Wiki Page, Math 13 - Perez Sandbox : Module 20 - Introduction to Hypothesis Testing (5d of 5)

Wiki Page
2026-01-17T21:15:31-08:00

In the next section, "Hypothesis Test for a Population Proportion," we learn the details of hypothesis testing for claims about a population proportion. Before we get into the details, we want to step back and think more generally about hypothesis testing. We close our introduction to hypothesis testing with a helpful analogy.

A Courtroom Analogy for Hypothesis Tests

When a defendant stands trial for a crime, he or she is innocent until proven guilty. It is the job of the prosecution to present evidence showing that the defendant is guilty beyond a reasonable doubt. It is the job of the defense to challenge this evidence to establish a reasonable doubt. The jury weighs the evidence and makes a decision.

When a jury makes a decision, it has only two possible verdicts:

Guilty: The jury concludes that there is enough evidence to convict the defendant. The evidence is so strong that there is not a reasonable doubt that the defendant is guilty.

Not Guilty: The jury concludes that there is not enough evidence to conclude beyond a reasonable doubt that the person is guilty. Notice that they do not conclude that the person is innocent. This verdict says only that there is not enough evidence to return a guilty verdict.

How is this example like a hypothesis test?

The null hypothesis is “The person is innocent.” The alternative hypothesis is “The person is guilty.” The evidence is the data. In a courtroom, the person is assumed innocent until proven guilty. In a hypothesis test, we assume the null hypothesis is true until the data proves otherwise.

The two possible verdicts are similar to the two conclusions that are possible in a hypothesis test.

Reject the null hypothesis: When we reject a null hypothesis, we accept the alternative hypothesis. This is like a guilty verdict. The evidence is strong enough for the jury to reject the assumption of innocence. In a hypothesis test, the data is strong enough for us to reject the assumption that the null hypothesis is true.

Fail to reject the null hypothesis: When we fail to reject the null hypothesis, we are delivering a “not guilty” verdict. The jury concludes that the evidence is not strong enough to reject the assumption of innocence, so the evidence is too weak to support a guilty verdict. We conclude the data is not strong enough to reject the null hypothesis, so the data is too weak to accept the alternative hypothesis.

How does the courtroom analogy relate to Type I and Type II errors?

Type I error: The jury convicts an innocent person. By analogy, we reject a true null hypothesis and accept a false alternative hypothesis.

Type II error: The jury says a person is not guilty when he or she really is. By analogy, we fail to reject a null hypothesis that is false. In other words, we do not accept an alternative hypothesis when it is really true.

Mark this page as done, then click the "Next" or > button to continue.

Content by the Open Learning Initiative and licensed under CC BY.