⚖️ Hypothesis Testing Framework

Setting Up Hypotheses

Null Hypothesis ($H_0$):

• Statement of "no effect," "no difference," or "no change"
• What we assume true unless evidence suggests otherwise
• Always uses = sign

Alternative Hypothesis ($H_a$):

• Statement we're testing for
• What we'd conclude if $H_0$ is rejected
• Can be one-sided (<, >) or two-sided (≠)

Types of Hypotheses: One-Sided vs Two-Sided

Two-sided test (most common initially):

• $H_0: p = 0.5$ vs $H_a: p \neq 0.5$
• Tests for any difference (either direction)
• Uses both tails of distribution

One-sided test (left):

• $H_0: p = 0.5$ vs $H_a: p < 0.5$
• Tests if parameter less than null value
• Uses left tail only

One-sided test (right):

• $H_0: p = 0.5$ vs $H_a: p > 0.5$
• Tests if parameter greater than null value
• Uses right tail only

Test Statistic

The test statistic measures how far the sample statistic is from the null value, in standard errors.

For proportions (z-test): $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

For means (t-test): $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

p-Value

The p-value is:

• Probability of observing test statistic as extreme or more extreme, given $H_0$ is true
• Measures evidence against $H_0$
• Smaller p-value → stronger evidence against $H_0$

Interpretation:

• p-value = 0.03 means: If $H_0$ were true, we'd see results this extreme 3% of the time

Significance Level (α)

The significance level is the threshold for rejecting $H_0$.

Common choices:

• α = 0.05 (most common; 5% risk of Type I error)
• α = 0.01 (more stringent; 1% risk)
• α = 0.10 (less stringent; 10% risk)

Decision Rule

Compare p-value to α:

• If p-value < α: Reject $H_0$ (statistically significant; evidence for $H_a$)
• If p-value ≥ α: Fail to reject $H_0$ (not enough evidence)

Worked Example

Claim: A coin is fair. Test at α = 0.05.

• $H_0: p = 0.5$ (fair coin)
• $H_a: p \neq 0.5$ (unfair coin, two-sided)
• Flip 100 times, get 62 heads

Calculate test statistic: $z = \frac{0.62 - 0.5}{\sqrt{\frac{0.5 \cdot 0.5}{100}}} = \frac{0.12}{\sqrt{0.0025}} = \frac{0.12}{0.05} = 2.4$

Find p-value: For z = 2.4 (two-sided): p-value ≈ 0.0164

Decision: p-value (0.0164) < α (0.05) → Reject $H_0$

Conclusion: There is significant evidence that the coin is not fair.

Conclusion in Context

Always state conclusion in terms of original problem:

• ✅ "At the 5% significance level, there is sufficient evidence that the mean GPA has increased."
• ❌ "We reject the null hypothesis."

Include context; address the original claim.

Common Mistakes

1. Confusing p-value with probability of $H_0$: p-value is conditional on $H_0$ being true
2. Stating wrong hypotheses: $H_a$ should match the research question
3. One-sided vs two-sided: Determine direction before collecting data
4. Ignoring assumptions: Check independence, randomness, and normality

AP Exam Tip

Free-response hypothesis test questions follow a four-step format:

1. State: $H_0$ and $H_a$ (or state conditions and parameter)
2. Plan: Name the test and check conditions
3. Do: Calculate test statistic and p-value
4. Conclude: Decision and interpretation in context

Show all work and use appropriate notation.