Wiki Page, Math 13 - Perez Sandbox : Module 20 - Wrap Up "Hypothesis Testing"-2

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2026-01-17T21:15:32-08:00

Let’s Summarize

In this section, we introduced the four-step process of hypothesis testing:

Step 1: Determine the hypotheses.

The hypotheses are claims about the population(s).

The null hypothesis is a hypothesis that the parameter equals a specific value.

The alternative hypothesis is the competing claim that the parameter is less than, greater than, or not equal to the parameter value in the null. The claim that drives the statistical investigation is usually found in the alternative hypothesis.

Step 2: Collect the data.

Because the hypothesis test is based on probability, random selection or assignment is essential in data production.

Step 3: Assess the evidence.

Use the data to find a P-value.

The P-value is a probability statement about how unlikely the data is if the null hypothesis is true.

More specifically, the P-value gives the probability of sample results at least as extreme as the data if the null hypothesis is true.

Step 4: Give the conclusion.

A small P-value says the data is unlikely to occur if the null hypothesis is true. We therefore conclude that the null hypothesis is probably not true and that the alternative hypothesis is true instead.

We often choose a significance level as a benchmark for judging if the P-value is small enough. If the P-value is less than or equal to the significance level, we reject the null hypothesis and accept the alternative hypothesis instead.

If the P-value is greater than the significance level, we say we “fail to reject” the null hypothesis. We never say that we “accept” the null hypothesis. We just say that we don’t have enough evidence to reject it. This is equivalent to saying we don’t have enough evidence to support the alternative hypothesis.

Our conclusion will respond to the research question, so we often state the conclusion in terms of the alternative hypothesis.

Inference is based on probability, so there is always uncertainty. Although we may have strong evidence against it, the null hypothesis may still be true. If this is the case, we have a Type I error. Similarly, even if we fail to reject the null hypothesis, it does not mean the alternative hypothesis is false. In this case, we have a Type II error. These errors are not the result of a mistake in conducting the hypothesis test. They occur because of random chance.

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