,this is a form of hypothesis testing.a) What are the null and alternative hypotheses?b) What are type I and type II errors in this case?c) Which type of error is more costly?d) Based on your answer to part (c), what value of a would you recommend for this test?e) If the sensitivity of the metal detector is increased, how would the probabilities of type I andtype II errors be affected?n If a is to be increased, should the sensitivity of the metal detector be increased or decreased?Problem 9:The calculated z for a hypothesis test is 1.75. What is the p-value if the test is (a) left-tailed, (h)right-tailed, and (c) two-tailed?Problem 10:A fashion industry analyst wants to prove that models featuring Liz Claiborne clothing earn onaverage more than models featuring clothes designed by Calvin Klein. For a given period oftime, a random sample of 32 Liz Claiborne models reveals average earnings of S4,238.00 and astandard deviation of $1,002.50. For the same period, an independent random sample of 37Calvin Klein models has mean earnings of $3,888.72 and a sample standard deviation of$876.05.a. Is this a one-tailed or a two-tailed test? Explain.b. Carry out the hypothesis test at the 0.05 level of significance.c. State your conclusion.d. What is the p-value? Explain its relevance.e. Redo the problem, assuming the results are based on a random sample of 10 Liz Claibornemodels and 11 Calvin Klein models.Problem 11:A physicians' group is interested in testing to determine whether more people in small townschoose a physician by word of mouth in comparison with people in large metropolitan areas. Arandom sample of 1,00 people in small towns reveals that 850 chose their physicians by wordof mouth; a random sample of 2,500 people living in large metropolitan areas reveals that 1,950chose a physician by word of mouth. Conduct a one-tailed test aimed at proving that thepercentage of popular recommendation of physicians is larger in small towns than in largemetropolitan areas. Use a = 0.01.Problem 12:Compaq Computer Corporation has an assembly plant in Houston, where the company'sDeskpro computer is built. Engineers at the plant are considering a new production facility andare interested in going online with the new facility if and only if they can be fairly sure that thevariance of the number of computers assembled per day using the new facility is lower than theproduction variance of the old system. A random sample of 40 production days using the oldproduction method gives a sample variance of 1,288; and a random sample of 15 production daysusing the proposed new method gives a sample variance of 1,112. Conduct the appropriate test ata = 0.05. 13:49all 91%Homework 3.pdfHomework 3.pdf vHOMEWORK 3Problem 1:Suppose you are sampling from a population with mean pu= 1,065 and standard deviation o=500. The sample size is n=100. What are the expected value and the variance of the samplemean X?Problem 2:The Toyota Prius uses both gasoline and electric power. Toyota claims its mileage per gallon is52. A random sample of 40 cars is taken and each sampled car is tested for its fuel efficiency.Assuming that 52 miles per gallon is the population mean and 24 miles per gallon is thepopulation standard deviation, calculate the probability that the sample mean will be between 52and 53.Problem 3:A car manufacturer wants to estimate the average miles-per-gallon highway rating for a newmodel. From experience with similar models, the manufacturer believes the miles-per-gallonstandard deviation is 4.6. A random sample of 100 high-way runs of the new model yields asample mean of 32 miles per gallon. Give a 95% confidence interval for the population averagemiles-per-gallon highway rating.Problem 4:The following gambling game, known as the wheel of fortune (or chuck-a-luck), is quitepopular at many carnivals and gambling casinos: A player bets on one of the numbers 1through 6. Three dice are then rolled, and if the number bet by the player appears i times,i=1, 2, 3, then the player wins i units; if the number bet by the player does not appear onany of the dice, then the player loses 1 unit. Is this game fair to the player? (Actually, thegame is played by spinning a wheel that comes to rest on a slot labeled by three of thenumbers I through 6, but this variant is mathematically equivalent to the dice version.Problem 5:The manufacturer of batteries used in small electric appliances wants to estimate the average lifeof a battery. A random sample of 12 batteries yields = 34.2 hours and s = 5.9 hours. Give a95% confidence interval for the average life of a battery.Problem 6:A machine produces safety devices for use in helicopters. A quality-control engineer regularlychecks samples of the devices produced by the machine, and if too many of the devices aredefective, the production process is stopped and the machine is readjusted. If a random sample of52 devices yields 8 defectives, give a 98% confidence interval for the proportion of defectivedevices made by this machine.Problem 7:Find the minimum required sample size for estimating the average return on real estateinvestments to within 0.5% per year with 95% confidence. The standard deviation of returns isbelieved to be 2% per year.Problem 8:Consider the use of metal detectors in airports to test people for concealed weapons. In essence,this is a form of hynothesis testing

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Problem 1: Suppose you are sampling from a population with mean p= 1,065 and standard deviation a = 500. The sample size is n=100. What are the expected value and the variance of the sample mean X?