24.2 Confidence Intervals

Point Estimation and the Idea of a Confidence Interval

A point estimate is a single number summarizing what we believe about a population parameter: the sample mean x̄ estimates μ, the sample proportion p̂ estimates p. Point estimates are simple but give no sense of uncertainty. A confidence interval (CI) surrounds the point estimate with a range of plausible values, quantifying the precision of the estimate. On the PA-CAT, Statistics items commonly ask you to interpret a reported CI or to choose the correct critical value.

General Form

A two-sided CI for a parameter has the form:

estimate ± critical value × standard error

For a population mean: x̄ ± z* (σ/√n) when σ is known, or x̄ ± t* (s/√n) when σ is estimated by s. For a proportion: p̂ ± z* √(p̂(1−p̂)/n). The margin of error (ME) is the critical value × standard error portion; the confidence level is the long-run proportion of intervals that would contain the parameter if the procedure were repeated many times.

Confidence Interval for a Mean (z or t?)

Use z when the population standard deviation σ is known. The critical value z* for 95% confidence is 1.96; for 90%, 1.645; for 99%, 2.576. Use t when σ is unknown and estimated by s. The t-distribution is bell-shaped and symmetric but wider than z, with heavier tails reflecting the added uncertainty from estimating σ. The t-distribution has n − 1 degrees of freedom for a single-sample mean. As n grows, t approaches z.

Example: a PA student measures resting heart rate in 25 healthy adults and obtains x̄ = 72 bpm, s = 10. Because σ is unknown, use t with 24 df. For 95% confidence, t* ≈ 2.064. The interval is 72 ± 2.064 × (10/√25) = 72 ± 4.13, or (67.9, 76.1) bpm.

Confidence Interval for a Proportion

For a proportion, the estimate is p̂ = x/n. The standard error is √(p̂(1−p̂)/n). Example: in a sample of 400 clinic patients, 80 report medication nonadherence: p̂ = 0.20. A 95% CI is 0.20 ± 1.96 × √(0.20 × 0.80 / 400) = 0.20 ± 0.039, or (0.161, 0.239). This large-sample (Wald) interval requires np̂ ≥ 10 and n(1−p̂) ≥ 10 and is the form expected on the PA-CAT.

Interpretation

A correct interpretation: "We are 95% confident that the true population mean lies between 67.9 and 76.1 bpm." This means that if we repeated this sampling procedure many times, about 95% of the resulting intervals would contain μ. It does not mean there is a 95% probability that μ lies in this particular interval—the parameter is fixed; the interval is random before sampling and fixed after.

Common misinterpretations the PA-CAT may test:

• "95% of sample means fall in the interval" — no; the interval targets the population parameter.
• "There is a 95% chance the next sample mean falls in the interval" — no.
• "95% of the population values fall in the interval" — no; that is a tolerance interval, not a CI.

Margin of Error and What Drives It

The margin of error shrinks with:

• Larger sample size n (ME ∝ 1/√n). To halve the ME, quadruple n.
• Lower confidence level (smaller critical value).
• Smaller variability (smaller σ or s).

Conversely, higher confidence demands a wider interval—there is a trade-off between precision and confidence. A 99% CI is wider than a 95% CI for the same data.

Sample Size Determination

For estimating a mean with margin of error ME at confidence level 1−α: n = (z σ / ME)²*, always rounded up. For a proportion: n = (z/ME)² × p(1−p*)**. If a prior estimate p* is unavailable, use p* = 0.5, which maximizes p(1−p) and gives the most conservative (largest) sample size.

Example: How many patients are needed to estimate the proportion with postoperative nausea within ±3 percentage points at 95% confidence, with no prior estimate? n = (1.96/0.03)² × 0.25 ≈ 1,067.1 → 1,068 patients.

Assumptions

CI formulas assume:

1. The data are an SRS (or a representative sample with known design effects).
2. The sampling distribution of the estimator is approximately normal (Central Limit Theorem for means when n ≥ 30; normal population when n is small; np̂ ≥ 10 and n(1−p̂) ≥ 10 for proportions).
3. Observations are independent (sample size ≤ 10% of population when sampling without replacement).
4. For the t-interval, the population is approximately normal, or n is large enough for the CLT to dominate.

When assumptions are seriously violated, the nominal confidence level is wrong. Outliers threaten the mean-based interval; consider a robust estimator or a nonparametric procedure (e.g., bootstrap CI).