Elementary Statistics – Central Limit Theorem and Linear Combination of Normal-Distribution Variables

Problem 1

A manufacturer of car batteries claims that the distribution of the life spans of its best battery has mean 54 months and standard deviation 6 months. A consumer group purchases a random sample of 50 batteries and tests the battery life spans. The sample mean lifespan is 52 months.

a. Assuming the manufacturer’s claim is true, what is the approximate probability of observing a sample mean of 52 months or less?

b. Does your answer to part a make you doubt the manufacturer’s claim? (This foreshadows how we’ll use the central limit theorem to do inference—namely, significance testing.)

c. For part a, Bubba determined the probability to be 0.3707 (based on a z-value of -0.33).What exactly did Bubba do wrong?

Problem 2

An elevator has a capacity of 30 people. The distribution of the weights of elevator passengers (and whatever they are carrying) has a mean of 168 pounds and a standard deviation of 10 pounds. Consider rides when the elevator is full with a random sample of people. What weight capacity should be listed on the elevator, so there’s only a 0.01 chance the elevator is overloaded? (This example is a bit contrived, but I wanted to refresh your memory on reverse-look-up problems.)

The question asks for the total weight that should be written on the elevator. The textbook does not discuss the distribution of a total. You have two options for solving this problem:

· Use what you know about the sample average weight (and its distribution) to solve the problem. How would you do this?

· The total weight (since n is reasonably large), like the average weight, has an approximate normal distribution. But the mean and standard deviation for the total weight are different. Let denote the total weight (I know the notation is grungy, but it’s just notation). Then the mean and standard deviation of the total weight are and . You can use these results to directly solve this problem (without working with the sample average). Use this method and verify that you get the same answer as your previous method. (From now on, you can choose the solution method that feels most authentic to you.)

Problem 3

Two Lawrence students enter a math competition (woo-hoo!). Each student must take an exam. Let the random variable X be Karen’s exam score. Suppose that X has a normal distribution with mean 75 points and standard deviation 5 points. (There is variability in Karen’s score, based on, for example, how much she studies, how she feels, the types of questions that are asked.) Let the random variable Y be Mikah’s exam score, and suppose that Y has a normal distribution with mean 69 points and standard deviation 4 points. Furthermore, suppose Karen’s score is independent of Mikah’s score (perhaps strangely, they don’t study together).

Karen and Mikah each take the exam separately, but they are considered a team. Their individual scores are added to determine their team score. That is, their team score is T = X + Y. Karen and Mikah receive new iPods if their team score is greater than 160 points. What is the probability they get iPods?

Problem 4

A lab technician regularly performs two different experiments, and there is variation in the time required for each experiment. The time for the first experiment follows an approximate normal distribution with mean 43.1 minutes and standard deviation 8.6 minutes. The time for the second experiment follows an approximate normal distribution with mean 50.2 minutes and standard deviation 10.1 minutes. Furthermore, the times for the two experiments are independent. What is the probability the second experiment takes longer than the first?