Hypothesis Testing

Formulation and Testing of Statistical Hypotheses

Structure of a Hypothesis Test

1. State hypotheses:
2. $H_0$: null hypothesis (conservative, equality claim)
3. $H_1$: alternative hypothesis (what you are testing for)
4. Choose significance level $\alpha$ (usually 0.05 or 0.01)
5. Compute test statistic from the data
6. Find the p-value (probability of observing data this extreme if $H_0$ is true)
7. Make a decision: reject $H_0$ if p-value $< \alpha$
8. State conclusion in context

Types of Hypotheses

Test for Population Mean ($\sigma$ known)

$$z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

Example 1: A factory claims average fill is 500 mL ($\sigma = 10$ mL). A sample of $n=36$ gives $\bar{x} = 497$ mL. Test at $\alpha = 0.05$ (two-tailed).

$H_0: \mu = 500$; $H_1: \mu \neq 500$.

$$z = \frac{497 - 500}{10/\sqrt{36}} = \frac{-3}{10/6} = \frac{-3}{1.667} \approx -1.80$$

p-value $= 2P(Z < -1.80) = 2 \times 0.0359 = 0.0718 > 0.05$.

Conclusion: Fail to reject $H_0$. Insufficient evidence that the mean fill differs from 500 mL at the 5% significance level.

Example 2: Same setting, but $\bar{x} = 495$ mL.

$$z = \frac{495-500}{10/6} = \frac{-5}{1.667} \approx -3.00$$

p-value $= 2\times P(Z < -3) = 2\times0.00135 = 0.0027 < 0.05$.

Conclusion: Reject $H_0$. There is strong evidence the mean fill is not 500 mL.

Test for Population Proportion

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

Example 3: A candidate claims 60% approval. A poll of $n=400$ finds $\hat{p} = 0.56$. Test $H_0: p = 0.6$ vs $H_1: p < 0.6$ at $\alpha = 0.05$.

$$z = \frac{0.56 - 0.60}{\sqrt{0.6\times0.4/400}} = \frac{-0.04}{0.0245} \approx -1.63$$

p-value $= P(Z < -1.63) \approx 0.052 > 0.05$. Fail to reject $H_0$.

Significance Level vs p-value

• Significance level $\alpha$: set before collecting data; the threshold below which we reject $H_0$.
• p-value: calculated from data; describes the strength of evidence against $H_0$.

Type I and Type II Errors

KEY TAKEAWAY: A hypothesis test quantifies the strength of evidence against the null hypothesis using the p-value. Rejecting $H_0$ means the data is inconsistent with $H_0$ at the chosen significance level.

EXAM TIP: Always state your conclusion in context, referring to the specific claim and the significance level. “We reject $H_0$” is incomplete — say what this means for the problem.

COMMON MISTAKE: Confusing “fail to reject $H_0$” with “accept $H_0$.” Failing to reject does not prove $H_0$ is true; it only means there is insufficient evidence against it.