Question

The central limit theorem is important in statistics because it allows us to use a Normal distribution to find probabilities involving the sample mean if the

a. sample size is reasonably large (for any population).

b. population is Normally distributed (for any sample size).

c. population is Normally distributed and the sample size is reasonably large.

d. population is Normally distributed and the population standard deviation is known (for any sample size).

e. population size is reasonably large (whether the population distribution is known or not).

Short answer

(a) The central limit theorem is important in statistics because it allows us to use a Normal distribution to find probabilities involving the sample mean if the sample size is reasonably large (for any population).

Step-by-step solution

Given Information

The central limit theorem is significant in statistics because it allows us to find probabilities involving the sample mean using a Normal distribution.

Explanation for correct option

According to the central limit theorem, if the sample size is big, the sampling distribution of the sample mean $\overline{x}$is approximately normal.

As a result, the best solution is (a)

 Step 3: Explanation for incorrect option

b. population is Normally distributed (for any sample size) is not the answer.

c. population is Normally distributed and the sample size is reasonably large is not the answer.

d. population is Normally distributed and the population standard deviation is known (for any sample size) is not the answer.

e. population size is reasonably large (whether the population distribution is known or not) is not the answer.