10.1 Mechanics of a hypothesis test

This structure is consistent for hypothesis testing even through the specifics of the hypotheis being tested and the test statistics being calculated will differ between types of tests.

  1. Set up the null and alternative hypotheses in words and notation.
    • In words: ``The population mean for [what is being studied] is different from [value of \(\mu_0\)].’’ (Note that the statement in words is in terms of the alternative hypothesis.)
    • In notation: \(H_0: \mu=\mu_0\) versus \(H_A: \mu \ne \mu_0\) (where \(\mu_0\) is specified by the context of the problem).
  2. Choose the significance level of the test, such as \(\alpha=0.05\).

  3. Compute the test statistic, such as \(t_{s} = \frac{\bar{Y}-\mu_0}{SE_{\bar{Y}}}\), where \(SE_{\bar{Y}}=s/\sqrt{n}\) is the standard error.

  4. Determine the tail(s) of the sampling distribution where the \(p\)-value from the test statistic will be calculated (for example, both tails, right tail, or left tail). (Historically, we would compare the observed test statistic, \(t_{s}\), with the critical value \(t_{\textrm{crit}}=t_{\alpha/2}\) in the direction of the alternative hypothesis from the \(t\)-distribution table with degrees of freedom \(df = n-1\).)

  5. State the conclusion in terms of the problem.
    • Reject \(H_0\) in favor of \(H_A\) if \(p\textrm{-value} < \alpha\).
    • Fail to reject \(H_0\) if \(p\textrm{-value} \ge \alpha\). (Note: We DO NOT accept \(H_0\).)
  6. Check assumptions of the test (next week).