Quantitative Methods Assignment 3 Problem Company Is Trying

Quantitative Methods Assignment 3problema Company Is Trying To Deter

Quantitative Methods – Assignment 3 Problem A company is trying to determine how to allocate its $145,000 advertising budget for a new product. The company is considering newspaper ads and television commercials as its primary means for advertising. The following table summarizes the costs of advertising in these different media and the number of new customers reached by increasing amounts of advertising. Media & No of Ads No of New Customers Reached Cost per Ad Newspaper: $1,000 Newspaper: $900 Newspaper: $800 Television: ,000 $12,000 Television: ,500 $10,000 Television: ,000 $8,000 For instance, each of the first 10 ads the company place in newspapers will cost $1000 and is expected to reach 900 new customers. Each of the next 10 newspaper ads will cost $900 and is expected to reach 700 new customers. Note that the number of new customers reached by increasing amount of advertising decreases as the advertising saturates the market. Assume the company will purchase no more than 30 newspaper ads and no more than 15 television ads. Formulate an LP model for this problem to maximize the number of new customers reached by advertising.

Question 2

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 company’s 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 inspectors. One measure of the overall condition of a company's cars is the mean of all scores received by the company (i.e., the company’s 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 company’s fleet was drawn and inspected).

i. [3 points] Describe the sampling distribution of the mean score, the mean of a sample of n = 100 rental cars.

ii. [3 points] Interpret the mean of this sampling distribution in the context of the problem.

iii. [3 points] Assume the true mean score (μ) and variance (σ²) for one rental car company are known. For this company, find the expected value of the sample mean score.

iv. [3 points] If the sample mean score (x̄) for this company is 45, and the true mean is μ, what is the standard error of the mean?

v. [4 points] Refer to part iii. The company claims that their true fleet mean score “couldn’t possibly be as high as 30.” The sample mean score for the company was 45. Does this result tend to support or refute the claim? Explain.

Question 3

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 p̂ represent the proportion who have paid to download music.

i. [4 points] Find the mean and standard deviation of the sampling distribution of p̂.

ii. [5 points] What does the Central Limit Theorem say about the shape of the sampling distribution of p̂?

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 physician’s diagnosis. Yet, when Olympic athletes are tested for illegal drug use (doping), the results of a single positive test are used to ban the athlete from competition. In Chance (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.

Question 4

[15 points] Suppose that X and Y are random observations taken from a population with mean μ and variance σ². Consider the following three point estimators, X̄ , Ȳ , Z̄ , of μ: X, Y, and Z.

a) [5 points] Show that all three estimators X̄, Ȳ, and Z̄ are unbiased estimators of μ.

b) [5 points] Which of the estimators X̄, Ȳ, and Z̄ is the most efficient? Explain.

c) [5 points] Find the relative efficiency of X̄ with respect to each of the other two estimators Ȳ and Z̄.

Question 5

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).

Question 6

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(0,1’s) as entries. (Use Random Number Generator in 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; use Rank and Percentile in 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 (χ5,α 2) with 5 Degrees of Freedom χ5,α 2 α = .99 α = .975 α = .95 α = .05 α = .025 α = .01 True Value (from χ2 Tables) Simulated Value Simulated Mean Simulated Variance 2 m 2 s 12 ()/2 Xxx =+ )/4 Yxx =+ )/3 Zxx =+ 1 x 2 x

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