Question

The height of women in the US is normally distributed with mean (mu) = 65 inches and standard deviation (sigma) = 2.5 inches. A random sample of 15 women is chosen from all women in the US. Is the sampling distribution of the sample ( x - bar) mean normally distributed? Why?

	A.	No because the standard deviation is too small
	B.	No because x < 30
	C.	Yes. Because x < 30
	D.	Yes, because the original x distribution was normally distributed

Answer 1

According to central limit theorem,

i) If distribution of random variable x is normal then the sampling distribution of sample mean is also normally distributed.

ii) If sample size n is large (n > 30) then the sampling distribution of sample mean is approximately normally distributed irrespective of the distribution of random variable x.

Here random variable X is height of women in the US.

X is normally distributed.

So, the sampling distribution of the sample mean is normally distributed.

Option D is correct.