1. Prof. Miller ran a marathon. Given that the mean finishing time for all runners was 259 minutes with a standard deviation of 46 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 23rd percentile (23% of finishers were under that time)
(c) the percent of runners who finished between 3 hours and 4 hours
(d) a random sample of 32 runners got a special vitamin supplement right before the race. Their mean time was 4 hr 2 min. What is the probability that a group of 32 runners chosen at random would have a mean time of 4 hr 2 minutes or less?
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 explanationsno points)
(e) Are C and D independent events? Explain. (no explanationsno points)
3. The mean length of a movie in 2014 was 121 min (https://www.randalolson.com/2014/01/25/movies-arent-actually-much-longer-than-they-used-to-be/) Suppose that the standard deviation is 16 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 only 5% of all movies last more than that time.
(C) A theatre randomly selects 14 movies to show. What is the probability that the mean run time for the 14 movies is more than 130 minutes?
4. Suppose that 30% of the births in the US are done by Cesarean section.
(a) If 8 births are selected at random, what is the probability that less than four of them have are Cesareans.
(b) If 80 births occur in New York in one day, what is the probability that less than forty of them are Cesareans (using the normal approximation)? Explain why this is higher or lower than the answer in part (a).
(c) If 20 births occur at a hospital in one month, what is the probability that at least one of the births is a Cesarean (I'd be impressed if you could solve this in two different ways. . . )
4. 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?
5. 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 1.7 years and that the distribution of lifespans is normal (not unreasonable!)
Find: (a) the probability that a $5 bill will last less than 3 years.