Let’s Summarize

Hypothesis Tests in General:

Hypothesis tests consist of four steps, which apply to all the hypothesis tests we will do in this course.

Step 1: Determine the hypotheses.

The hypotheses are statements about the parameter(s) in question. The null hypothesis, [latex]H_0[/latex], is always a statement of equality and usually means no change or difference. The alternative hypothesis, [latex]H_a[/latex], is always an inequality, either [latex]<, >, \ \mathrm{or} \ \neq[/latex], and is based on the research question. If the alternative hypothesis contains [latex]> \ \mathrm{or} \ <[/latex], then the test is called a one-sided test. If the alternative hypothesis contains [latex]\neq[/latex], then the test is called a two-sided test.

Step 2: Collect the data.

The data must come from a random sample that is representative of the population in question.

Step 3: Assess the evidence.

The P-value is the evidence. The P-value is the probability that we would get sample results at least as extreme as those observed if the null hypothesis is true. If the P-value is smaller than the significance level (also known as the alpha level), the results are unusual enough for us to reject the null hypothesis. Otherwise, we “fail to reject” the null hypothesis.

• When doing a hypothesis test for a mean, a t-distribution is used when the sample size is small or you are using the sample standard deviation. This is called a t-test for a mean.
• When doing a hypothesis test for a mean, a normal distribution is used when the sample size is large and the population standard deviation is known. This is called a z-test for a mean.
• When doing a hypothesis test for a proportion, a normal distribution is used. However, np and nq must be both greater than 5. This is called a z-test for a proportion.

Step 4: Give the conclusion.

Our conclusion is stated in terms of the alternative hypothesis. Either there is or there is not enough evidence to say that the alternative hypothesis is true. We always use the context of the problem in the conclusion and always include the P-value. Finally, we never say that the null hypothesis is true, only that we reject or fail to reject it.

Hypothesis tests are based on random samples, so the conclusions are really statements about probabilities, and it is possible for the conclusions to be wrong. If our test results in rejecting a null hypothesis that is actually true, it is called a Type I error. If our test results in failing to reject a null hypothesis that is actually false, it is called a Type II error.