Introduction to Hypothesis Testing

Hypothesis testing is a procedure for deciding whether sample data provide sufficient evidence against a claim about a population parameter.

We frame the problem with two competing claims:

• The null hypothesis $H_0$ specifies a baseline or status-quo value of the parameter. It is the claim we assume to be true unless the data say otherwise.
• The alternative hypothesis $H_1$ is the claim we are seeking evidence for.

For testing a single proportion $p,$ the null hypothesis is

$$H_0: p = p_0$$

for some specific value $p_0,$ and the alternative takes one of three forms:

• Upper-tailed: $H_1: p > p_0$
• Lower-tailed: $H_1: p < p_0$
• Two-tailed: $H_1: p \neq p_0$

The direction of $H_1$ is determined by what the researcher is trying to detect.

Illustrative example: A factory claims its defect rate is $10\%.$ A consumer group suspects the true rate is higher and will sample items to test the claim. The relevant hypotheses are

$$H_0: p = 0.10 \quad \text{vs.} \quad H_1: p > 0.10.$$