9.6 Chapter Summary

A hypothesis test consists of two competing hypotheses, $H_0$ and $H_1.$

The null hypothesis, $H_0$, is generally the “current” theory; the alternative hypothesis, $H_1$, is the “new” theory.

A hypothesis test results in a decision being made between two options: Reject $H_0,$ or fail to eject $H_0.$ The decision to reject $H_0$ is made only if there is strong evidence against $H_0.$

A Type I error occurs when $H_0$ is true, but the decision is to reject $H_0.$ The chance of making a Type I error is called the level of significance and is denoted by $\alpha .$

A Type II error occurs when $H_0$ is false, but the decision is to fail to reject $H_0.$ The chance of making a Type II error is denoted by $\beta .$

The direction of extreme identifies those values that give stronger evidence against $H_0.$ The direction of extreme is to the left if smaller values give stronger evidence against $H_0;$ it is to the right if larger values give stronger evidence against $H_0;$ it is two sided if smaller and larger values give stronger evidence against $H_0.$

In doing a hypothesis test, from the sample data a test statistic is computed. A decision rule defines when the null hypothesis will be rejected, and is phrased as “if the test statistic is equal to the critical value or is more extreme, then $H_0$ is rejected.” A decision rule is equivalent to picking a value for $\alpha .$

A $p$-value is the probability of observing the test statistic, or anything more extreme, assuming $H_0$ is true. If the $p$-value is $\le \alpha ,$ then $H_0$ is rejected.