Hypothesis testing is a formal procedure for deciding whether sample data provides sufficient evidence to reject a claim about a population parameter.
Example: Testing whether a coin is fair:
$$H_0: p = 0.5 \quad \text{(fair coin)} \qquad H_a: p \neq 0.5 \quad \text{(biased coin)}$$
The significance level $\alpha$ is the threshold probability for rejecting $H_0$.
For a test of population proportion:
$$z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}$$
For a test of population mean (known $\sigma$):
$$z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$
The p-value is the probability of obtaining a sample result at least as extreme as observed, assuming $H_0$ is true.
| p-value | Decision |
|---|---|
| $p < \alpha$ | Reject $H_0$ — evidence supports $H_a$ |
| $p \geq \alpha$ | Do not reject $H_0$ — insufficient evidence |
A cereal manufacturer claims each packet weighs 500 g on average. A consumer group samples 36 packets: $\bar{x} = 495$ g, $\sigma = 12$ g.
$H_0: \mu = 500$, $H_a: \mu < 500$, $\alpha = 0.05$.
$$z = \frac{495 - 500}{12/\sqrt{36}} = \frac{-5}{2} = -2.5$$
$p$-value for $z = -2.5$ in a one-tailed test $\approx 0.006$.
Since \$0.006 < 0.05$, reject $H_0$. There is significant evidence the mean weight is less than 500 g.
REMEMBER: Rejecting $H_0$ does not prove $H_a$ is true — it means the evidence is inconsistent with $H_0$ at the chosen significance level.
EXAM TIP: Always state hypotheses symbolically ($H_0$ and $H_a$) and in words. Write a conclusion sentence that references the context, significance level, and decision.
