91Ó°ÊÓ

Triathlon times, Part II. In Exercise 4.4 we saw two distributions for triathlon times: \(N(\mu=4313, \sigma=\) 583 ) for Men, Ages \(30-34\) and \(N(\mu=5261, \sigma=807)\) for the Women, Ages \(25-29\) group. Times are listed in seconds. Use this information to compute each of the following: (a) The cutoff time for the fastest \(5 \%\) of athletes in the men's group, i.e. those who took the shortest \(5 \%\) of time to finish. (b) The cutoff time for the slowest \(10 \%\) of athletes in the women's group.

Spray paint, Part II. As described in Exercise 4.14 , the area that can be painted using a single can of spray paint is slightly variable and follows a nearly normal distribution with a mean of 25 square feet and a standard deviation of 3 square feet. (a) What is the probability that the area covered by a can of spray paint is more than 27 square feet? (b) Suppose you want to spray paint an area of 540 square feet using 20 cans of spray paint. On average, how many square foet must each can be able to cover to spray paint all 540 square feet? (c) What is the probability that you can cover a 540 square feet area using 20 cans of spray paint? (d) If the area covered by a can of spray paint had a slightly skewed distribution, could you still calculate the probabilities in parts (a) and (c) using the normal distribution?

Defective rate. A machine that produces a special type of transistor (a component of computers) has a \(2 \%\) defective rate. The production is considered a random process where each transistor is independent of the others. (a) What is the probability that the \(10^{t h}\) transistor produced is the first. with a defect? (b) What is the probability that the machine produces no defective transistors in a batch of \(100 ?\) (c) On average, how many transistors would you expect to be produced before the first with a defect? What is the standard deviation? (d) Another machine that also produces transistors has a \(5 \%\) defective rate where each transistor is produced independent of the others. On average how many transistors would you expect to be produced with this machine before the first with a defect? What is the standard deviation? (e) Based on your answers to parts (c) and (d), how does increasing the probability of an event affect the mean and standard deviation of the wait time until success?

Bernoulli, the standard deviation. Use the probability rules from Section 3.5 to derive the standard deviation of a Bernoulli random variable, i.e. a random variable \(X\) that takes value 1 with probability \(\mathrm{p}\) and value 0 with probability \(1-p\). That is, compute the square root of the variance of a generic Bernoulli random variable.

Game of dreidel. A dreidel is a four-sided spinning top with the Hebrew letters nun, gimel, hei, and shin, one on each side. Each side is equally likely to come up in a single spin of the dreidel. Suppose you spin a dreidel three times. Calculate the probability of getting (a) at least one nun? (b) exactly 2 nuns? (c) exactly 1 hei? (d) at most 2 gimels?