26134 · Responsible Evidence-based Decisions
Estimating Population Values: Confidence Intervals
Week 7 completes Module 2: estimating a population mean with an interval, not just a point. You learn the confidence level 1 − α, when to use Z (σ known) versus the t-distribution (σ unknown), how to read the t-table, and how to interpret a confidence interval correctly. Building and interpreting a CI is a core exam item, and the interpretation carries its own marks.
What this chapter covers
- 01Point estimate (x̄ for μ) vs interval estimate (a confidence interval)
- 02Confidence level 1 − α, and the general form point estimate ± critical value × standard error
- 03CI for a mean, σ known: x̄ ± z_(α/2)·σ/√n (z* = 1.645, 1.96, 2.576 for 90/95/99%)
- 04CI for a mean, σ unknown: x̄ ± t_(α/2, n−1)·s/√n, reading the t-table with n − 1 df
- 05Correct interpretation: the procedure captures μ in 1 − α of repeated samples
- 06The common misinterpretation to avoid (not 'a 95% chance μ is in this interval')
- 07Width vs confidence: higher confidence → wider; larger n → narrower
- 08CI for a variance uses chi-square (asymmetric bounds)
95% confidence interval for a mean (σ unknown)
- +1Choose the distribution. σ is unknown and estimated by s, so use the t-distribution with df = n − 1 = 24. For 95% confidence, α = 0.05 and the critical value is t_(0.025, 24) = 2.064.
- +1Standard error. SE = s/√n = 12/√25 = 12/5 = 2.4 seconds.
- +1Margin of error. E = t* × SE = 2.064 × 2.4 = 4.95 seconds.
- +1Interval. CI = x̄ ± E = 68 ± 4.95 = (63.05, 72.95) seconds. We are 95% confident the true mean wait time lies in this range.
Key terms
- Point vs interval estimate
- A point estimate is a single best guess of a parameter (x̄ for μ); an interval estimate (confidence interval) gives a range expected to contain the parameter with a stated confidence level, acknowledging sampling uncertainty.
- Confidence level (1 − α)
- The long-run proportion of confidence intervals, built the same way from repeated samples, that would contain the true parameter. α is the risk of building an interval that misses it; common choices are 90%, 95% and 99%.
- Margin of error
- The half-width of a confidence interval, critical value × standard error (e.g. t_(α/2, n−1)·s/√n). It sets how precise the estimate is; it shrinks with larger n and grows with higher confidence.
- t-distribution
- A bell-shaped, symmetric distribution with heavier tails than the normal, indexed by degrees of freedom (n − 1). It is used for inference about a mean when σ is unknown and estimated by s, and it approaches the normal as df grows.
- Critical value
- The multiplier from the Z- or t-table for a chosen confidence level: z_(α/2) = 1.96 for 95% (σ known), or t_(α/2, n−1) for σ unknown. It scales the standard error into the margin of error.
- Confidence-interval interpretation
- The correct reading is about the procedure: 95% of intervals built this way capture μ. It is NOT correct to say there is a 95% probability that μ lies in one specific interval, because μ is fixed and the interval is the random quantity.
Estimating Population Values: Confidence Intervals FAQ
When do I use t instead of Z for a confidence interval?
Use Z when the population standard deviation σ is known (rare in practice); use the t-distribution with n − 1 degrees of freedom when σ is unknown and you estimate it with the sample s (the usual case). The t has heavier tails to account for the extra uncertainty from estimating σ, so its intervals are slightly wider.
What does '95% confident' actually mean?
It refers to the procedure, not this one interval: if you repeatedly sampled and built a 95% CI each time, about 95% of those intervals would contain the true parameter. Because μ is fixed and the interval is random, it is wrong to say 'there is a 95% probability μ is in this particular interval'.
Why does a higher confidence level give a wider interval?
Higher confidence means a smaller α and a larger critical value, which multiplies the standard error into a bigger margin of error. To be more sure of capturing μ you must cast a wider net; a 100% interval would be (−∞, +∞) and carry no information. You can offset the widening by increasing n.
Can AI help me with confidence intervals in 26134?
Yes, as a study aid. Sia can walk you through choosing Z or t, computing the standard error and margin of error, reading the t-table, and phrasing the interpretation correctly, one step at a time. Use it to rehearse; it does not do your graded assessment, and the UTS academic-integrity policy applies.
Exam move
Confidence intervals are a guaranteed exam item, so drill the full pattern: decide Z vs t (σ known vs unknown), compute SE = s/√n, look up the critical value at α/2, form point estimate ± margin, then interpret. Practise the interpretation sentence until it is automatic and correct — 'the procedure captures μ in 95% of samples', not 'a 95% chance μ is here' — because the wrong phrasing loses the interpretation mark. Rehearse reading the t-table at the right df and α/2, and note CONFIDENCE.T for the open-book setting. Keep the two CI formulas and the critical z-values (1.645, 1.96, 2.576) on your printed notes. When the choice of distribution confuses you, ask Sia to route it with you on a fresh sample; confirm assessment details on Canvas.
Working through Estimating Population Values: Confidence Intervals in 26134? Sia is AskSia’s AI Statistics tutor — ask any 26134 Estimating Population Values: Confidence Intervals question and get a clear, step-by-step explanation grounded in how 26134 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
