Okay, on my AP Statistics test on the chapter on significance tests, the following free response question appeared:
When the manufacturing process is working properly, NeverREady batteries have lifetimes that follow a slightly right-skewed distribution with μ = 7 hours. A quality control supervisor selects the simple random sample of n batteries every hour and measures the lifetime of each. If she is convinced that the mean lifetime of all batteries produced that hour is less than 7 hours at the 5% significance level, then all of those batteries are discarded.
(b) Since testing the lifetime of a battery requires draining the battery completely, the supervisor wants to sample as few batteries as possible from each hour's production. She is considering a sample size of n = 4. Explain why this sample size may lead to problems in carrying out the significance test of (a).
Now, (a) was just defining the parameter of interest (μ = the true mean lifetime (in hours) of NeverReady batteries) and labeling the null (μ = 7) & alternative (μ < 7) hypotheses.
I answered that it would result in a lower power of the significance test.
Now, why was this the incorrect answer?
Is it because it was already stated above that the population distribution was slightly right skewed and therefore not Normal?
In addition, when I searched the question up online and I found 3 answer keys with the answer to the problem: 2 of them state it's because of the lower power, and 1 of them states that its because the population distribution is neither large (n = 4 < 30) nor Normal (because its slightly right-skewed).
Which one is right? Is it the Normal/Large Sample condition or is it the lower power that may cause problems when doing a significance test? I'm assuming the former but only 1/3 answer keys would have the correct answer in that case. It seems to me that both answers are problems a small sample size would have on a significance test.