Introduction to Statistics

Chapter 10: Hypothesis Tests

Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Sections 10.1–10.4

Chapter Summary

Chapter 9 estimated parameters with intervals; this chapter tests claims about them. A hypothesis test starts with a claim — “the mean battery life is 500 hours,” “fewer than 30% of voters support the measure” — and asks whether the sample data contradict it strongly enough to reject it. The null hypothesis (H₀) is the status-quo statement of “no effect / no difference,” and the alternative (H₁) is what you’re looking for evidence of. You assume H₀ is true, then compute how surprising your sample would be under that assumption. That surprise is measured by the P-value: the probability of getting results at least as extreme as yours if H₀ were true. A tiny P-value means your data would be shocking under H₀ — so you reject it.

You’ll run tests for a population mean (z when σ is known, t when unknown), a population proportion, and a population variance/standard deviation (using the chi-square distribution). Two kinds of mistakes frame everything: a Type I error (rejecting a true H₀ — a false alarm) and a Type II error (failing to reject a false H₀ — a missed detection). Realistic example: a cereal company claims boxes contain 500 g on average. A sample of 40 boxes averages 497 g with s = 6 g. The t-test gives a P-value around 0.003 — far below α = 0.05 — so you’d reject the claim and conclude the mean fill is below 500 g. Notice the discipline: we never “prove” H₁ or “accept” H₀; we only reject or fail to reject, because absence of evidence isn’t evidence of absence.

Study Guide

Key terms and definitions

  • Null hypothesis (H₀) — the statement of no effect, no difference, or status quo (always contains =).
  • Alternative hypothesis (H₁) — the claim to be supported by evidence; determines the test direction: left-tailed (<), right-tailed (>), or two-tailed (≠).
  • Test statistic — how many standard errors the sample estimate is from the hypothesized value.
  • P-value — the probability of observing results at least as extreme as the sample’s, assuming H₀ is true. Small P-value → evidence against H₀.
  • Level of significance (α) — the cutoff for “small enough” (commonly 0.05). Reject H₀ when P-value ≤ α.
  • Type I error — rejecting a true H₀; P(Type I) = α. Type II error — failing to reject a false H₀; P(Type II) = β.
  • Power — 1 − β; the probability of correctly rejecting a false H₀.
  • Practical vs. statistical significance — a result can be statistically significant (tiny P-value from a huge sample) without mattering in practice.

Essential formulas

  • Test for μ, σ known: z₀ = (x̄ − μ₀) ÷ (σ/√n).
  • Test for μ, σ unknown: t₀ = (x̄ − μ₀) ÷ (s/√n), df = n − 1.
  • Test for p: z₀ = (p̂ − p₀) ÷ √[p₀(1 − p₀)/n]. Requirement: np₀(1 − p₀) ≥ 10.
  • Test for σ: χ²₀ = (n − 1)s² ÷ σ₀², df = n − 1 (requires a normal population).
  • Decision rule: reject H₀ if P-value ≤ α; otherwise do not reject H₀.

Step-by-step procedures

  • Run a hypothesis test: (1) state H₀ and H₁ and the significance level; (2) verify requirements (random sample, normality conditions); (3) compute the test statistic; (4) find the P-value; (5) decide: reject H₀ if P-value ≤ α; (6) state the conclusion in context (“There is sufficient evidence to conclude the mean fill is less than 500 g”).
  • Read a P-value: sketch the distribution, shade the tail(s) at least as extreme as your test statistic — two-tailed tests double the one-tail area.

Common mistakes to avoid

  • Saying “accept H₀” — the correct phrase is “do not reject H₀” (or “fail to reject”).
  • Interpreting the P-value as P(H₀ is true). It’s P(data this extreme | H₀ true) — a statement about the data, not the hypothesis.
  • Choosing a one-tailed test after seeing the data. The direction comes from the research question, decided in advance.
  • Confusing statistical significance with practical importance — always ask “how big is the effect?”

Self-check: I can…

  • [ ] State null and alternative hypotheses and identify the test direction.
  • [ ] Explain what a P-value measures and use it to make a decision.
  • [ ] Perform z- and t-tests for a population mean.
  • [ ] Perform a test for a population proportion.
  • [ ] Perform a chi-square test for a population variance/standard deviation.
  • [ ] Define Type I and Type II errors and explain power.
  • [ ] State a conclusion in plain, contextual language.

StatCrunch How-To

  • Test for a mean, σ known: Stat > Z Stats > One Sample > With Data or With Summary > select “Hypothesis test for μ”, enter H₀ value, choose the H₁ direction, enter α > Compute. Read the test statistic and P-value.
  • Test for a mean, σ unknown: Stat > T Stats > One Sample > same choices as above.
  • Test for a proportion: Stat > Proportion Stats > One Sample > With Data or With Summary > “Hypothesis test for p” > enter p₀, direction, α > Compute.
  • Test for a variance/SD: Stat > Variance Stats > One Sample > With Data or With Summary > “Hypothesis test” > Compute.
  • Tip: the output plot shades the P-value region — use it to double-check your test direction.

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