Writing Null and Alternative Hypotheses and Interpreting Test Decisions

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What is the null hypothesis, $H_0$?

A claim about a population, often representing no difference or a status quo. It includes equality through $=$, $\leq$, or $\geq$.

What is the alternative hypothesis, $H_a$?

The claim that contradicts $H_0$ and expresses the difference, direction, or effect being investigated. It uses $\neq$, $<$, or $>$.

How do you choose the direction of $H_a$?

Translate the research question: “different” suggests $\neq$, “less than” suggests $<$, and “more than” suggests $>$. The null uses the complementary relationship and includes equality.

What does “do not reject $H_0$” mean?

The sample evidence was not sufficient to reject the null hypothesis. It does not mean that $H_0$ has been proven true.

How are the p-value and $\alpha$ used in a decision?

$\alpha$ is selected before the test, and the p-value is calculated from sample data. Reject $H_0$ if the p-value is less than or equal to $\alpha$; otherwise, do not reject it.

Quiz questions

1. What do the null and alternative hypotheses describe?Answer: Competing claims about a population parameter.
2. A study asks whether the average college GPA differs from 2.0. Which hypotheses express this question?Answer: $H_0: \mu = 2.0$ and $H_a: \mu \neq 2.0$
3. A study tests whether more than 6.6% of students take advanced placement exams. Which hypotheses match this claim?Answer: $H_0: p \leq 0.066$ and $H_a: p > 0.066$
4. A hypothesis test uses $\alpha = 0.05$ and has a p-value of $0.05$. What decision follows the stated rule?Answer: Reject $H_0$ because the p-value is equal to $\alpha$.
5. A hypothesis test does not reject $H_0$. What does this decision mean?Answer: The sample evidence was insufficient to reject $H_0$; it does not prove that $H_0$ is true.