9.1 $H_0$ and $H_1$

Goal:

$\bullet$ Introduce the competing hypotheses.

A hypothesis test is the business of making a decision based on a sample, such as the drug example given earlier. A hypothesis test always contains two competing hypotheses, called the null hypothesis $H_{\mathrm{0}}$ and alternative hypothesis $H_{\mathrm{1}}$. Both hypotheses will be statements³⁶³⁶A statement evaluates to true or false. about one or more populations, such that one hypothesis is true, and the other false.

In running a hypothesis test, from the sample data a researcher will compute a test statistic. Then, using the test statistic, the researcher will make one of two decisions: Reject $H_0,$ or fail to reject $H_0.$ The decision of rejecting $H_0$ is made only when the test statistic gives significant evidence against $H_0,$ i.e., the test statistic is highly unlikely to be observed if $H_0$ is true. Failing to reject $H_0$ means the observed test statistic is not highly unusual if $H_0$ is true. You can think of $H_0$ as “current” theory, and $H_1$ as the “new” theory to replace the current. In science, a current theory isn’t abandoned in favor of a new theory unless there is strong evidence to do so.³⁷³⁷Willingness to revise thinking based on new data is vital in all of the academic disciplines, as doing otherwise would be extremely stupid.

For example, using the drug example given earlier, $H_0$ and $H_1$ would be as follows:

The pharmaceutical company should not abandon the current drug unless there is strong evidence that the new drug is better, i.e., $H_0$ should not be rejected unless there is strong evidence to do so.

The choice of hypotheses and what constitutes “significant evidence against $H_0$” are choices made by the researcher (preferably before collecting data). In making a decision, there are two ways in which a good decision can occur:

• •

  $H_0$ is true, and the researcher fails to reject $H_0.$

• •

  $H_0$ is false, and the researcher decides to reject $H_0.$

And, there are two ways for a bad decision to occur:

• •

  $H_0$ is true, and the researcher decides to reject $H_0.$ This is called a Type I error.

• •

  $H_0$ is false, and the researcher fails to reject $H_0.$ This is called a Type II error.

Note that if $H_0$ is true, then it is impossible to make a Type II error; similarly, if $H_0$ is false, then a Type I error cannot occur. Once the researcher chooses the hypotheses, only two of the four outcomes above are possible, e.g., if $H_0$ is true, then either a good decision is made (fail to reject $H_0$) or a Type I error is made (reject $H_0$).