Question

Power +, Inc. produces AA batteries used in remote-controlled toy cars. The mean life of these batteries follows a normal distribution with a mean of 36 hours and a standard deviation of 5.3 hours. As a part of its quality assurance program, Power +, Inc. tests samples of 9 batteries. a. What can you say about the shape of the distribution of the sample mean? Sample mean Round answers to four decimal places. b. What is the standard error of the distribution of the sample mean? Note: The standard error is the standard deviation of the distribution of the sample mean. That is, the standard error is the population standard deviation divided by the square root of the sample size. Standard error 0.7667 c. What proportion of the samples will have a mean useful life of more than 37 hours? Probability d. What is the probability that a randomly selected AA battery will have a useful life of at most 34 hours? Probability e. What proportion of the samples will have a mean useful life between 34 and 37 hours? Probability

Answer 1

Solution: It is given:

a. What can you say about the shape of the distribution of the sample mean?

Answer: The shape of the distribution of the sample mean will be normal distribution because the population from which the sample is chosen is also normal.

b. What is the standard error of the distribution of the sample mean?

c. What proportion of the samples will have a mean useful life of more than 37 hours?

Answer: It is required to find:

Now using the z-score formula, we have:

Now using the standard normal table, we have:

Therefore, the proportion of the samples that will have a mean useful life of more than 37 hours is 0.2857

d. What is the probability that a randomly selected AA battery will have a useful life of at most 34 hours?

Answer: We are required to find:

Using the z-score formula, we have:

Now using the standard normal table, we have:

Therefore, the probability that a randomly selected AA battery will have a useful life of at most 34 hours is 0.3531

e. What proportion of the samples will have a mean useful life between 34 and 37 hours?

Answer: It is required to find:

Now using the z-score formula, we have:

Now using the standard normal table, we have:

Therefore, the proportion of the samples that will have a mean useful life between 34 and 37 hours is 0.5855