BUSS1020 · Quantitative Business Analysis
Hypothesis Testing: One-Sample Tests
Hypothesis Testing: One-Sample Tests (Week 8, Berenson Ch 9) gives you a formal procedure for deciding whether sample evidence contradicts a claim about a population. You set up the null and alternative hypotheses, choose a significance level α, compute a test statistic for a single mean (Z if σ known, t if unknown) or a single proportion, and decide using the critical-value or p-value approach. You also reason about Type I and Type II errors and power, and you apply the ethics of honest statistical reporting.
What this chapter covers
- 01Null H₀ and alternative H₁; one- vs two-tailed tests
- 02Significance level α; rejection region
- 03Type I error (α), Type II error (β), and power = 1 − β
- 04Test for a mean, σ known: Z = (X̄ − μ₀)/(σ/√n)
- 05Test for a mean, σ unknown: t = (X̄ − μ₀)/(S/√n), df = n − 1
- 06Test for a proportion: Z = (p − π₀)/√[π₀(1−π₀)/n]
- 07Critical-value approach vs p-value approach
- 08Ethics: report all tests, avoid p-hacking, statistical vs practical significance
One-sample test for a proportion
- 1 markSample proportion p = 102/300 = 0.34.
- 2 marksHypotheses (one-tailed, right): H₀: π = 0.30 versus H₁: π > 0.30.
- 1 markCheck CLT conditions: nπ₀ = 300 × 0.30 = 90 ≥ 5 and n(1−π₀) = 300 × 0.70 = 210 ≥ 5 — both satisfied.
- 1 markStandard error under H₀ = √[π₀(1−π₀)/n] = √[(0.30 × 0.70)/300] = √(0.21/300) = √0.0007 ≈ 0.02646.
- 2 marksTest statistic Z = (p − π₀)/SE = (0.34 − 0.30)/0.02646 = 0.04/0.02646 ≈ 1.51.
- 1 markCritical value for a right-tailed test at α = 0.05 is 1.645. Since 1.51 < 1.645, do not reject H₀: there is insufficient evidence the click-through rate has increased.
Key terms
- Null hypothesis H₀
- The default claim about a parameter that is assumed true unless the data provide strong evidence against it; it always contains an equality.
- Significance level α
- The probability of a Type I error you are willing to tolerate, i.e. the chance of rejecting a true H₀; common values are 0.05 and 0.01.
- p-value
- The probability, if H₀ were true, of observing a test statistic at least as extreme as the one obtained; reject H₀ when the p-value is less than α.
- 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). Reducing one tends to increase the other for fixed n.
- Power
- Power = 1 − β is the probability of correctly rejecting a false H₀; it rises with a larger sample size or a larger true effect.
Hypothesis Testing: One-Sample Tests FAQ
How do I choose between a one-tailed and a two-tailed test?
Let the research question set the alternative. If it asks whether a parameter changed in a SPECIFIC direction (increased, decreased, under-fills), use a one-tailed test; if it asks whether it merely DIFFERS from a value, use a two-tailed test. Decide before seeing the data — switching tails afterward is a form of p-hacking.
What's the relationship between a hypothesis test and a confidence interval?
They are two sides of the same coin. A two-tailed test at level α rejects H₀: parameter = value exactly when the (1 − α) confidence interval excludes that value. So a 95% CI that does not contain the null value implies rejection at α = 0.05.
Why can't I 'accept' the null hypothesis?
Because failing to reject H₀ only means the evidence wasn't strong enough to overturn it — it doesn't prove H₀ is true. The correct wording is 'fail to reject H₀' or 'insufficient evidence', which matters for marks in written answers.
What does the ethics content actually ask?
It asks you to recognise honest reporting: disclose all tests you ran rather than only the significant ones, don't choose α after the fact to manufacture significance, and distinguish a statistically significant result from one that is large enough to matter in practice.
Exam move
Learn hypothesis testing as a fixed five-step ritual — state H₀ and H₁, choose α and the test, compute the statistic, find the critical value or p-value, and conclude in context — and apply the SAME ritual to every test for the rest of the unit. Drill the wording: hypotheses contain the parameter symbol, conclusions are in business English, and you never 'accept' H₀. Practise both the mean (Z and t) and proportion versions, and be ready to define Type I/II errors and power precisely. Because Part B leans on this skill, write out full solutions, not just final numbers.
