ECON1012 · Data Analytics
Hypothesis Testing
Hypothesis Testing (Module 7, Week 7) is the second core inference procedure in ECON 1012, after estimation: instead of building a range for a parameter, you decide whether the sample gives enough statistical evidence for a belief about it. The module teaches the course's six-component recipe — hypotheses, test statistic, significance level, decision rule, computed value, conclusion — used for every test from here to regression. You learn to write H₀ (which always carries the equality μ = μ₀) against a one- or two-tailed H₁, standardise with Z when σ is known or t with n − 1 degrees of freedom when it is not, and reach the same decision by rejection region or by p-value (reject H₀ when p-value < α). Type I and Type II errors round out the week, and the practice exam's case-study marking asks for every one of the six steps written out.
What this chapter covers
- 01The six components of every test: hypotheses → test statistic → significance level α → decision rule → computed value → conclusion
- 02H₀ always carries the equality (H₀: μ = μ₀); H₁ takes ≠, < or > and bears the burden of proof
- 03Tail choice from the claim: 'differs' → μ ≠ μ₀ (two-tail) · 'more than' → μ > μ₀ · 'less than' → μ < μ₀
- 04Test statistic: Z = (X̄ − μ₀)/(σ/√n) when σ is known; t = (X̄ − μ₀)/(s/√n) with df = n − 1 when only s is available
- 05Rejection regions: two-tail |z₀| > z_α/2 (±1.96 at α = 0.05); one-tail uses z_α (1.645 at α = 0.05)
- 06p-value routes: left-tail p = P(Z < z₀) · right-tail p = P(Z > z₀) · two-tail p = 2·P(Z > |z₀|); reject H₀ when p-value < α
- 07Evidence bands: p < 0.01 overwhelming · 0.01–0.05 strong · 0.05–0.10 weak · > 0.10 no evidence
- 08Type I error = reject a true H₀ (P = α); Type II error = don't reject a false H₀ (P = β); the two are inversely related
A six-step two-tail Z-test, its p-value, and the error check
- 1 mark(a) Step 1 — hypotheses: 'differs' has no direction, so the test is two-tailed. H₀: μ = 55 vs H₁: μ ≠ 55.
- 1 mark(a) Step 2 — test statistic: σ is known, so standardise with Z = (X̄ − μ₀)/(σ/√n).
- 2 marks(a) Steps 3 and 4 — significance level α = 0.05; two-tail decision rule: reject H₀ if z₀ < −1.96 or z₀ > 1.96 (z₀.₀₂₅ = 1.96).
- 2 marks(a) Step 5 — value of the test statistic: standard error σ/√n = 12/√36 = 12/6 = 2, so z₀ = (50.5 − 55)/2 = −4.5/2 = −2.25.
- 2 marks(a) Step 6 — conclusion: since z₀ = −2.25 < −1.96, reject H₀ in favour of H₁. There is sufficient evidence to infer that the mean session length differs from 55 minutes, at a significance level of 5%.
- 2 marks(b) Two-tail p-value = 2·P(Z > |−2.25|) = 2 × (1 − 0.9878) = 2 × 0.0122 = 0.0244. Since 0.0244 < 0.05 the decision matches part (a); at α = 0.01, 0.0244 > 0.01, so H₀ is not rejected — the p-value is the smallest α at which H₀ can be rejected.
- 2 marks(c) The test rejected H₀, so the only mistake it could have made is a Type I error — rejecting a true H₀ — with probability α = 0.05. A Type II error is impossible here: it can occur only when H₀ is not rejected.
Key terms
- Null hypothesis (H₀)
- The statement that the parameter equals the hypothesised value (H₀: μ = μ₀). Equality is always part of H₀, and it is presumed true unless the sample provides sufficient evidence against it.
- Alternative hypothesis (H₁)
- The claim that carries the burden of proof, taking one of three forms: μ ≠ μ₀ (two-tail), μ < μ₀ (left-tail) or μ > μ₀ (right-tail). Its direction fixes where the rejection region sits; the course also writes it H_A.
- Significance level α
- The probability of rejecting the null hypothesis when it is true — i.e. the probability of a Type I error. Chosen before testing (commonly 0.05) and used to set the critical value(s) of the rejection region.
- Rejection region
- The set of test-statistic values that lead to rejecting H₀: z₀ > z_α for a right-tail test, z₀ < −z_α for a left-tail test, |z₀| > z_α/2 for a two-tail test; analogous rules use t critical values with df = n − 1 when σ is unknown.
- p-value
- The evidence against the null hypothesis — the lower the p-value, the stronger the evidence — equivalently the minimum significance level required to reject H₀. Decision rule: reject H₀ when p-value < α.
- Type I and Type II errors
- A Type I error rejects a true H₀ (probability α); a Type II error fails to reject a false H₀ (probability β). For a given sample size the two probabilities are inversely related — lowering α raises β.
Hypothesis Testing FAQ
When do I use t instead of Z in an ECON 1012 hypothesis test?
Ask one question: is the population standard deviation σ known? If σ (or σ²) is given, standardise with Z = (X̄ − μ₀)/(σ/√n). If you only have the sample standard deviation s, use t = (X̄ − μ₀)/(s/√n) with df = n − 1 and read the critical value from the t table — both Z and t tables are provided in the final exam. MCQ options are regularly built on exactly this choice, so make it your first written line.
Why can't I write 'accept H₀' when the test does not reject?
Because failing to reject only means the sample lacked sufficient evidence against H₀ — it does not demonstrate that H₀ is true. H₀ is presumed true from the start and H₁ carries the burden of proof, so the only two conclusions are 'reject H₀ in favour of H₁' or 'do not reject H₀', each paired with a plain-English sentence: 'there is (not) sufficient evidence to infer that …, at a significance level of 5%'.
How do I decide between a one-tailed and a two-tailed test?
Read the direction of the claim. 'Differs', 'changed', 'is not' → two-tail (H₁: μ ≠ μ₀, with α split into two tails of α/2). 'More than', 'increased', 'exceeds' → right-tail (H₁: μ > μ₀). 'Less than', 'decreased', 'fell' → left-tail (H₁: μ < μ₀). The null hypothesis is identical in all three cases — H₀: μ = μ₀, because equality always belongs to H₀ — only the alternative and the rejection region move.
What does the p-value actually tell me?
Two equivalent readings from the module: it measures the evidence against H₀ (the lower, the stronger), and it is the minimum significance level at which H₀ would be rejected — so reject when p-value < α. The course's evidence bands: p < 0.01 overwhelming, 0.01–0.05 strong, 0.05–0.10 weak, above 0.10 no evidence. For a two-tail test remember to double the tail area: p = 2·P(Z > |z₀|).
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Exam move
Marks in this module are won by discipline, not difficulty. Write all six steps every time — the practice exam's case-study marking instructs you to show and explain each one; skipped steps are skipped marks. The two slips that cost most: reaching for Z when σ is unknown (only s given → t with df = n − 1), and standardising with σ instead of the standard error σ/√n. Never write 'accept H₀' — the course wording is 'do not reject', paired with a plain-English sentence stating the significance level. Remember a Type I error is only possible when you reject and a Type II only when you fail to reject, with P(Type I) = α. Finish with the Module 7 quiz on myLearning — the exam MCQs mirror the module quizzes.
