ECON20003 · Quantitative Methods 2
Estimation & Hypothesis Testing of a Population Mean
Estimation & Hypothesis Testing of a Population Mean sets up the spine of the whole subject. You learn what makes a good estimator (unbiased, efficient, consistent), how the sample mean X̄ behaves across repeated samples (its sampling distribution is N(μ, σ²/n), exactly if the population is normal and approximately otherwise by the CLT), and how to turn that into a confidence interval or a hypothesis test. The key fork is Z versus t: use Z when σ is known and t (with df = n − 1) when σ is estimated by S. Every test is written as the five-step ritual.
What this chapter covers
- 01Estimator properties: unbiasedness, efficiency, consistency
- 02Sampling distribution of X̄: N(μ, σ²/n); SE = σ/√n or S/√n
- 03Z when σ known vs t (df = n − 1) when σ unknown
- 04Confidence interval for μ: X̄ ± t_{α/2,n−1}·(S/√n)
- 05The five-step hypothesis-test ritual
- 06Type I error (α), Type II error (β) and power = 1 − β
- 07p-value = smallest α at which H₀ is rejected
95% confidence interval for a mean (σ unknown)
- 1 markIdentify the situation. σ is unknown and estimated by S, so use the t-distribution with df = n − 1 = 15.
- 1 markFind the critical value from the t-table: t₀.₀₂₅,₁₅ = 2.131.
- 1 markCompute the standard error: S/√n = 8/√16 = 8/4 = 2.0 orders.
- 1 markCompute the margin of error: t × SE = 2.131 × 2.0 = 4.262 orders.
- 1 markAssemble the interval: X̄ ± margin = 50 ± 4.262, i.e. (45.74, 54.26).
- 1 markInterpret: we are 95% confident the true mean daily orders lies between about 45.7 and 54.3 — the confidence level refers to the long-run procedure, not a probability about this one fixed interval.
Key terms
- Sampling distribution of X̄
- The distribution of the sample mean over all possible samples of size n: centred at μ with standard error σ/√n. It is exactly normal if the population is normal, and approximately normal for large n by the CLT.
- Z vs t statistic
- Z = (X̄ − μ₀)/(σ/√n) when σ is known; t = (X̄ − μ₀)/(S/√n) with df = n − 1 when σ is unknown. The t-distribution has heavier tails that thin toward the normal as df grows.
- Type I and Type II error
- A Type I error rejects a true H₀ (probability α, the significance level); a Type II error fails to reject a false H₀ (probability β). Power = 1 − β is the chance of detecting a real effect.
- p-value
- The smallest significance level α at which H₀ would be rejected — equivalently the probability, if H₀ were true, of a test statistic at least as extreme as the one observed. Small p means strong evidence against H₀.
Estimation & Hypothesis Testing of a Population Mean FAQ
When do I use Z instead of t?
Use Z only when the population standard deviation σ is genuinely known (rare in practice). The moment you estimate the spread with the sample S — which is almost always — switch to t with df = n − 1. For large n the two give nearly identical answers, but the exam rewards naming the correct distribution.
What does 'fail to reject H₀' actually mean?
It means the data do not provide enough evidence against H₀ at the chosen α — NOT that H₀ is proven true. A non-significant result is consistent with H₀ but also with small effects you lacked the power to detect, so always phrase the conclusion as 'insufficient evidence to reject', never 'accept H₀'.
Exam move
Memorise the five-step template and write every single mean problem in it, even easy ones, because the exam awards marks line by line. Practise switching cleanly between the confidence-interval form and the test form — they use the same SE and the same critical value.
