(I) Company A is considering the acquisition of two separate but large companies, Company B and Company C, having sales and assets equal to its own. The following table gives the probabilities of returns for each of the three companies under various economic conditions. The table also gives the probabilities of returns for each possible combination: Company A plus Company B, and Company A plus Company C.(For the table, please see the attach assignment.)

(a) For each of Companies A, B and C, find the mean return and the standard deviation of returns. (b) Find the mean return and the standard deviation of returns for the combination of Company A plus Company B. (c) Find the mean return and the standard deviation of returns for the combination of Company A plus Company C. (d) Compare the mean returns for each of the two possible combinations – Company A plus Company B and Company A plus Company C. Is either mean higher? How do they compare to Company A’s mean return? (e) Compare the standard deviation of the returns for each of the two possible combinations – Company A plus Company B and Company A plus Company C. Which standard deviation is smaller? Which possible combination involves less risk? How does the risk carried by this combination compare to the risk carried by Company A alone? (f) Which acquisition would you recommend – Company A plus Company B or Company A plus Company C?

(II) A consumer advocate claims that 80 percent of cable television subscribers are not satisfied with their cable service. In an attempt to justify this claim, a randomly selected sample of cable subscribers will be polled on this issue.

(a) Suppose that the advocate’s claim is true, and suppose that a random sample of five cable subscribers is selected. Assuming independence, use an appropriate formula to compute the probability that four or more subscribers in the sample are not satisfied with their service. (b) Suppose that the advocate’s claim is true, and suppose that a random sample of 25 cable subscribers is selected. Assuming independence, find(1) The probability that more than 20 subscribers in the sample are not satisfied with their service.(2) The probability that 15 or fewer subscribers in the sample are not satisfied with their service. (c) Suppose that when we survey 25 randomly selected cable television subscribers, we find that 15 are actually not satisfied with their service. Using a probability you found in this exercise as the basis for your answer, do you believe the consumer advocate’s claim? Explain.

(III) A local law enforcement agency claims that the number of times that a patrol car passes through a particular neighborhood follows a Poisson process with a mean of three times per nightly shift. Let X denote the number of times that a patrol car passes through the neighborhood during a nightly shift.

(a) Calculate the probability that no patrol car pass through the neighborhood during a nightly shift. (b) Suppose that during a randomly selected night shift no patrol car pass through the neighborhood. Based on your answer in part(a), do you believe the agency’s claim? Explain. (c) Assuming that nightly shifts are independent and assuming that the agency’s claim is correct, find the probability that exactly one patrol car will through the neighborhood on each of four consecutive nights.

(IV) A weather forecaster predicts that the May rainfall in a local area will be between three and six inches but has no idea where within the interval the amount will be. Let X be the amount of May rainfall in local area, and assume that X is uniformly distributed in the interval three to six inches. (a) Write the formula for the probability density function of X. (b) Graph the probability density function of X. (c) What is the probability that May rainfall will be at least four inches? At least ve inches? At most 4.5 inches? (d) Calculate the expected May rainfall. (e) What is the probability that the observed May rainfall will fall within two standard deviations of the mean? Within one standard deviation of the mean?

(V) Stanford-Binet IQ Test scores are normally distributed with a mean score of 100 and a standard deviation of 16. (a) Find the probability that a randomly selected person has an IQ test score(1) between 72 and 128.(2) within 1.5 standard deviations of the mean. (b) Suppose that you take the Stanford-Binet IQ Test and receive a score of 136. What percentage of people would receive a score higher than yours?

(VI) Suppose that yearly health care expenses for a family of four are normally distributed with a mean expense equal to $3,000 and a standard deviation of $500. An insurance company has decided to offer a health insurance premium reduction if a policyholder’s health care expenses do not exceed a specied dollar amount. What dollar amount should be established if the insurance company wants families having the lowest 33 percent of yearly health care expenses to be eligible for the premium reduction?

(VII) In a survey of marketing professionals about various scenarios involving ethical issues, among 205 randomly selected marketing researchers who participated in the survey, 117 said they disapprove of an actions taken in a certain scenario. Suppose that, before the survey was taken, a marketing manager claimed that at least 65 percent of all marketing researchers would disapprove of that scenario.

(a) Assuming that the manager’s claim is correct, calculate the probability that 117 or fewer of205 randomly selected marketing researchers would disapprove of the scenario. Use the normal approximation to the binomial. (b) Based on you result of part (a), do you believe the marketing manager’s claim? Explain.