HOMEWORK # 5 (DUE ON OCTOBER 31, 1997(FRIDAY)

4.51, 4.57, 4.61, 4.62, 4.65, 4.71, 4.72, 4.75, 4.77, 5.6, 5.12, 5.20, 5.29, 5.40

HOMEWORK # 6 (DUE ON NOVEMBER 4, 1997(TUESDAY)

5.49, 5.55, 5.66, 5.71, 6.1, 6.8, 6.16, 6.22, 6.24, 6.28, 6.30, 6.41

1. An employee of a firm has an option to invest $1,000 in the company's bonds. At the end of 1 year, the company will buy back the bonds at a price determined by its profits for the year. From the past years, the company predicts it will buy the bonds back at the following prices with the associated probabilities (x = price paid for bonds):

x	$0	$500	$1,000	$1,500	$2,000
p(x)	.01	.22	.30	.22	.25

a. What is the probability the employees will receive $1,000 or less for the investment?

b. What is the expected price paid for the bonds?

c. What is the employee's expected profit?

d. Find the variance and standard deviation for this probability distribution.

2. In a poll conducted by Parents magazine, 60% of parents said they wished they had received more education (Parents, August 1988). A random sample of twenty parents are selected.

a. What is the probability distribution of x, the number of parents who hold this view in this sample? Justify your answer.

b. What is the expected number of parents who will hold this view in this sample? What is the standard deviation of the random variable in part (a)

c. Find the probability that exactly 3 will not hold this view in the sample.

d. Use the normal approximation to find P(6<x<9).

3. The speeds of all cars traveling on a stretch of Interstate Highway I-95 are normally distributed with a mean of 68 miles per hour and a standard deviation of 3 miles.

a. Find the percentage of travelers who are violating the 65 miles speed limit.

b. If a police officer decides to give a ticket to the fastest 10% of the drivers, what should be the minimum speed he would use to write a speeding ticket?

c. If a random sample of 36 cars traveling on this highway has been selected, what is the probability that their average speed will exceed the 65 miles speed limit?

4. Based on the sample data collected in the Denver area, Nicholas Kiefer (1985) found that in some cases the exponential distribution is an adequate approximation for the distribution of the time (in weeks) an individual is unemployed. Use q = 13 to answer the following questions.

a. What is the mean time workers are unemployed according to the exponential distribution?

b. Find the probability that a worker who just lost her job will be unemployed for at least two weeks.

c. What is the probability that an unemployed worker will find a new job within 12 weeks?

d. What is the probability that only two of the five unemployed workers will find a new job within 12 weeks?

5. A particular type of birth-control pill is 90% effective. A random sample of 20 persons are selected.

a. What is the probability distribution of x, the number of unplanned births for this sample? Justify your answer?

b. How many unplanned births would you expect in this sample? What is the standard deviation of the random variable defined in part (a)?

c. Find the probability that exactly 3 out of 20 will have unplanned births?

6. In 1987, the average annual rate of interest paid by savings and loan institutions in Pennsylvania was 7.26%. Assume a normal distribution and a standard deviation of 1.50% to answer the following questions.

a. What is the probability that a randomly selected Pennsylvania savings and loan institution paid between 7.00% and 8.00% interest on deposits?

b. What is the probability that a randomly selected 9 such institutions on the average paid between 7.00% and 8.00% interest on deposits?

c. What is the probability that only 5 of the 9 such institutions paid between 7.00% and 8.00% interest on deposits?

7. In deciding how many customer service representatives to hire an in planning their schedules, it is important for a firm marketing electronic typewriters to study the repair times for the machines. Such a study revealed that repair times have an exponential distribution with q=22 minutes

a. What is the expected repair time for an electronic typewriter?

b. Find the probability that a repair time will last less than 10 minutes.

c. The charge for typewrite repair is $50 for each half hour, or part thereof, for labor. What is the probability that a repair job will result in a charge for labor of $100?

8. The board of examiners that administered the real estate brokers' examination in a certain state found that the mean score on the test was 435 and the standard deviation was 72. If the board wants to set the passing score so that only the best 30% of all applicants pass, what is the passing score? Assume that the scores are normally distributed.

9. A taxi service based at an airport can be characterized as a transportation system with one source terminal and a fleet of vehicles that take passengers from the terminal to different designations. Each vehicle returns to the terminal after some random trip time and makes another trip. To improve the vehicle-dispatching decisions involved in such system, Sims and Templeton (1985) assumed travel times of successive trips are independent exponential random variables to model the system. Assume q = 20 minutes.

a. What is the mean trip time for the taxi service?

b. What is the probability that a particular trip will take less than 30 minutes.

c. What is the probability that a particular trip will take between 10 to 15 minutes.

d. Two taxis have just been dispatched. What is the probability that both will be gone for more than 30 minutes? That at least one of the taxis will return within 30 minutes?