What's the Difference Between the Null Hypothesis and Alternative Hypothesis?

The null hypothesis, $H_0$, states that there is no effect, difference, association, or change in the population. The alternative hypothesis, $H_1$, states the competing claim that an effect, difference, association, or change exists.

How the Two Hypotheses Work

A hypothesis test begins by treating $H_0$ as the default position. Sample data are then used to calculate a p-value, which measures how likely a result at least as extreme as the observed result would be if $H_0$ were true.

The p-value is compared with the significance level, $\alpha$:

For example, suppose a school tests whether two populations have different mean journey times:

$$H_0:\mu_A=\mu_B$$ $$H_1:\mu_A\neq\mu_B$$

If the test gives $p=0.032$ at the $5\%$ significance level, then $0.032<0.05$. Therefore, reject $H_0$; there is sufficient evidence that the population mean journey times differ.

The form of $H_1$ determines whether the test is one-tailed or two-tailed. A claim of “different” uses $\neq$, while directional claims such as “greater than” or “less than” use $>$ or $<$. The equality belongs in , because the null hypothesis defines the reference model.

For a chi-squared test of independence, the hypotheses are usually written in words:

• $H_0$: the two categorical variables are independent.
• $H_1$: the two categorical variables are associated.

A common misconception is to say that $H_0$ has been “proved false” or “accepted.” A hypothesis test provides evidence, not proof. If the result is not significant, write “there is insufficient evidence to reject $H_0$,” not “accept $H_0$.”

IB Exam Technique

This is IB Mathematics: Applications and Interpretation SL 4.11 and may appear in GDC-active Paper 1 or Paper 2. State both hypotheses before testing, compare the p-value explicitly with $\alpha$, and finish with a conclusion in context; a calculator result alone does not complete the test.