6.5 Hypothesis Tests

A hypothesis test is a formal statistical method for deciding whether there is enough evidence in a sample to support a particular claim about a population. Think of it like a courtroom: you start by assuming the person is innocent (no change, no effect), and you only change your mind if the evidence is strong enough.

1. Set up the null and alternative hypotheses.
2. Choose a significance level.
3. Collect data using a random sampling method (data must be independent).
4. Carry out the test — do the calculations.
5. Interpret the result in the context of the original problem.

The null hypothesis is the starting assumption — the "nothing has changed" position. It always contains an equals sign. You assume H₀ is true unless the data gives you enough evidence to reject it.

The alternative hypothesis is the claim you are trying to find evidence for. It represents the idea that something has changed or that a specific effect exists.

The test statistic is the value you calculate from your sample data. You use it to decide whether to reject H₀. For example, it might be the number of heads in 20 coin tosses, or the sample mean from a group of measurements.

The significance level (written as α, pronounced "alpha") is the probability threshold you set before the test. It is the maximum probability of wrongly rejecting H₀ that you are willing to accept. Common values are 5% (0.05), 1% (0.01), and 10% (0.10).

• A small significance level (e.g., 1%) means you need very strong evidence before rejecting H₀.
• A larger significance level (e.g., 10%) makes it easier to reject H₀, but you risk being wrong more often.

The rejection region (also called the critical region) is the set of values of the test statistic for which you would reject H₀. If your observed value falls in this region, you reject the null hypothesis.

The acceptance region is the set of values for which you do not reject H₀. If the test statistic falls here, there is not enough evidence to reject H₀.