Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Sections 9.1–9.3
Chapter Summary
A single sample gives us a point estimate — one best guess at a parameter, like x̄ = 72.3 estimating the population mean. But a single number hides how uncertain that guess is. A confidence interval fixes that by giving a range of plausible values: “we are 95% confident the true mean lies between 70.1 and 74.5.” The interval has the form estimate ± margin of error, and the margin of error is where sample size, variability, and confidence level all come together.
The interpretation takes practice: “95% confident” means that if we repeated this procedure many times, about 95% of the intervals built this way would capture the true parameter. It does not mean there’s a 95% chance the parameter is inside this particular interval — the parameter is fixed; it’s the interval that varies from sample to sample. You’ll build intervals for a population mean (using z when σ is known, t when it isn’t — the usual real-world case) and for a population proportion, and you’ll learn to work backward to find the sample size needed for a desired precision. Realistic example: a college wants to estimate the proportion of students who work part-time within ±3 percentage points at 95% confidence. With no prior estimate, use p̂ = 0.5: n = 0.25 × (1.96/0.03)² ≈ 1,068 students. Precision costs sample size — that’s the trade-off running the whole chapter.
Study Guide
Key terms and definitions
- Point estimate — a single value used to estimate a parameter (x̄ estimates μ; p̂ estimates p).
- Confidence interval — a range of plausible values for a parameter, with an associated confidence level.
- Confidence level (1 − α) — the proportion of intervals from repeated sampling that would contain the parameter (e.g., 95%).
- Margin of error (E) — the “±” part of the interval; half its width.
- Critical value — z_{α/2} (σ known) or t_{α/2} (σ unknown, df = n − 1).
- t-distribution — bell-shaped like the normal but with heavier tails; approaches the normal as df grows. Used when σ is unknown.
Essential formulas
- CI for μ, σ known: x̄ ± z_{α/2} · (σ/√n).
- CI for μ, σ unknown: x̄ ± t_{α/2} · (s/√n), with df = n − 1.
- CI for p: p̂ ± z_{α/2} · √[p̂(1 − p̂)/n]. Requirement: np̂(1 − p̂) ≥ 10.
- Margin of error for μ: E = z_{α/2} · (σ/√n).
- Sample size for μ: n = [z_{α/2} · σ ÷ E]² (round up).
- Sample size for p: n = p̂(1 − p̂) · [z_{α/2} ÷ E]² (round up); use p̂ = 0.5 if no prior estimate.
Step-by-step procedures
- Build a CI for a mean: (1) is σ known? If yes use z; if no use t with df = n − 1 (check normality/small-sample conditions). (2) Find the critical value. (3) Compute E = critical value × standard error. (4) Form x̄ ± E. (5) Interpret: “We are C% confident the population mean is between ___ and ___.”
- Build a CI for a proportion: (1) compute p̂ = x/n; (2) check np̂(1 − p̂) ≥ 10; (3) compute E = z_{α/2}·√[p̂(1−p̂)/n]; (4) form p̂ ± E; (5) interpret.
- Find required sample size: (1) identify E and the confidence level; (2) plug into the sample-size formula; (3) always round UP (rounding down gives slightly less precision than requested).
Common mistakes to avoid
- Saying “there’s a 95% probability μ is in this interval.” Correct: “we are 95% confident the interval captures μ.”
- Using z when σ is unknown — the realistic case needs the t-distribution.
- Forgetting the requirements: t-interval needs approximate normality for small n; proportion interval needs np̂(1 − p̂) ≥ 10.
- Rounding the sample size down — always round up.
Self-check: I can…
- [ ] Explain the difference between a point estimate and a confidence interval.
- [ ] Correctly interpret a confidence level.
- [ ] Construct and interpret a CI for a population mean (z and t cases).
- [ ] Construct and interpret a CI for a population proportion.
- [ ] Determine the sample size needed for a given margin of error.
- [ ] Explain how confidence level, sample size, and variability affect the margin of error.
StatCrunch How-To
- CI for a mean, σ known: Stat > Z Stats > One Sample > choose With Data (select the column) or With Summary (enter n, x̄, σ) > select “Confidence interval”, enter the level > Compute.
- CI for a mean, σ unknown: Stat > T Stats > One Sample > With Data or With Summary (enter n, x̄, s) > “Confidence interval” > enter level > Compute.
- CI for a proportion: Stat > Proportion Stats > One Sample > With Data (successes in a column) or With Summary (enter # of successes and # of observations) > “Confidence interval” > Compute.
- Tip: StatCrunch checks nothing for you — verify the requirements (normality, np̂(1−p̂) ≥ 10) yourself before trusting the output.
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