ETX5900 · Business Statistics
Confidence Interval Estimation: Mean & Proportion
Confidence Interval Estimation is Week 5 of ETX5900 Business Statistics at Monash University, the first inference topic in the unit (Berenson Ch 8). It turns a single sample statistic into a range of plausible values for the unknown population parameter, always in the shape estimate ± (critical value × standard error). The one decision that changes the answer is the critical value: z when the population σ is known, t (df = n−1) when σ is estimated by the sample s, and z again for a proportion.
What this chapter covers
- 01Point estimate vs interval estimate; the universal estimate ± margin-of-error shape
- 02Confidence level (1−α) and what “95% confident” really means
- 03z-interval for a mean when σ is KNOWN: x̄ ± z_{α/2}·σ/√n
- 04t-interval for a mean when σ is UNKNOWN: x̄ ± t_{n−1,α/2}·s/√n, df = n−1
- 05z-interval for a proportion: p ± z_{α/2}·√(p(1−p)/n)
- 06Margin of error and how it responds to confidence level, n and σ/s
- 07Normal-approximation condition for a proportion (np ≥ 5 and n(1−p) ≥ 5)
- 08Reading critical values from the provided standard-normal and Student-t tables
- 09CI ↔ hypothesis-test duality: a value outside the interval is rejected at level α
- 10Correct interpretation wording (confident about the procedure, not a probability for the parameter)
95% confidence interval for a mean when σ is unknown (t-interval), then a benchmark decision
- +1Choose the interval. We are estimating a population MEAN and only the sample s is given (σ unknown), so use the t-interval x̄ ± t_{n−1,α/2}·s/√n with df = n−1 = 24.
- +1Critical value. For 95% confidence, α = 0.05 and α/2 = 0.025, so from the Student-t table t_{24, 0.025} = 2.064.
- +1Standard error. s/√n = 10/√25 = 10/5 = 2.
- +2Margin of error and interval. E = 2.064 × 2 = 4.128 ≈ 4.13, so the CI is 68 ± 4.13 = (63.87, 72.13) mL.
- +1Interpret and decide. We are 95% confident the true mean fill lies between 63.87 and 72.13 mL. The claimed 75 mL lies ABOVE the upper limit, so 75 is OUTSIDE the interval → reject H₀: μ = 75 at the 5% level; the evidence points to a mean below 75 mL.
Key terms
- Point estimate
- A single best-guess number for a population parameter — the sample mean x̄ for μ, or the sample proportion p for π. It carries no indication of its own sampling uncertainty.
- Confidence interval (CI)
- A range of plausible values for a population parameter, built as estimate ± margin of error and attached to a confidence level (1−α). All three ETX5900 intervals share this shape.
- Confidence level (1−α)
- The long-run proportion of such intervals that would capture the parameter over repeated sampling (e.g. 95%). α is the leftover error split into two tails of α/2 for a two-sided interval.
- Margin of error (E)
- The ± half-width of a CI, E = critical value × standard error. It grows with a higher confidence level and larger σ/s, and shrinks as n rises through the √n; halving E needs roughly four times the sample size.
- Standard error
- The standard deviation of the sampling distribution of the estimate: σ/√n (or s/√n) for a mean, and √(p(1−p)/n) for a proportion.
- Critical value (z or t)
- The table multiplier for the margin. Use z_{α/2} from the standard-normal table when σ is known (or for a proportion); use t_{n−1,α/2} from the Student-t table with df = n−1 when σ is estimated by s. The t value is always larger than the matching z.
- Student-t distribution
- A bell-shaped distribution with heavier tails than the normal, indexed by degrees of freedom df = n−1. It applies to a mean when σ is unknown and approaches the normal as n grows large.
Confidence Interval Estimation: Mean & Proportion FAQ
When do I use z instead of t for a confidence interval for the mean?
Use z only when the population standard deviation σ is genuinely given. The moment you must estimate the spread from the sample — you are handed s, not σ — switch to t with df = n−1. The t value is larger than the matching z (heavier tails), which widens the interval to account for the extra uncertainty in estimating σ; as n grows, t converges to z.
What does “95% confident” actually mean?
It is a statement about the procedure, not this one interval. If you repeated the sampling many times and built a 95% interval each time, about 95% of those intervals would contain the true parameter. For the single interval in front of you the parameter is either in it or not, so we say we are ‘95% confident’ rather than ‘there is a 95% probability the parameter lies in this interval’.
Can AI help me with confidence intervals in ETX5900?
Yes — Sia can explain the method step by step: why σ-known picks z and only-s picks t, how to read the standard-normal and Student-t tables at the right α and df, and how the margin of error responds to n and the confidence level. Work your own practice numbers through it to build the routine. Sia explains and coaches; it does not sit your exam or promise guaranteed answers, grades or a pass.
Exam move
Drill the three intervals as one routine — decide mean-or-proportion, then σ-known-or-not, to pick z vs t; look up the critical value at the stated confidence (and df = n−1 for t); then compute standard error → margin → the two limits, and finish with one interpretation sentence naming the parameter and the confidence level. Because the exam provides the formula sheet and statistical tables, practice reading the right table column rather than memorising numbers, and rehearse the interpretation wording (confident about the procedure, not a probability for the parameter). Learn the CI ↔ test duality so you can judge a benchmark value directly from the interval, and keep the standard error to 4–5 significant figures so a borderline decision does not flip on rounding.
Working through Confidence Interval Estimation: Mean & Proportion in ETX5900? Sia is AskSia’s AI Statistics tutor — ask any ETX5900 Confidence Interval Estimation: Mean & Proportion question and get a clear, step-by-step explanation grounded in how ETX5900 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
