Hypothesis Testing
Hypothesis testing is a statistical method for making decisions about a population based on sample data. You set up two competing hypotheses (null and alternative), collect data, calculate a test statistic, and determine whether the evidence is strong enough to reject the null hypothesis.
Key Takeaways
- The null hypothesis (H₀) represents the status quo — "nothing is happening" or "no difference exists." The alternative hypothesis (Hₐ) is what you're trying to find evidence for.
- The p-value is the probability of getting your observed result (or more extreme) if the null hypothesis were true. A small p-value means the data is unlikely under H₀.
- If p-value < significance level (α, usually 0.05), you reject H₀. If p-value ≥ α, you fail to reject H₀ (you never "accept" H₀).
- Type I error: rejecting H₀ when it's actually true (false positive). Type II error: failing to reject H₀ when it's actually false (false negative).
How Hypothesis Testing Works (Step by Step)
Hypothesis testing is the backbone of statistical inference. Every time you see a medical study claiming "this drug reduces blood pressure" or a headline saying "students who sleep more get better grades," hypothesis testing is the method behind the conclusion.
Here's how it works in four steps:
Step 1: State the hypotheses. The null hypothesis (H₀) is always the "no effect" or "no difference" statement. For example, H₀: μ = 100 (the population mean equals 100). The alternative hypothesis (Hₐ) is what you're investigating: Hₐ: μ ≠ 100 (two-tailed), Hₐ: μ > 100 (right-tailed), or Hₐ: μ < 100 (left-tailed).
Step 2: Check conditions and choose a test. For a one-sample z-test, you need a random sample, independence (n < 10% of population), and either a normal population or large sample size (n ≥ 30 by the Central Limit Theorem). For proportions, check that np ≥ 10 and n(1-p) ≥ 10.
Step 3: Calculate the test statistic and p-value. The test statistic measures how far your sample result is from the null hypothesis value, in standard error units. For a one-sample z-test for means: z = (x̄ - μ₀) / (σ/√n). The p-value is then the probability of getting a test statistic as extreme or more extreme than what you observed.
Step 4: Make a decision. If the p-value is less than your significance level (α), reject H₀ and conclude there is significant evidence for the alternative. If the p-value is greater than or equal to α, fail to reject H₀ — the data doesn't provide enough evidence against the null.
Critical point: "fail to reject" is NOT the same as "accept." You never prove the null hypothesis is true — you only determine whether there's sufficient evidence against it.
Hypothesis Testing on the AP Statistics Exam
Hypothesis testing is worth roughly 20-25% of the AP Statistics exam. It appears in both multiple-choice and free-response sections, and the FRQ typically requires a full 4-step procedure.
The AP graders look for four specific components in your FRQ response: (1) State the hypotheses using correct notation (H₀ and Hₐ with the parameter), (2) Name the test and check conditions, (3) Calculate the test statistic and p-value, and (4) State your conclusion in context — link back to the real-world scenario.
For the conclusion, use this template: "Since the p-value (0.023) is less than α (0.05), we reject H₀. There is significant evidence that [restate alternative in context]." Or: "Since the p-value (0.18) is greater than α (0.05), we fail to reject H₀. There is not sufficient evidence that [alternative in context]."
Common test types on the AP exam: one-sample z-test for proportions, one-sample t-test for means, two-sample z-test for proportions, two-sample t-test for means, chi-square test for independence/goodness-of-fit, and linear regression t-test. Know when to use each one.
Common Mistakes Students Make
- Saying "accept H₀" instead of "fail to reject H₀." This is an automatic deduction on the AP exam. You never prove the null is true — you only determine if there's enough evidence against it.
- Forgetting to check conditions. The AP graders award points for naming the test and verifying conditions (random sample, independence, normality). Skipping this step costs you points.
- Not stating the conclusion in context. Don't just say "reject H₀." Say "There is significant evidence that the mean test score is greater than 70" (or whatever the specific scenario is).
Related Topics
Frequently Asked Questions
A p-value is the probability of observing data as extreme as (or more extreme than) what you actually got, assuming the null hypothesis is true. A small p-value (like 0.01) means your data would be very unlikely if the null were true, which is evidence against the null hypothesis.
A Type I error (false positive) occurs when you reject a true null hypothesis — you conclude there's an effect when there isn't one. A Type II error (false negative) occurs when you fail to reject a false null hypothesis — you miss a real effect. The probability of a Type I error equals your significance level (α).
Use a z-test when you know the population standard deviation (σ) — which is common for proportion tests. Use a t-test when you don't know σ and must estimate it from the sample — which is almost always the case for mean tests. On the AP exam, proportion tests use z, and mean tests use t.