Chapter 7 Probability of selecting a sample with a particular mean: the z-test

If you know both the standard deviation of the population (\(\sigma\)) and the mean of the population (\(\mu\)), you can calculate the probability of obtaining a sample with a particular mean. You did this previously.

To review, you used the population mean and the expected standard error (based on population standard deviation and sample size) to convert the raw mean to a z-score. Then, you used the z-table to compute the proportion of the area under the curve.

We are going to use the same technique to find the probability of selecting a sample of a given size and having its mean fall that far from the population mean. This probability is called p. p is the probability of finding a z-score that is more extreme than your score, assuming that the sample comes out of the population.

We can use p to determine whether a sample is the same as its underlying population. If p is very low, then we are unlikely to have selected this sample by chance. That means that we can conclude our sample is different from the population.