Hypothesis Testing for Means and Proportions

Hypothesis Testing: Means and Proportions in Depth

Full Procedure

1. Define population parameter ($\mu$ or $p$).
2. State $H_0$ (equality) and $H_1$ (inequality, direction).
3. Choose $\alpha$ (significance level).
4. Compute test statistic.
5. Find p-value (using $z$ or $t$ table/calculator).
6. Compare p-value with $\alpha$: if p $< \alpha$, reject $H_0$.
7. State conclusion in context.

Test for Mean ($\sigma$ known)

$$z_{\text{obs}} = \frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}} \sim N(0,1) \text{ under } H_0$$

Example 1: A machine fills bags. Claimed $\mu = 500$ g, $\sigma = 8$ g. Sample: $n=64$, $\bar{x} = 498$ g. Test at $\alpha=0.05$ (two-tailed).

$z = \dfrac{498-500}{8/8} = \dfrac{-2}{1} = -2$.

p-value $= 2P(Z < -2) = 2(0.0228) = 0.0456 < 0.05$.

Reject $H_0$. There is evidence the mean fill is not 500 g.

Test for Proportion

$$z_{\text{obs}} = \frac{\hat{p}-p_0}{\sqrt{p_0(1-p_0)/n}}$$

Example 2: A coin is tossed 200 times: 115 heads. Test $H_0: p=0.5$ vs $H_1: p > 0.5$ at $\alpha=0.05$.

$\hat{p} = 0.575$. $z = \dfrac{0.575-0.5}{\sqrt{0.25/200}} = \dfrac{0.075}{0.0354} \approx 2.12$.

p-value $= P(Z > 2.12) = 1 - 0.9830 = 0.0170 < 0.05$.

Reject $H_0$. Sufficient evidence that $p > 0.5$ at the 5% level.

Interpreting p-values

The p-value is the probability of obtaining a test statistic at least as extreme as the observed value, under the assumption that $H_0$ is true.

• p-value is not the probability that $H_0$ is true.
• A small p-value indicates that the data would be unusual if $H_0$ were true.

p-value and significance level:
- $p < \alpha$: Data is inconsistent with $H_0$ at significance level $\alpha$ $\Rightarrow$ reject $H_0$
- $p \geq \alpha$: Insufficient evidence to reject $H_0$

Two-Tailed vs One-Tailed Tests

Practical Significance vs Statistical Significance

A statistically significant result (small p-value) does not necessarily mean the effect is large or practically important.

With very large $n$, even tiny differences from $\mu_0$ become statistically significant.

Always comment on the size of the observed difference in context.

KEY TAKEAWAY: A hypothesis test quantifies evidence against the null hypothesis. The p-value and significance level together determine whether the null hypothesis is rejected.

EXAM TIP: When concluding, say “There is sufficient evidence at the $\alpha$% significance level to conclude…” or “There is insufficient evidence…”. Never say “we accept $H_0$”.

COMMON MISTAKE: Using a one-tailed test when the direction of the difference was determined by looking at the data (data snooping). The direction in $H_1$ must be specified before seeing the data.