Significance Testing
Before we actually work our way through inference for means and proportions, we need to review the
hypothesis testing procedure introduced in Chapter 11 and to understand how questions involving
inference will be scored on the AP exam. In the previous chapter, we identified the steps in the
hypothesis-testing procedure as follows:
I. State hypotheses in the context of the problem. The first hypothesis, the null hypothesis , is the
hypothesis we are actually testing. The null hypothesis usually states that there is nothing going on: the
claim is correct or there is no distinction between groups. It is symbolized by H 0 .
The second hypothesis, the alternative hypothesis , is the theory that the researcher wants to confirmby rejecting the null hypothesis. The alternative hypothesis is symbolized by H (^) A . There are three
forms for the alternative hypothesis: ≠, >, or <. That is, if the null is H 0 : μ 1 – μ 2 = 0, then H (^) A could
be:
(In the case of the one-sided alternative H (^) A : μ 1 – μ 2 > 0, the null hypothesis is sometimes
written H 0 : μ 1 – μ 2 ≤ 0.)
II. Identify which procedure you intend to use and show that the conditions for its use are satisfied.
If you are going to state a significance level, α, it can be done here.
III. Compute the value of the test statistic and the P -value.
IV. Using the value of the test statistic and/or the P -value, give a conclusion in the context of the
problem.
If you stated a significance level, the conclusion can be based on a comparison of the P -value
with α. If you didn’t state a significance level, you can argue your conclusion based on the relative
size of the P -value alone: if it is small, you have evidence against the null hypothesis; if it is not
small, you do not have evidence against the null hypothesis.
The conclusion can be (1) that we reject H 0 (because of a sufficiently small P -value) or (2) that
we do not reject H 0 (because the P -value is too large). We never accept the null hypothesis: we
either reject it or fail to reject it. If we reject H 0 , we can say that we have evidence in favor of H (^) A.
Significance testing involves making a decision about whether or not an observed result is
statistically significant. That is, is the result sufficiently unlikely, if the null hypothesis were true, so
as to provide good evidence for rejecting the null hypothesis in favor of the alternative? The four
steps in the hypothesis testing process outlined above are the four steps that are required on the AP
exam when doing inference problems. In brief, every test of a hypothesis should have the following
four steps:
I. State the null and alternative hypotheses in the context of the problem, defining all symbols.
II. Identify the appropriate test and check that the conditions for its use are present.