MATH 220

Assignment #5

Question 1: The average time it takes a group of adults to complete a certain achievement test is 46.2 minutes.The standard deviation is 8 minutes.Assume the variable is normally distributed.

a)Find the probability that a randomly selected adult will complete the test in less than 43 minutes.

b)Find the probability that if 50 randomly selected adults take the test, the mean time it takes the group to complete the test is less than 43 minutes.

c)Does it seem reasonable that an adult would finish the test in less than 43 minutes?Explain.

d)Does it seem reasonable that the mean of the 50 adults could be less than 43 minutes?Explain.

Question 4: (a) 0.3446 (b) 0.0023 (c) Yes, since it is within one standard deviation of the mean. (d) Very unlikely, since the probability would be less than 1%.

Question 2: Solve this problem.

Risks and insurance.The idea of insurance is that we all face risks that are unlikely but carry high cost. Think of a fire destroying your home. So we form a group to share the risk: we all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn down. An insurance company looks at the records for millions of homeowners and sees that the mean loss from fire in a year is= $250 per house and that the standard deviation of the loss is= $1000. (The distribution of losses is extremely right-skewed: most people have $0 loss, but a few have large losses.) The company plans to sell fire insurance for $250 plus enough to cover its costs and profit.

(a) Explain clearly why it would be unwise to sell only 12 policies. Then explain why selling many thousands of such policies is a safe business.

(b) If the company sells 25,000 policies, what is the approximate probability that the average loss in a year will be greater than $270?

Question 3: A study by a federal agency concludes that polygraph tests given to truthful persons have probability 0.2 of suggesting that the person is deceptive. A firm asks 12 job applicants about thefts from previous employers, using a polygraph to assess their truthfulness. Suppose that all 12 answer truthfully. Let X be the number of applicants who are classified deceptive.

a)Describe the probability mass function of X.

b)What is the probability that the polygraph says at least 1 is deceptive?

c)What is the mean number among 12 truthful persons who will be classified as deceptive? What is the standard deviation of this number?

d)What is the probability that the number classified deceptive is less than the mean?

Question 4: Solve this problem.

A selective college would like to have an entering class of 950 students. Because not all students who are offered admission accept, the college admits more than 950 students. Past experience shows that about 75% of the students admitted will accept. The college decides to admit 1200 students. Assuming that students make their decisions independently, the number who accept has theB(1200, 0.75) distribution. If this number is less than 950, the college will admit students from its waiting list.

(a) What are the mean and the standard deviation of the numberXof students who accept?

(b) Use the Normal approximation to find the probability that at least 800 students accept.

(c) The college does not want more than 950 students. What is the probability that more than 950 will accept?

(d) If the college decides to increase the number of admission offers to 1300, what is the probability that more than 950 will accept?