In Section 3.3, we introduced hypothesis testing, a formal procedure to evaluate a statement about a population or populations. In particular, we tested hypotheses involving two population means using a permutation test. This test makes no assumptions about the distributions underlying the two populations. We now consider hypothesis testing in situations where we can make some assumptions about the distribution of a population or populations.
The underlying procedure for analyzing a hypothesis test remains the same as before: we compute a test statistic from the data, and then compute a P-value, measuring how often chance alone would give a test statistic as extreme as the observed statistic, assuming the null hypothesis is true. If the P-value is small, then the results cannot easily be explained by chance alone, and we reject the null hypothesis.
To compute the P-value, we need to find a reference distribution, the null distribution—the distribution that the test statistic follows if the null hypothesis is true. We then compute a one-sided P-value using that distribution. For a two-sided test, we multiply the smaller of the one-sided P-values by two.
Example 8.1 For college-bound seniors in 2009, SAT math scores are normally distributed with a mean of 515 and a standard deviation of 116. You suspect that seniors in the town of Sodor are much brighter than the country ...