Question

For which combination of population and sample size listed below will you find the sampling distribution of the sample mean approximately normally distributed?

a) Population is Right Skewed and n = 10

b) Population is Right Skewed and n = 40

c) Population is Bell Shaped and n = 10

d) B and C only

e) A, B and C

Answer 1

Solution:

The correct choice is d. b & c only.

Reason:

In option c, the statement under consideration is the Population is Bell shaped and the sample size is 10.

It is reasonable and can be easily proved that a population with a normal distribution will have the sample means that are normally distributed irrespective of the sample sizes, i.e. even if the samples are of very small sizes. Now in this choice it is said that the population is Bell-shaped which suggests a population with normal distribution. So the small sample size(=10) does not matter here; the sampling distribution of the sample mean will be normally distributed.

Now, in option b, the statement under consideration is the Population is Right-skewed and the sample size is 40.

The distribution of sample means will be approximately normal as long as the the sample size is large enough which we have also seen in the form of Central Limit Theorem. So evidently, for this it is not necessary that the population distribution has to be normal. So in the choice the distribution is said to be Right-skewed which is a fair assumption while dealing with this in the light of Central Limit Theorem.

Now the question is how large a sample size do we need in order to assume that sample means are normally distributed when the population is not normal, right-skewed in this case? Generally samples of sizes greater than or equal to 30 will have a fairly normal distribution irrespective of the shape of the distribution of the population.

Here in choice b the sample size = 40(>30) but in choice a the sample size = 10(<30). So our assumption of Normal distribution of sample mean is legitimate here in choice b but not in choice a.

Hence d is definitely the correct choice for this question.