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 
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.
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