91Ó°ÊÓ

Area under the curve, Part II. What percent of a standard normal distribution \(N(\mu=0, \sigma=1)\) is found in each region? Be sure to draw a graph. (a) \(Z>-1.13\) (b) \(Z<0.18\) (c) \(Z>8\) (d) \(|Z|<0.5\)

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.

CAPM. The Capital Asset Pricing Model (CAPM) is a financial model that assumes returns on a portfolio are normally distributed. Suppose a portfolio has an average annual return of \(14.7 \%\) (i.e. an average gain of \(14.7 \%\) ) with a standard deviation of \(33 \%\). A return of \(0 \%\) means the value of the portfolio doesn't change, a negative return means that the portfolio loses money, and a positive return means that the portfolio gains money. (a) What percent of years does this portfolio lose money, i.e. have a return less than \(0 \% ?\) (b) What is the cutoff for the highest \(15 \%\) of annual returns with this portfolio?

Find the SD. Cholesterol levels for women aged 20 to 34 follow an approximately normal distribution with mean 185 milligrams per deciliter \((\mathrm{mg} / \mathrm{dl})\). Women with cholesterol levels above \(220 \mathrm{mg} / \mathrm{dl}\) are considered to have high cholesterol and about \(18.5 \%\) of women fall into this category. What is the standard deviation of the distribution of cholesterol levels for women aged 20 to \(34 ?\)

Chicken pox, Part I. The National Vaccine Information Center estimates that \(90 \%\) of Americans have had chickenpox by the time they reach adulthood. \({ }^{32}\) (a) Suppose we take a random sample of 100 American adults. Is the use of the binomial distribution appropriate for calculating the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood? Explain. (b) Calculate the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood. (c) What is the probability that exactly 3 out of a new sample of 100 American adults have not had chickenpox in their childhood? (d) What is the probability that at least 1 out of 10 randomly sampled American adults have had chickenpox? (e) What is the probability that at most 3 out of 10 randomly sampled American adults have not had chickenpox?

Chickenpox, Part II. We learned in Exercise 4.18 that about \(90 \%\) of American adults had chickenpox before adulthood. We now consider a random sample of 120 American adults. (a) How many people in this sample would you expect to have had chickenpox in their childhood? And with what standard deviation? (b) Would you be surprised if there were 105 people who have had chickenpox in their childhood? (c) What is the probability that 105 or fewer people in this sample have had chickenpox in their childhood? How does this probability relate to your answer to part (b)?

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?

Eye color, Part II. Exercise 4.13 introduces a husband and wife with brown eyes who have 0.75 probability of having children with brown eyes, 0.125 probability of having children with blue eyes, and 0.125 probability of having children with green eyes. (a) What is the probability that their first child will have green eyes and the second will not? (b) What is the probability that exactly one of their two children will have green eyes? (c) If they have six children, what is the probability that exactly two will have green eyes? (d) If they have six children, what is the probability that at least one will have green eyes? (e) What is the probability that the first green eyed child will be the \(4^{\text {th }}\) child? (f) Would it be considered unusual if only 2 out of their 6 children had brown eyes?

Sickle cell anemia. Sickle cell anemia is a genetic blood disorder where red blood cells lose their flexibility and assume an abnormal, rigid, "sickle" shape, which results in a risk of various complications. If both parents are carriers of the disease, then a child has a \(25 \%\) chance of having the disease, \(50 \%\) chance of being a carrier, and \(25 \%\) chance of neither having the disease nor being a carrier. If two parents who are carriers of the disease have 3 children, what is the probability that (a) two will have the disease? (b) none will have the disease? (c) at least one will neither have the disease nor be a carrier? (d) the first child with the disease will the be \(3^{r d}\) child?

Exploring permutations. The formula for the number of ways to arrange \(n\) objects is \(n !=n \times(n-\) 1) \(\times \cdots \times 2 \times 1\). This exercise walks you through the derivation of this formula for a couple of special cases. A small company has five employees: Anna, Ben, Carl, Damian, and Eddy. There are five parking spots in a row at the company, none of which are assigned, and each day the employees pull into a random parking spot. That is, all possible orderings of the cars in the row of spots are equally likely. (a) On a given day, what is the probability that the employees park in alphabetical order? (b) If the alphabetical order has an equal chance of occurring relative to all other possible orderings, how many ways must there be to arrange the five cars? (c) Now consider a sample of 8 employees instead. How many possible ways are there to order these 8 employees' cars?