Hypothesis Testing

Formulation and Testing of Hypotheses

The Hypothesis Testing Framework

Hypothesis testing is a formal procedure for deciding whether sample data provides sufficient evidence to reject a claim about a population parameter.

Step 1: State the Hypotheses

Null hypothesis $H_0$: the default claim (assumed true until evidence otherwise)
Alternative hypothesis $H_a$ (or $H_1$): the claim to be tested

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)}$\$

Step 2: Choose a Significance Level

The significance level $\alpha$ is the threshold probability for rejecting $H_0$.

Common choice: $\alpha = 0.05$ (5%)

Step 3: Calculate the Test Statistic

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 $\sigma$):

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

Step 4: Find the p-value

The p-value is the probability of obtaining a sample result at least as extreme as observed, assuming $H_0$ is true.

Small p-value (e.g. $p < 0.05$): strong evidence against $H_0$
Large p-value (e.g. $p > 0.05$): insufficient evidence to reject $H_0$

Step 5: Make a Decision

p-value	Decision
$p < \alpha$	Reject $H_0$ — evidence supports $H_a$
$p \geq \alpha$	Do not reject $H_0$ — insufficient evidence

Worked Example

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.

p-value	Decision
\(p < \alpha\)	Reject \(H_0\) — evidence supports \(H_a\)
\(p \geq \alpha\)	Do not reject \(H_0\) — insufficient evidence

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