IB Maths AI 4.7 Notes

Hypothesis testing

In this section, we study statistical hypothesis testing. The aim is to decide whether sample evidence is strong enough to support a claim about a population. We will focus on writing hypotheses, interpreting significance levels and $p$-values, using the $\chi^2$ test, and using the $t$-test.

A hypothesis is a statement about a population parameter or about a relationship in a population.

• The null hypothesis is written as $H_0$. This is the default claim, usually stating that there is no difference, no effect, no association, or that a parameter has a stated value.
• The alternative hypothesis is written as $H_1$. This is the claim we investigate if the sample evidence is strong enough against $H_0$.

For example, if we want to test whether a coin is fair, we could write:

• $H_0$: the coin is fair
• $H_1$: the coin is not fair

If we are comparing two population means, we might write:

• $H_0:\mu_1=\mu_2$
• $H_1:\mu_1\ne\mu_2$

For a one-tailed test, the alternative may be directional, such as

• $H_1:\mu_1\gt \mu_2$
• $H_1:\mu_1\lt \mu_2$.

When testing any hypothesis, a significance level needs to be established. This is the cut-off used to decide whether the evidence against $H_0$ is strong enough. It is usually denoted by $\alpha$. Common significance levels are $10\%$, $5\%$, and $1\%$.

If the result is significant at the $5\%$ level, this means there is a less than 5% probability that the observed result would be due to random chance if $H_0$ were true. At this stage, we say that $H_0$ is rejected.

The p-value measures how extreme the sample result is, assuming that $H_0$ is true. A small $p$-value means the observed result is unlikely under the null hypothesis, so there is evidence against $H_0$.

Decision rule:

• if $p\le\alpha$, reject $H_0$
• if $p\gt \alpha$, do not reject $H_0$

It is important to say 'do not reject $H_0$' rather than 'accept $H_0$', because the test does not prove that $H_0$ is true.