ECON1012 · Data Analytics
Estimation & Confidence Intervals
Estimation (Module 6, Week 6) is where the Inferential Statistics block of ECON 1012 at Adelaide University turns Week 5's sampling distributions into a usable tool: estimating an unknown population mean μ from a single sample. The module opens with point estimators and the three properties the course examines — unbiasedness, consistency and relative efficiency (x̄ beats the sample median because its variance σ²/n is smaller) — then builds interval estimators: x̄ ± z·σ/√n when σ is known, and x̄ ± t·s/√n with df = n − 1 when only the sample s is available. It closes with what drives interval width and how to choose the sample size n that delivers a target bound B. On the final exam this is hand-calculation territory: critical values come from the provided Z and t tables, and finding z in the body of the Z table is itself a tested skill.
What this chapter covers
- 01Point vs interval estimators — a single best guess vs a range with a stated confidence level
- 02Good-estimator checklist: unbiased · consistent · relatively efficient (x̄: σ²/n beats the median: 1.57σ²/n)
- 03CI for μ, σ known: x̄ ± z·σ/√n — 90% → z = 1.645 · 95% → 1.96 · 99% → 2.575
- 04CI for μ, σ unknown: x̄ ± t·s/√n with df = n − 1 (t is symmetric around 0 with fatter tails than Z)
- 05Width W = 2·z·σ/√n: higher confidence → wider · larger σ → wider · quadruple n → half the width
- 06Sample size for a bound B: n = (z·σ/B)² — always round UP; no σ estimate? use σ ≈ range/4
- 07Interpretation: 95% of repeated-sample intervals capture μ — never 'μ has a 95% chance of being inside'
t-interval for a mean, then sample-size planning
- 1 mark(a) σ is unknown (only s is given), so use the t-distribution with df = n − 1 = 15. From the t table, t₀.₀₂₅,₁₅ = 2.131.
- 1 mark(a) Standard error of the mean: s/√n = 8/√16 = 8/4 = 2 minutes.
- 1 mark(a) Bound: B = 2.131 × 2 = 4.262 minutes.
- 2 marks(a) 95% CI: 34 ± 4.262 → (29.74, 38.26) minutes.
- 1 mark(a) Interpretation: if we repeatedly drew samples of 16 deliveries and built an interval from each, about 95% of those intervals would contain the true mean μ.
- 1 mark(b) No σ is given for planning, so approximate σ ≈ range/4 = (95 − 15)/4 = 80/4 = 20 minutes.
- 1 mark(b) n = (z·σ/B)² = (1.96 × 20/4)² = (39.2/4)² = 9.8² = 96.04.
- 1 mark(b) Always round UP so the bound is guaranteed: n = 97 deliveries.
Key terms
- Point estimator
- A single value calculated from sample data that estimates an unknown population parameter — e.g. x̄ as the point estimate of μ. On its own it carries no information about how far off it might be.
- Interval estimator
- Draws inferences about a population by estimating a parameter with a range rather than a single value; the confidence level 1 − α is the long-run proportion of such intervals that capture the parameter.
- Unbiased estimator
- An estimator whose expected value equals the parameter it estimates. x̄ is unbiased for μ, and the sample variance s² is unbiased for σ² precisely because of its n − 1 divisor.
- Relative efficiency
- Of two unbiased estimators, the one with the smaller variance is relatively efficient. For a symmetric population, x̄ (variance σ²/n) beats the sample median (variance ≈ 1.57σ²/n).
- t-distribution
- Bell-shaped and symmetric around zero like the standard normal but with fatter tails; its spread is governed by degrees of freedom n − 1 and it approaches Z as df grows — beyond roughly 200 df the Z table is used instead.
- Bound B
- The half-width of a confidence interval — the 'to within ±B units' in a question. B = z·σ/√n (or t·s/√n), and the full interval width is W = 2B.
Estimation & Confidence Intervals FAQ
When do I use t instead of Z in ECON 1012?
Use Z only when the population standard deviation σ is actually given; the moment you are working from the sample standard deviation s, switch to t with df = n − 1. Exam MCQ options are often built exactly one z-vs-t slip apart (1.96 versus a nearby t value), so make the check explicit before touching the tables. For very large df — the course convention is above 200 — t and Z give effectively the same critical values.
What does '95% confident' actually mean in this course?
It is a statement about the method, not about one interval: if we repeatedly drew samples of the same size from the same population, 95% of the resulting intervals would contain the population mean. Writing 'there is a 95% probability that μ is inside my interval' is a marked error in ECON 1012 — the parameter is fixed, and it is the interval that varies from sample to sample.
Does ECON 1012 Module 6 cover confidence intervals for proportions?
The Module 6 concept material constructs intervals for a population mean only — σ known (Z) and σ unknown (t). Proportions appear in Week 5 through the sampling distribution of p̂, not as a Module 6 interval formula. If you want certainty about exactly what your sitting expects, check the current unit outline and the myLearning module pages.
Do I need to memorise Z and t critical values for the final exam?
No — the invigilated final provides Z and t tables, and you may bring one double-sided A4 note sheet plus a non-wireless calculator. Even so, knowing 1.645, 1.96 and 2.575 by heart saves real time across the 25 MCQs, and you still need the table skill of finding z in the body of the Z table (not the margins) for less common confidence levels.
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Exam move
Drill the z-versus-t decision until it is reflexive — σ given → z, only s given → t with df = n − 1 — because whole MCQ option sets are built around that one slip. The other marked errors: using σ instead of the standard error σ/√n, rounding a sample size down instead of up, and interpreting a 90% interval as 'μ is inside with 90% probability' rather than the repeated-sampling statement. Learn the width logic as hard facts: higher confidence → wider, larger σ → wider, quadrupling n halves the width, and the population mean's size changes nothing (a favourite distractor). Re-attempt the randomised Module 6 Quiz on myLearning, and put both interval formulas, the critical-value trio and 'round n UP' on your A4 note sheet.
