a)[20 points]National Car Rental Systems Inc.,commissioned the Canadian Automobile Association (CAA) to conduct a survey of the general condition of the cars rented to the public by Hertz, Avis, National and Budget Rent-a Car. CAA officials evaluate each companys cars using a demerit points system. Each car starts with a perfect score of 0 points and incurs demerit points for each discrepancy noted by the inspectors. One measure of the overall condition of a companys cars is the mean of all scores received by the company (i.e., the companys fleet mean score). To estimate the fleet mean score of each rental car company, 10 major airports were randomly selected and 10 cars from each company were randomly rented for inspection from each airport by CAA officials (i.e., a random sample of n = 100 cars from each companys fleet was drawn and inspected).
i.[3 points]Describe the sampling distribution of , the mean score of a sample of n = 100 rental cars.
ii.[3 points]Interpret the mean of in the context of this problem.
iii.[3 points]Assume that and for one rental car company. For this company, find .
iv.[3 points]If what is ?
v.[4 points ]Refer to partiii. The company claims that their true fleet mean score couldnt possibly be as high as 30. The sample mean score tabulated by the CAA for this company was 45. Does this result tend to support or refute the claim? Explain.
b)[4 points]Will the sampling distribution of always be approximately normally distributed? Explain.
a)[15 points]According to a recent Pew Internet and American Life Project Survey (October 2010), 67% of adults who use the Internet have paid to download music. In a random sample of n = 1,000 adults who use the Internet, let represent the proportion who have paid to download music.
i.[4 points]Find the mean and standard deviation of the sampling distribution of .
ii.[5 points]What does the Central Limit Theorem say about the shape of the sampling distribution of .
iii.[3 points]What is the probability that less than 75% of adults who use the Internet have paid to download music?
iv.[3 points]What is the probability that more than 50% of these adults have paid to download music?
b)[5 points]Due to inaccuracies in drug-testing procedures (e.g., false positives and false negatives) in the medical field, the results of a drug test represent only one factor in a physicians diagnosis. Yet, when Olympic athletes are tested for illegal drug use (i.e., doping), the results of a single positive test are used to ban the athlete from competition. InChance(Spring 2004), University of Texas biostatisticians D. A. Berry and L. Chastain demonstrated the application of Bayes Rule for making inferences about testosterone abuse among Olympic athletes. They used the following example: In a population of 1,000 athletes, suppose 100 are illegally using testosterone. Of the users, suppose 50 would test positive for testosterone. Of the nonusers, suppose 9 would test positive. If an athlete tests positive for testosterone, use Bayes Rule to find the probability that the athlete is really doping.
[15 points]Suppose that and are random observations taken from a population with mean and variance. Consider the following three point estimators,X,Y,Z, of:
; and .
a)[5 points] Show that all three estimators X, Y, and Z are unbiased estimators of .
b)[5 points]Which of the estimators X, Y and Z is the most efficient? Explain.
c)[5 points]Find the relative efficiency ofXwith respect to each of the other two estimatorsYandZ.
[20 points]The amount of time spent (in minutes) for the completion of the 4th assignment in Statistics by a random sample of 10 students gave the following results: 215, 182, 193, 208, 210, 176, 197, 188, 218, 213.
a)[5 points]Calculate the sample mean and sample standard deviation.
b)[5 points]Specify the appropriate assumptions and find a 95% confidence interval for the population mean time spent by students on the 4th Assignment.
c)[5 points]Find a 90% confidence interval for the population mean time spent by students on the 4th Assignment.
d)[5 points]Compare your findings in (b) and (c).
EXCEL problem. (15 points)
a) Simulate 10,000 draws from a standard normal distribution for each of 5 variables. Square each of these; you should have 5 columns, each of length 10000, with squared N(0,1s) as entries. (UseRandom Number Generatorin Data Analysis.)
b) Create a new column of length 10,000 by adding the five entries across each row and dividing this sum by 5.
c) Sort the last column from low to high; useRank and Percentilein Data Analysis. Using this sorted series, fill in the following Table as the answer to this problem, along with an EXCEL output of the first twenty rows of the six variables. Also, compute the average and variance of this last row to be included in the Table. Finally, include a histogram of the series as part of the output.
Simulation of the Chi-square Distribution (5,2) with 5 Degrees of Freedom
|5,2||= .99||= .975||= .95||= .05||= .025||= .01|
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