Hypothesis Testing Steps

Why This Matters

Hypothesis testing is the backbone of statistical inference—it's how statisticians move from sample data to conclusions about populations. On the AP Statistics exam, you're not just being tested on whether you can plug numbers into formulas. You're being tested on whether you understand the logic of inference: why we assume the null hypothesis is true, how we measure evidence against it, and what our conclusions actually mean in context. Every free-response question involving inference expects you to demonstrate this reasoning process.

The steps of hypothesis testing connect directly to core concepts like probability, sampling distributions, Type I and Type II errors, and the interpretation of p-values. Understanding each step—and why it matters—will help you tackle everything from one-sample z-tests to chi-square tests for independence. Don't just memorize the sequence; know what each step accomplishes and how errors at any stage can invalidate your conclusions.

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Setting Up the Test

Before any calculations happen, you need to establish what you're testing and what would count as convincing evidence. This foundation determines everything that follows—get it wrong, and your entire analysis falls apart.

State the Null and Alternative Hypotheses

• The null hypothesis ($H_0$) represents the "no effect" or "no difference" claim—it's what you assume is true until evidence suggests otherwise
• The alternative hypothesis ($H_a$) captures what you're trying to find evidence for—this determines whether you use a one-sided or two-sided test
• Write hypotheses using population parameters, not sample statistics (use $\mu$, $p$, or $p_1 - p_2$, never $\bar{x}$ or $\hat{p}$)

Choose the Significance Level ($\alpha$)

• The significance level $\alpha$ is your threshold for "surprising enough to reject $H_0$"—typically 0.05 unless otherwise specified
• $\alpha$ equals the probability of a Type I error—rejecting $H_0$ when it's actually true (a false positive)
• Choosing $\alpha$ involves trade-offs: lowering $\alpha$ reduces Type I error risk but increases Type II error risk and decreases power

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Checking Conditions and Selecting Methods

The validity of your inference depends entirely on whether the conditions for the test are met. This is where many students lose points—skipping conditions or stating them without verification.

Select the Appropriate Test Statistic

• Match your test to your data type and question: z-tests for proportions, t-tests for means, chi-square for categorical distributions
• The test statistic measures how far your sample result is from what $H_0$ predicts, standardized by the standard error
• Common formulas include $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ for proportions and $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ for means

Verify Conditions for Inference

• Random condition: data must come from a random sample or randomized experiment—this ensures the sample is representative
• Independence condition (10% rule): sample size must be $\leq 10\%$ of the population when sampling without replacement
• Large counts or normality condition: for proportions, check $np_0 \geq 10$ and $n(1-p_0) \geq 10$; for means, check sample size or population normality

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Calculating and Evaluating Evidence

This is where the math happens—but remember, the calculations serve the reasoning. You're quantifying how surprising your sample result would be if $H_0$ were true.

Calculate the Test Statistic

• Plug your sample data into the appropriate formula to get a standardized value measuring distance from the null
• Show your work clearly: write the formula, substitute values, and compute—this earns process points on FRQs
• The test statistic tells you how many standard errors your sample is from the hypothesized parameter

Determine the P-Value

• The p-value is the probability of getting a result as extreme or more extreme than yours, assuming $H_0$ is true
• Use the appropriate distribution: standard normal (z) for proportions, t-distribution with $df = n-1$ for means, chi-square for categorical tests
• Direction matters: for one-sided tests, find the tail probability; for two-sided tests, double it (or find both tails)

Compare P-Value to $\alpha$

• If p-value $\leq \alpha$, reject $H_0$—the evidence against the null hypothesis is statistically significant
• If p-value $> \alpha$, fail to reject $H_0$—there's insufficient evidence to support the alternative
• Never say "accept $H_0$"—you can only fail to reject it; absence of evidence isn't evidence of absence

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Drawing Conclusions

The final steps connect your statistical results back to the real-world question. This is where you demonstrate understanding—not just calculation ability.

State Your Decision

• Use correct language: "reject $H_0$" or "fail to reject $H_0$"—never "accept" either hypothesis
• Link the decision to your p-value and $\alpha$: "Since p-value = 0.023 < $\alpha$ = 0.05, we reject $H_0$"
• Acknowledge uncertainty: your decision could be wrong—rejecting $H_0$ risks Type I error; failing to reject risks Type II error

Interpret Results in Context

• Write a conclusion in plain language that someone without statistics training could understand
• Reference the alternative hypothesis and the context: "There is convincing evidence that the true proportion of defective items is greater than 0.02"
• Avoid causal language unless the study was a randomized experiment—observational studies only support association claims

Assess Practical Significance

• Statistical significance $\neq$ practical significance—a tiny difference can be "significant" with a large enough sample
• Consider effect sizes and real-world implications: Is the detected difference large enough to matter in practice?
• Discuss limitations: sample size, scope of inference, potential confounding variables, and generalizability

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Quick Reference Table

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Self-Check Questions

1. What is the difference between the null hypothesis and the alternative hypothesis, and why do we assume $H_0$ is true when calculating the p-value?

2. A student writes "We accept $H_0$ because the p-value is 0.12." Identify two errors in this statement and explain how to correct them.

3. Compare and contrast Type I and Type II errors: Which one does $\alpha$ control, and how does increasing sample size affect each?

4. Why must you check conditions before performing a hypothesis test, and what happens to your conclusions if the conditions aren't met?

5. An FRQ presents a hypothesis test with p-value = 0.001 and asks whether the result is practically significant. What additional information would you need to answer this question, and why might statistical significance not imply practical importance?