Now let's consider another possible shape for the population distribution of sleep times.
- In the applet below, press the radio button for Pop 2.
- Then follow the instructions in the scroll box
(k) Did the assumed shape of the population distribution of sleep times make a substantial difference in the shape, center, or spread of the distribution of sample means? In the approximate p-value? That is, did it really matter what the shape of the population distribution was? Explain.
As promised by the Central Limit Theorem, as long as the sample size is large enough, the shape of the population doesn't really matter. As long as the sampling distribution of the sample means is approximately normal, then our approximation of the p-value should be similar regardless of the population shape.
This is all fine except when we go to standardize our observation, the quantity 
doesn't quite follow a normal distribution (unless the sample size is pretty large). Now pay attention to the graph in the lower right corner. It's subtle, but you may notice that the "tails" of this distribution are just a bit "heavier" than they would be for a normal distribution, you have a bit more values beyond 2, 3, 4, "standard errors" than you would expect from a normal distribution.
Because we rarely, if ever, know the population standard deviation in a study, with quantitative data we will compute p-values (and confidence intervals) we using information from what is called the t-distribution rather than the normal distribution.
