Appropriate Sample Size with little data

Question

What assumptions should I make to be able to answer this question? Without knowing any further data, how would I go about answering this statistics question?

I thought the rule of thumb based on the Central Limit Theorem n greater than = 30.

Answer 1

One must assume that the $n$ employees, however many, are selected at random. Sometimes in practice, large samples are not as carefully drawn as larger ones. But if we assume samples of all sizes are carefully chosen, then the largest feasible $n$ is best. So I would say use (d) $n = 50.$

In this particular case, the sample size $n = 50$ is more than 10% of the population size $N = 300.$ So if 50 are selected it is best to use the 'finite population correction' in making a confidence interval or doing a formal statistical test. For maximum information, one would want to be sure to take the sample without replacement; that is, make sure the same person is not interviewed more than once.

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Note: The sample size $n = 30$ is often quoted as a threshold for various desirable properties in statistics. I don't think it should be used as a rule of thumb for assuming the sample mean is normally distributed according to the Central Limit Theorem. If the population distribution is strongly skewed (as for reaction times or waiting times), then it may take more than 50 obserevations to get nearly normally distributed $\bar X.$ If the population distribution is already nearly normal in shape, 10 may suffice.

The histograms below show results of repeated experiments with two kinds of distributions:

At left, the experiment is to roll $n=10$ fair dice and take the mean of the spots showing per die. The distribution for each die is symmetrical. The fit to normal is quite good.

At right, the experiment is to test batches of $n=50$ electronic devices noting the mean time to failure in each batch. The distribution for each component is severely right-skewed. The fit to normal is a a little off, the distribution of sample means is a little skewed to the right.