26134 · Responsible Evidence-based Decisions
Testing Ideas I: Hypothesis-Testing Foundations
Week 8 opens Module 3 with the one-sample hypothesis test, the template every later test reuses. You set null and alternative hypotheses, compute a test statistic, and decide using either a rejection region or a p-value against a significance level α, while distinguishing Type I and Type II errors. A one-sample z- or t-test with a stated decision is a reliable exam item.
What this chapter covers
- 01Null hypothesis H₀ (contains the equality) vs alternative H₁ (two-, left- or right-tailed)
- 02Test statistic as a standardised distance from the H₀ value
- 03Significance level α and the rejection region beyond the critical value
- 04p-value: probability of a statistic at least as extreme as observed if H₀ is true; reject if p ≤ α
- 05One-tailed vs two-tailed tests (α vs α/2 in each tail)
- 06z-test for a mean, σ known: z = (x̄ − μ₀)/(σ/√n)
- 07t-test for a mean, σ unknown: t = (x̄ − μ₀)/(s/√n), df = n − 1
- 08Type I error (reject a true H₀, prob α) vs Type II error (fail to reject a false H₀, prob β); power = 1 − β
One-sample z-test for a mean (σ known)
- +1Hypotheses and test. H₀: μ = 5.0 versus H₁: μ ≠ 5.0 (two-tailed). σ is known, so use a z-test.
- +1Standard error. SE = σ/√n = 1.2/√36 = 1.2/6 = 0.2 minutes.
- +1Test statistic. z = (x̄ − μ₀)/SE = (5.4 − 5.0)/0.2 = 0.4/0.2 = 2.0.
- +1Decision. Critical values are ±1.96. Since |z| = 2.0 > 1.96, reject H₀. Equivalently the two-tailed p-value = 2 × (1 − Φ(2.0)) = 2 × 0.0228 = 0.0456 < 0.05. There is evidence at the 5% level that mean handling time differs from 5.0 minutes.
Key terms
- Null vs alternative hypothesis
- H₀ is the status-quo claim and always contains the equality (e.g. μ = 5); H₁ is what you seek evidence for, and is two-tailed (≠), left-tailed (<) or right-tailed (>). You either reject H₀ or fail to reject it — you never 'accept' it.
- Test statistic
- A standardised distance of the sample estimate from the value assumed under H₀, such as z = (x̄ − μ₀)/(σ/√n) or the t-equivalent. You compare it with a critical value or convert it to a p-value.
- Significance level (α)
- The probability of rejecting a true null hypothesis (a Type I error), fixed before the test — commonly 0.05, 0.01 or 0.10. It sets the size of the rejection region and the p-value threshold.
- p-value
- The probability, assuming H₀ is true, of observing a test statistic at least as extreme as the one obtained. Reject H₀ when p ≤ α. A small p-value means the data would be unusual if H₀ held.
- Rejection region
- The set of test-statistic values beyond the critical value(s) that lead to rejecting H₀. For a two-tailed test at α it is split into α/2 in each tail; for a one-tailed test all α sits in one tail.
- Type I vs Type II error
- A Type I error rejects a true H₀ (probability α); a Type II error fails to reject a false H₀ (probability β). Power = 1 − β is the chance of correctly detecting a real effect. Lowering α raises β unless n increases.
Testing Ideas I: Hypothesis-Testing Foundations FAQ
How do I decide between a one-tailed and two-tailed test?
It comes from the alternative hypothesis. If H₁ is 'differs from' (≠), use a two-tailed test and split α into α/2 in each tail. If H₁ is directional — 'greater than' (>) or 'less than' (<) — use a one-tailed test with all α in that one tail. The claim wording tells you which.
What is the difference between the rejection-region and p-value methods?
They are two routes to the same decision. The rejection-region method compares the test statistic to a critical value from the table; the p-value method computes how extreme the statistic is and compares p to α. Reject H₀ if the statistic is in the rejection region, or equivalently if p ≤ α — they always agree.
What do Type I and Type II errors mean in plain terms?
A Type I error is a false alarm: rejecting a null that is actually true (probability α). A Type II error is a miss: failing to detect a real effect, so not rejecting a false null (probability β). You control α directly; reducing β without raising α usually requires a larger sample.
Can AI help me with hypothesis testing in 26134?
Yes, as a study aid. Sia can help you state H₀ and H₁, choose z or t, compute the test statistic, and decide with either a critical value or a p-value, step by step, while checking your reasoning. Use it to rehearse the method; it does not do your graded quizzes or exam, and the UTS academic-integrity policy applies.
Exam move
Learn the one-sample test as a fixed six-step template, because Week 9 just swaps the statistic: state H₀ and H₁, choose z (σ known) or t (σ unknown), compute the standard error, compute the test statistic, find the critical value or p-value, then decide and interpret. Practise both decision routes on the same problem so they agree — that cross-check catches arithmetic slips. Be careful with tails: H₀ holds the equality and the tail follows H₁. Rehearse writing the plain-English conclusion, since it earns a mark. Keep the z- and t-test formulas and the common critical values on your printed exam notes. When tails or the reject/fail-to-reject logic confuse you, ask Sia to walk a fresh test with you; confirm assessment details on Canvas.
Working through Testing Ideas I: Hypothesis-Testing Foundations in 26134? Sia is AskSia’s AI Statistics tutor — ask any 26134 Testing Ideas I: Hypothesis-Testing Foundations question and get a clear, step-by-step explanation grounded in how 26134 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
