BUSA 2000 Statistical Analysis for Business Short Exercise 6 Question 1. The random variable x is known to be uniformly distributed between 1.0 and 1.5. a. Show the graph of the probability density function. b. Compute P(x = 1.25). c. Compute P(1.0 x 1.25). d. Compute E(x). e. Compute Var(x).Answer: The uniform probability density function is: f (x) = 1/(b - a) for a < x < b

Question 2. Given that z is a standard normal random variable, compute the following probabilities. a. P(z -1.0) b. P(z -1) c. P(z -1.5) d. P(-2.5 z) e. P(-3< z 0) Answer: These probabilities can be obtained using Excel's NORM.S.DIST function or the standard normal probability table in the Appendix B-Table 1. (Appendix B-TABLE 1: Cumulative Probabilities for the standard Normal Distribution)

Question 3: Given that z is a standard normal random variable, compute the following probabilities. a. P(0 z .83) b. P(-1.57 z 0) c. P(z > .44) d. P(z 2.23) Answer: These probabilities can be obtained using Excel's NORM.S.DIST function or the standard normal probability table in the text.

Question 4: The mean cost of domestic airfares in the United States rose to an all-time high of $385 per ticket (Bureau of Transportation Statistics website, November 2, 2012). Airfares were based on the total ticket value, which consisted of the price charged by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of $110. a. What is the probability that a domestic airfare is $550 or more? b. What is the probability that a domestic airfare is $250 or less? c. What if the probability that a domestic airfare is between $300 and $500? d. What is the cost for the highest 3% of domestic airfares?

Question 5: Consider the following exponential probability density function. f ( x ) = 1 ex/ for x > 0 f ( x )= 1 8 ex /8 for x > 0 a. Find P(x 6). b. Find P(x 4). c. Find P(x 6). d. Find P(4 x 6).

Question 6. The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 12 seconds. a. Sketch this exponential probability distribution. b. What is the probability that the arrival time between vehicles is 12 seconds or less? c. What is the probability that the arrival time between vehicles is 6 seconds or less? d. What is the probability of 30 or more seconds between vehicle arrivals?

Question 7. Selling a House. A business executive, transferred from Chicago to Atlanta, needs to sell her house in Chicago quickly. The executive's employer has offered to buy the house for $210,000, but the offer expires at the end of the week. The executive does not currently have a better offer but can afford to leave the house on the mar ket for another month. From conversations with her realtor, the executive believes the price she will get by leaving the house on the market for another month is uniformly distributed between $200,000 and $225,000. a. If she leaves the house on the market for another month, what is the mathematical expression for the probability density function of the sales price? b. If she leaves it on the market for another month, what is the probability that she will get at least $215,000 for the house? c. If she leaves it on the market for another month, what is the probability that she will get less than $210,000? d. Should the executive leave the house on the market for another month? Why or why not?

Question 8. Production Defects. Motorola used the normal distribution to determine the probability of defects and the number of defects expected in a production process. Assume a production process produces items with a mean weight of 10 ounces. Calculate the probability of a defect and the expected number of defects for a 1,000unit production run in the following situations. a. The process standard deviation is .15, and the process control is set at plus or minus one standard deviation. Units with weights less than 9.85 or greater than 10.15 ounces will be classified as defects. b. Through process design improvements, the process standard deviation can be reduced to .05. Assume the process control remains the same, with weights less than 9.85 or greater than 10.15 ounces being classified as defects. c. What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean?

Question 9. NCAA Scholarships. The NCAA estimates that the yearly value of a full athletic scholarship at instate public universities is $19,000 (The Wall Street Journal). Assume the scholarship value is normally distributed with a standard deviation of $2,100. a. For the 10% of athletic scholarships of least value, how much are they worth? b. What percentage of athletic scholarships are valued at $22,000 or more? c. For the 3% of athletic scholarships that are most valuable, how much are they worth?

Question 10. Mean Time Between Failures. The mean time between failures (MTBF) is a common metric used to measure the performance of manufacturing systems. MTBF is the elapsed time between failures of a system during normal operations. The failures could be caused by broken machines or computer errors, among other failures. Suppose that the MTBF for a new automated manufacturing system follows an exponential distribution with a mean of 12.7 hours. a. What is the probability that the automated manufacturing system runs for more than 15 hours without a failure? b. What is the probability that the automated manufacturing system runs for eight or fewer hours before failure? c. What is the probability that the automated manufacturing system runs for more than six hours but less than 10 hours before a failure?

Can you please help with these?