Question

Suppose we are investigating the oxygen-levels at the summits of mountains in the states of New York, Vermont, New Hampshire, and Maine. Assume there are about 5,000 summits in consideration. We have no reason to suspect that the oxygen levels are normally distributed, but they are somewhat symmetrically distributed. If we measure the oxygen levels at a sample of 130 summits, explain what the Central Limit Theorem can do for us and where the sampling distribution comes in.

Answer 1

Solution:

We know that according to the central limit theorem, the sampling distribution of the sample means is approximately normally distributed when the sample size is sufficiently large (> 30), even though the population from which the sample is drawn is not normally distributed.

Since in given situation, we don't know whether the oxygen levels are normally distributed or not but they are somewhat symmetrically distributed and the sample size under consideration is 130, therefore, according to the central limit theorem, the distribution of the sample means will be approximately normally distributed with mean and standard deviation given below: