Question

Let ?̅ = sample mean in a SRS of ? objects from a large population. For each of the following, find ?(?̅), ???(?̅), ??(?̅) and indicate whether a normal distribution would be a good approximation to the shape.
a. severely right-skewed population, ? = 183, ? = 106, ? = 35
b. normal population, ? = 515, ? = 116, ? = 50
c. symmetric, unimodal population, ? = 10, ? = 3, ? = 14
d. mildly left-skewed population, ? = −16, ? = 22, ? = 100

Answer 1

There's a small ambiguity that it is not mentioned whether we are to determine that the normal distribution is a good approximation to the shape of the distribution of mean or the population. So, for that I will answering for both the cases.

Using the above relations the values are:

a.

The population is severly right skewed, so the Normal distribution is definitly not suitable for the population. For the distribution of sample mean, note that 35 can somtimes be considered a large sample, and due to CLT, the sample mean would be asymptotically normal. But since the distribution is severly right skewed, maybe a sample size of 35 would not be a very suitable choice. So, I would suggest that the normal distribution is not a suitable choice for the distribution of mean either.

b.

The population is normal, thus, the sample mean also follows normal. Hence, a normal approximation is suitable for both the population distribution and the distribution of the sample mean.

c.

The population distribution is symmetric, unimodal and has finite second moment. Also, it's standard deviation (which is 3) matches with that of the standard normal. So, we can say that the normal distribution is a good approximation for the population distribution, and hence also for the distribution of the sample mean.

d.

The population is mildly left-skewed, so the choice of whether a normal distribution is a good approximation for it, is subjective. The population has finite second moment. So, if there's only a slight departure from symmetry, we can say that the normal approximation holds good, but in case the departure is apparent and visible, the normal distribution fails to be a good approximation. However, for all the cases, the distribution of the sample mean can be approximated by a normal curve. It is so, since we have a large sample (n=100) and thus, the CLT holds.