1. Prof. Miller recently ran a marathon. Given that the mean finishing time for all runners was 258 minutes with a standard deviation of 44 minutes and that finishing times were approximately normally distributed, find
(a) the percent of finishers who ran faster than Prof. Miller’s time of 3 hr 30 min.
(b) a time that was the 33rd percentile (33% of finishers were under that time)
(c) the percent to runners who finished between 3 hours and 4 hours
2. Consider the experiment of rolling two dice (one die is blue and the other is red) and the following events:
A: ‘The sum of the dice is 6’ and B: ‘Both dice have even numbers’ and C: “The difference (absolute value) of the dice is 2” and D: “the blue die has an even number”
Find (a) p(A and B) (HINT: You cannot assume these are independent events.)
(b) p(A or B)
(c) Are A and B mutually exclusive events? Explain.
(d) Are A and B independent events? Explain. (no explanations no points)
(e) Are C and D independent events? Explain. (no explanations no points)
3. 20% of the population has type A blood.
(a) If 8 people are selected at random, what is the probability that at least three of them have type A blood.
(b) If 80 donors come to give blood one day, what is the probability that at least thirty of them have Type A blood (using the normal approximation)? Explain why this is higher or lower than the answer in part (a).
(c) If 20 people come to give blood, what is the probability that at least one of the donors is of Type A? (I’d be impressed if you could solve this in two different ways. . . )
4. The Federal Reserve reports that the mean lifespan of a five dollar bill is 4.9 years. Let’s suppose that the standard deviation is 1.8 years and that the distribution of lifespans is normal (not unreasonable!)
Find: (a) the probability that a $5 bill will last less than 6 years.
(b) the probability that a $5 bill will last between 3 and 5 years.
(c) the 98th percentile for the lifespan of these bills (a time such that 98% of bills last less than that time).
(d ) the probability that a random sample of 35 bills has a mean length of more than 5.5 years.
5. A health insurance company charges policyholders a $1250 annual premium for health insurance for hospitalization. The company estimates that each time a patient is hospitalized costs the company $2500. Furthermore, they have estimated that 85% of patients will not be hospitalized, 10% will be hospitalized once a year, and no one will be hospitalized more than twice.
(a) Find the insurance company’s expected profit per policyholder.
(b) What is the expected profit if they enroll 80,000 policyholders?
6. An airline knows that the mean weight of all pieces of passengers’ luggage is 49.3 lb with a standard deviation of 8.4 lb. What is the probability that the weight of 66 bags in a cargo hold is more than the plane’s total weight capacity of 3,500 lb?
7. The mean length of a movie in 2014 was 122 min (http://www.randalolson.com/2014/01/25/movies-arent-actually-much-longer-than-they-used-to-be/) Suppose that the standard deviation is 11 minutes and that the distribution of movie times is normal.
(A) What is the probability that a randomly selected movie lasts more than 110 minutes?
(B) Find a time such that 95% of all movies last less than that time (the 95th percentile)
(C) A theatre randomly selects 12 movies to show. What is the probability that the mean run time for the 12 movies is less than 125 minutes?
EXTRA CREDIT: Suppose that battery lives are normally distributed with a mean of 11.9 hours and a standard deviation of 1.9 hours. What is the minimum sample size that would be required so that the probability of obtaining a sample mean below 11.6 is less than 2.5%?