BUSS1020 · Quantitative Business Analysis
Confidence Intervals
Confidence Intervals (Week 7, Berenson Ch 8.1–8.6) turns a single sample estimate into a range that quantifies its uncertainty, in the form point estimate ± (critical value × standard error). You build intervals for a population mean (using Z when σ is known, t when it is unknown) and for a population proportion, choose the right critical value for a confidence level, and compute the sample size needed for a target margin of error. Correct interpretation — what '95% confident' really means — is itself examinable.
What this chapter covers
- 01General form: point estimate ± (critical value)(standard error)
- 02CI for μ when σ is known (Z) versus unknown (t)
- 03Critical values: 90% → 1.645, 95% → 1.96, 98% → 2.33, 99% → 2.58
- 04The t-distribution and degrees of freedom n − 1
- 05CI for a population proportion π
- 06Margin of error and interval width drivers
- 07Required sample size for a mean and for a proportion
- 08Correct interpretation of confidence; application to auditing
Confidence interval for a mean and required sample size
- 1 mark(a) σ is unknown, so use the t-distribution with df = n − 1 = 24. The critical value t₀.₀₂₅,₂₄ ≈ 2.064.
- 1 mark(a) Standard error S/√n = 3.50/√25 = 3.50/5 = 0.70.
- 1 mark(a) Margin of error = 2.064 × 0.70 = 1.4448 ≈ 1.44.
- 2 marks(a) 95% CI: 14.60 ± 1.44 → ($13.16, $16.04).
- 2 marks(b) Use n = Z²σ²/e² with Z = 1.96, σ = 3.50, e = 1.00: n = (1.96² × 3.50²)/1.00² = (3.8416 × 12.25)/1 = 47.06.
- 1 mark(b) Always round UP to guarantee the margin: n = 48 transactions.
Key terms
- Confidence interval
- A range, point estimate ± (critical value × standard error), constructed so that a stated percentage of such intervals across repeated samples would contain the true parameter.
- Critical value
- The multiplier (Z or t) corresponding to the chosen confidence level; for 95% the Z critical value is 1.96.
- Margin of error
- The half-width of a confidence interval, critical value × standard error; it shrinks with a larger sample, a smaller standard deviation, or a lower confidence level.
- t-distribution
- A bell-shaped distribution with heavier tails than the normal, used for inference about a mean when σ is unknown; its shape depends on the degrees of freedom n − 1.
- Required sample size
- The smallest n that achieves a target margin of error e; for a mean n = Z²σ²/e², for a proportion n = Z²π(1−π)/e², always rounded up.
Confidence Intervals FAQ
When do I use Z and when do I use t for a mean?
Use Z when the population standard deviation σ is known (rare in practice); use t with df = n − 1 when σ is unknown and you estimate it with the sample S. In BUSS1020 the t-case is the common one for real data.
What does '95% confident' actually mean?
It refers to the procedure: if you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true parameter. It is NOT a 95% probability that the one fixed parameter lies in your particular interval — the parameter is fixed, the interval is random.
Why use π = 0.5 for sample size when the proportion is unknown?
Because π(1−π) is largest at π = 0.5, plugging in 0.5 gives the most conservative (largest) required sample size, guaranteeing your target margin no matter the true proportion.
Exam move
Anchor everything on the single template 'estimate ± critical × standard error', then learn the three variations (mean with σ known, mean with σ unknown, proportion). Make the Z-versus-t decision automatic by asking 'is σ known?' first. Memorise the four common Z critical values and know how to read t from its table by degrees of freedom. Practise the sample-size formulas separately, remembering to use Z and to round up. This is the first heavily examined Part B topic, so rehearse writing a clean interpretation sentence under each interval.
