Question

A large number of samples each of size 15 taken from a normally population will have a mean that is best approximated by a:

		Normal distribution
		T distribution with 15 degrees of freedom
		T distribution with 14 degrees of freedom
		Chi-square distribution with 14 degrees of freedom

Answer 1

Given that a large number of samples each of size 15 is taken from a normally distributed population, we should find the best approximation of sample mean.

Now the central limit theorem states that if we have a population with mean and standard deviation and if we take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true even when the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30). If the population is normal, then the theorem holds true even for samples smaller than 30.

Thus in order for the result of the CLT to hold, the sample must be sufficiently large . There are two exceptions to this: If the population is normal, then the result holds for samples of any size (i..e, the sampling distribution of the sample means will be approximately normal even for samples of size less than 30).

Let ​ be the random samples each of size 15 drawn from a population with mean and standard deviation ​.

Since it is specified that the random samples of size 15 is drawn from normal population, using central limit theorem explained above, the distribution of the sample means will be approximately normally distributed.

For the random samples we took from the population, the mean of the sample means is:

  

and the standard deviation of the sample means is:

  

Thus the distribution of mean of samples each of size 15 approximately follows ​.