Complete question is;

A manufacturer of cell phone batteries claims that the average number of recharge cycles for its batteries is 400. A consumer group will obtain a random sample of 100 manufacturer's batteries and will calculate the mean number of recharge cycles. Which of the following statements is justified by the central limit theorem?

The Central Limit Theorem tells us that for large sample sizes (in practice n > 30) the sampling distribution of the sample mean will be approximately normal.

A. The distribution of the number of recharge cycles for the sample is approximately normal because the population mean of 400 is greater than 30.

B. The distribution of the number of recharge cycles for the sample is approximately normal because the sample size of 100 is greater than 30.

C. The distribution of the number of recharge cycles for the population is approximately normal because the sample size of 100 is greater than 30.

D. The distribution of the sample means of the number of recharge cycles is approximately normal because the sample size of 100 is greater than 30.

E. The distribution of the sample means of the number of recharge cycles is approximately normal because the population mean of 400 is greater than 30.

Answer:

D. The distribution of the sample means of the number of recharge cycles is approximately normal because the sample size of 100 is greater than 30.

Step-by-step explanation:

We are told that the Central Limit Theorem tells us that for large sample sizes (in practice n > 30) the sampling distribution of the sample mean will be approximately normal. Thus we can pick the correct option as follows:

Option A: This is not the correct answer because it is the sample size (n) that should be greater than 30 and not the population mean as stated.

Option B: This is not the correct answer because the distribution of the sample should correspond to the population shape for large samples because it's only the sampling distribution that is approximately normal.

Option C: This option is not correct because the sample is not meant to have any impact on the population.

Option D: This option is the correct answer because it fulfills the condition of the central limit theorem earlier described.

Option E: This is not the correct answer because just like in option A, it is the sample size (n) that should be greater than 30 and not the population mean as stated.