Q.4 - The Practice of Statistics for AP

Question

The central limit theorem is important in statistics because it allows us to use the Normal distribution to make inferences concerning the population mean 

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

(b) if the population is normally distributed and the sample size is reasonably large. 

(c) if the population is normally distributed (for any sample size). 

(d) if the population is normally distributed and the population variance is known (for any sample size). 

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

Step-by-Step Solution

Answer

The correct answer is (a) if the sample size is reasonably large (for any population). 

Step 1 Given Information

The central limit theorem is important in statistics because it allows us to use the Normal distribution to make inferences concerning the population mean 

Step 2 Explanation

If the population distribution is not Normal, the central limit theorem (CLT) states that when $n$ is large, the sampling distribution of x is approximately Normal. 

We know that according to the central limit theorem if the sample size of a sampling distribution is $30$ or more; then the sampling distribution which has a sample mean $\overline{x}$is approximately Normal.

Hence, the correct answer is (a).