A relay microchip in a telecommunications satellite has a life expectancy that follows a normal distribution with a mean of 93 months and a standard deviation of 3.1 months. When this computer-relay microchip malfunctions, the entire satellite is useless. A large London insurance company is going to insure the satellite for 50 million dollars. Assume that the only part of the satellite in question is the microchip. All other components will work indefinitely. (a) For how many months should the satellite be insured to be 98% confident that it will last beyond the insurance date? (Round your answer to the nearest month.) months (b) If the satellite is insured for 84 months, what is the probability that it will malfunction before the insurance coverage ends? (Round your answer to four decimal places.) (c) If the satellite is insured for 84 months, what is the expected loss to the insurance company? (Round your answer to the nearest dollar.) (d) If the insurance company charges $3 million for 84 months of insurance, how much profit does the company expect to make? (Round your answer to the nearest dollar.) Sketch the area under the standard normal curve over the indicated interval and find the specified area. (Round your answer to four decimal places.) The area to the left of z = 0.67 is Sketch the area under the standard normal curve over the indicated interval and find the specified area. (Round your answer to four decimal places.) USE SALT The area between z = -1.35 and z = 2.09 is 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 61% of all customers will take free samples. Furthermore, of those who take the free samples, about 38% 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, 315 customers passed by your counter. (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) = 0.38, while P(sample) = 0.61. (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). Step 1 (a) What is the probability that more than 180 will take your free sample? We are asked to find the probability that more than 180 people of the 315 that walk by will take a free sample in a supermarket. We are told that about 61% of all customers take free samples. First we must test to see if we can use the normal approximation to the binomial distribution. Here we will define success as "someone takes a free sample." In this scenario, the number of trials is n = q=1-p= Since np The probability of success is p = and nq 122.85, we ---Select--- use the normal approximation to the binomial distribution because these values are both greater than 5. which means that the probability of a failure, q, is Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.) P(z 2.78) = USE SALT Shade the corresponding area under the standard normal curve. 04 0.3 0,4 0.3 AA 0.2 0.1 0.2 0.1 Z -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 0.4 0.3 0.2 0.4 0.3 0.2 A A 0.1 0.1 -2 -1 1 2 0-3 -2 -1 1 2 3 Z