91Ó°ÊÓ

Sketch the areas under the standard normal curve over the indicated intervals, and find the specified areas. $$ \text { To the right of } z=0 $$

Based on long experience, an airline found that about \(6 \%\) of the people making reservations on a flight from Miami to Denver do not show up for the flight. Suppose the airline overbooks this flight by selling 267 ticket reservations for an airplane with only 255 seats. (a) What is the probability that a person holding a reservation will show up for the flight? (b) Let \(n=267\) represent the number of ticket reservations. Let \(r\) represent the number of people with reservations who show up for the flight. Which expression represents the probability that a seat will be available for everyone who shows up holding a reservation? \(P(255 \leq r) ; \quad P(r \leq 255) ; \quad P(r \leq 267) ; \quad P(r=255)\) (c) Use the normal approximation to the binomial distribution and part (b) to answer the following question: What is the probability that a seat will be available for every person who shows up holding a reservation?

Assume that \(x\) has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities. $$ P(x \geq 90) ; \mu=100 ; \sigma=15 $$

Sketch the areas under the standard normal curve over the indicated intervals, and find the specified areas. $$ \text { To the left of } z=0.72 $$

Find the \(z\) value described and sketch the area described. Find \(z\) such that \(97.5 \%\) of the standard normal curve lies to the left of \(z\).

Find the \(z\) value described and sketch the area described. Find \(z\) such that \(5 \%\) of the standard normal curve lies to the right of \(z\).

Sketch the areas under the standard normal curve over the indicated intervals, and find the specified areas. $$ \text { Between } z=-1.98 \text { and } z=-0.03 $$

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(z \leq-0.13) $$

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(z \leq 1.20) $$

A relay microchip in a telecommunications satellite has a life expectancy that follows a normal distribution with a mean of 90 months and a standard deviation of \(3.7\) 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) Inverse Normal Distribution For how many months should the satellite be insured to be \(99 \%\) confident that it will last beyond the insurance date? (b) If the satellite is insured for 84 months, what is the probability that it will malfunction before the insurance coverage ends? (c) If the satellite is insured for 84 months, what is the expected loss to the insurance company? (d) If the insurance company charges \(\$ 3\) million for 84 months of insurance, how much profit does the company expect to make?