Why is the central limit theorem important in statistics? a) Because for a large sample size n, it says the population is approximately normal. b) Because for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population. c) Because for a large sample size n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population. d) Because for any sample size n, it says the sampling distribution of the sample mean is approximately normal. e) none of the above

According to the Central Limit Theorem if we have a population with a known mean and standard deviation and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

Then, the mean of the distribution of sample means is given by, the population mean.

And the standard deviation of the distribution of sample means is given by,

[tex]SD_{\bar x}=\frac{SD}{\sqrt{n}}[/tex]

So, the most basic and main objective of the Central limit theorem is to approximate the sampling distribution of a statistic by the Normal distribution even when we do not known the distribution of the population.

Thus, the correct option is (c).