19. In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. What are the chances that a person who is murdered actually knew the murderer? The answer to this question explains why a lot of police detective work begins with relatives and friends of the victim! About 64% of people who are murdered actually knew the person who committed the murder. Suppose that a detective file in New Orleans has 65 current unsolved murders. Find the following probabilities. (Round your answers to four decimal places.) (a) at least 35 of the victims knew their murderers

(b) at most 48 of the victims knew their murderers

(c) fewer than 30 victims did not know their murderers

(d) more than 20 victims did not know their murderers 20. In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. Do you take the free samples offered in supermarkets? About 60% of all customers will take free samples. Furthermore, of those who take the free samples, about 33% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 313 customers passed by your counter. (Round your answers to four decimal places.) (a) What is the probability that more than 180 will take your free sample?

(b) What is the probability that fewer than 200 will take your free sample?

(c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.33, while P(sample) = 0.60.

(d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c). 21. In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. What's your favorite ice cream flavor? For people who buy ice cream, the all-time favorite is still vanilla. About 20% of ice cream sales are vanilla. Chocolate accounts for only 7% of ice cream sales. Suppose that 171 customers go to a grocery store in Cheyenne, Wyoming, today to buy ice cream. (Round your answers to four decimal places.) (a) What is the probability that 50 or more will buy vanilla?

(b) What is the probability that 12 or more will buy chocolate?

(c) A customer who buys ice cream is not limited to one container or one flavor. What is the probability that someone who is buying ice cream will buy chocolate or vanilla? Hint: Chocolate flavor and vanilla flavor are not mutually exclusive events. Assume that the choice to buy one flavor is independent of the choice to buy another flavor. Then use the multiplication rule for independent events, together with the addition rule for events that are not mutually exclusive, to compute the requested probability.

(d) What is the probability that between 50 and 60 customers will buy chocolate or vanilla ice cream? Hint: Use the probability of success computed in part (c). 22. (a) Suppose n = 41 and p = 0.40. Can we approximate p by a normal distribution? Why? (Use 2 decimal places.) np= nq= (b) What are the values of p and p? (Use 3 decimal places.)

p = p = (c) (c) Suppose n = 45 and p = 0.35. Can we approximate p by a normal distribution? Why? (Use 2 decimal places.) np= nq= (d) What are the values of p and p? (Use 3 decimal places.)

p= p =