Elementary Statistics – Central Limit Theorem and Linear Combination of Normals Examples

Example 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.

Assuming the manufacturer’s claim is true, what is the approximate probability of observing a sample mean of 52 or less? Does this make you doubt their claim?

Example 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?

Example 3

Two Lawrence students enter a math competition. 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. 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.

Karen and Mikah each take the exam separately, but they are considered a team. Because Karen is a senior, her score is weighted slightly more than Mikah’s score. Their team score is T = 0.6X + 0.4Y.

Karen and Mikah receive a trophy if their team score is greater than 80. What is the probability they get a trophy?

Example 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?