Exam 7: Normal Curves and Sampling Distributions

A relay microchip in a telecommunications satellite has a life expectancy that follows a normal distribution with a mean of 70 months and a standard deviation of 3.1 month. 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. If the satellite is insured for 71 months, what is the expected loss to the insurance company? Round to the nearest ten-thousand dollars.

What is a sampling distribution?

Assuming that the heights of college women are normally distributed with mean 67 inches and standard deviation 3 inches, what percentage of women are taller than 58 inches?

Suppose an airline found that about 3% of the people making reservations on a flight from Miami to Denver do not show up for the flight. Suppose that the airline overbooks this flight by selling 222 ticket reservations for an airplane with only 216 seats. Let n = 222 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: Expression I: $\quad P ( r \geq 216 )$ Expression II: $\quad P ( r \geq 222 )$ Expression III : $\quad P ( r \leq 216 )$ Expression $T : P ( r \leq 222 )$ Expression $\mathrm { V } : \quad P ( r = 222 )$

Suppose that the average life of a refrigerator before replacement is $\mu$ = 14 years with a (99.7% of data) range from 8 to 20 years. Let x = age at which a refrigerator is replaced. Assume that x has a distribution that is approximately normal. Find a good approximation for the standard deviation of x values.

Assume that about 30% of all U.S. adults try to pad their insurance claims. Suppose that you are the director of an insurance adjustment office. Your office has just received 140 insurance claims to be processed in the next few days. What is the probability that fewer than 38 of the claims have been padded?

A person's level of blood glucose and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean $\mu$ = 80 and standard deviation of $\sigma$ = 30 What is the probability that, for an adult after a 12-hour fast, x is less than 62?

Assume that x has a normal distribution, with the specified mean and standard deviation. Find the indicated probabilities. P(x $\le$ 24) $\mu$ = 28; $\sigma$ = 5

Let x = red blood cell (RBC) count in millions per cubic millimeter of whole blood. Suppose for healthy females, x has an approximately normal distribution with mean $\mu$ = 4.9 and standard deviation $\sigma$ =0.5 Convert the following x interval from a laboratory test to a z interval. 6.9 < x

The heights of 18-year-old men are approximately normally distributed with mean 68 inches and standard deviation 3 inches. What is the probability that the height of an 18-year-old man selected at random is between 70 inches and 71 inches?

Suppose that x has a distribution with $\mu$ = 26 and $\sigma$ =6. If a random sample is taken of size n = 37, find $\sigma$ _x.

Find the area under the standard normal curve over the interval specified below. To the right of z =2

Assume that 53% of all customers will take free samples. Furthermore, of those who take the free samples, assume that about 32% will buy what they have sampled. Suppose that you set up a counter in a supermarket offering free samples of a new product and that the day you were offering free samples, 403 customers passed by your counter. 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 we are given the conditional probability P(buy, given sample) = 0.32, while P(sample) = 0.53.]

A person's level of blood glucose and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean $\mu$ = 78 and standard deviation of $\sigma$ = 25. What is the probability that, for an adult after a 12-hour fast, x is more than 49?

Let x = red blood cell (RBC) count in millions per cubic millimeter of whole blood. Suppose for healthy females, x has an approximately normal distribution with mean $\mu$ = 4.3 and standard deviation $\sigma$ = 0.8 Convert the following x interval from a laboratory test to a z interval. 1.1 > x

A person's level of blood glucose and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean $\mu$ = 71 and standard deviation of $\sigma$ = 24. What is the probability that, for an adult after a 12-hour fast, x is between 35 and 110?

Shade the area under the standard normal curve corresponding to the probability below. P(-1 $\le$ z $\le$ 0)

Find z such that 40.3% of the standard normal curve lies between -z and z.

A certain company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a battery is normally distributed, with a mean of 44 months and a standard deviation of 7 months. If the company does not want to make refunds for more than 10% of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)?

Suppose that x has a distribution with $\mu$ = 21 and $\sigma$ = 5. If a random sample is taken of size n = 58, find . $\mu _ { x } ^ { - }$