QBUS5001 · Foundation In Data Analytics For Business
Hypothesis Testing (One Population)
Module 8 formalises decision-making under uncertainty with the six-step framework: state H₀ and H₁, choose α, compute the test statistic, find the critical value or p-value, decide, and conclude in context. You test a single population mean (Z when σ is known, t with n−1 degrees of freedom when it is not) and a single proportion (Z), choosing one- or two-tailed from the alternative.
The conceptual spine is the two error types: a Type I error rejects a true H₀ (probability α) and a Type II error retains a false one (probability β). The p-value rule — reject when p ≤ α — ties everything together.
What this chapter covers
- 01The six-step hypothesis-testing framework
- 02Null and alternative hypotheses; one- vs two-tailed
- 03Significance level α and the rejection region
- 04Test for a mean, σ known: Z = (x̄ − μ₀)/(σ/√n)
- 05Test for a mean, σ unknown: T = (x̄ − μ₀)/(s/√n) ~ t(n−1)
- 06Test for a proportion: Z = (p − π₀)/√(π₀(1−π₀)/n)
- 07Type I and Type II errors; the role of α and β
- 08The p-value decision rule: reject if p ≤ α
One-sample t-test on a proportion-style claim using the mean
- 1 markStep 1 — Hypotheses: H₀: μ = 250 versus H₁: μ ≠ 250 (two-tailed).
- 1 markStep 2 — Significance level: α = 0.05.
- 2 marksStep 3 — Test statistic: σ unknown → use t. SE = s/√n = 9/6 = 1.5, so T = (246−250)/1.5 = −2.6667 with df = 35.
- 1 markStep 4 — Critical value / region: reject if |T| > t(0.025, 35) ≈ 2.030.
- 1 markStep 5 — Decision: |−2.6667| = 2.6667 > 2.030, so reject H₀.
- 1 markStep 6 — Conclusion: at the 5% level there is significant evidence the true mean fill differs from (is below) 250 ml.
Key terms
- Null hypothesis (H₀)
- The default claim of no effect or no difference that the test assumes true until the data provide sufficient evidence against it.
- Significance level (α)
- The probability threshold for rejecting H₀, equal to the probability of a Type I error; common choices are 0.05 and 0.01.
- Test statistic
- A standardised quantity (Z or t) measuring how far the sample result is from H₀ in standard-error units; compared against a critical value or converted to a p-value.
- Type I error
- Rejecting a true null hypothesis; its probability is α, the significance level chosen by the analyst.
- Type II error
- Failing to reject a false null hypothesis; its probability is β, and power = 1 − β is the chance of correctly detecting a real effect.
Hypothesis Testing (One Population) FAQ
How do I decide one-tailed versus two-tailed?
The alternative hypothesis decides. “Differs from” gives a two-tailed test (≠); “greater than” or “less than” gives a one-tailed test (> or <). Reading the question wording carefully here is worth a mark and changes the critical value.
Does the confidence interval always agree with the two-tailed test?
Yes. If the hypothesised value lies outside the (1−α) confidence interval, you reject H₀ at α, and vice versa. They are two views of the same calculation, which is why quoting both is a strong exam strategy.
When do I use t instead of Z for a one-sample mean test?
Use t with n−1 degrees of freedom whenever the population standard deviation σ is unknown and estimated by the sample s — the usual case. Use Z only when σ is genuinely known.
Exam move
Write the six steps as a fixed template and answer every test by filling it in, so you never lose the hypothesis and conclusion marks. Pre-build a small decision tree for the test statistic (mean with σ known → Z; mean with σ unknown → t; proportion → Z) and practise stating conclusions in plain business language, since the exam consistently rewards the in-context sentence over the bare reject/retain verdict.
