Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A confidence interval is an interval estimate: a range of plausible values for a population parameter, built from one sample.
- For one population proportion, the procedure is the one-sample z-interval for a population proportion.
- Define the parameter with three parts: the proportion, the response (what counts as a success) and the population.
- Every interval has the shape point estimate ± margin of error.
Key relationships
| Quantity | Formula or rule |
|---|---|
| Point estimate | p̂ = x / n (x = number of successes) |
| Standard error | SE(p̂) = √[p̂(1 − p̂)/n] |
| Critical value | z* = invNorm((1 + C)/2): 1.645 (90%), 1.960 (95%), 2.576 (99%) |
| Margin of error | MOE = z* × SE |
| Interval | p̂ ± MOE |
| From an interval (a, b) | p̂ = (a + b)/2; MOE = (b − a)/2 |
| Sample size for a chosen MOE | n = (z*/MOE)² × p̂(1 − p̂), rounded up; use p̂ = 0.5 if no estimate |
Conditions to check (in context, with numbers)
- Random: say how the sample was chosen at random.
- 10%: when sampling without replacement, n ≤ 0.10N.
- Large counts: np̂ ≥ 10 and n(1 − p̂) ≥ 10, using the observed successes and failures.
Mistakes to avoid
- Calling p̂ the parameter, or defining p as a proportion of the sample.
- Writing “SRS ✓” with no link to the context.
- Using n ≥ 30 for the normality check instead of the success and failure counts.
- Using invNorm(C): for 90% this gives 1.282, the 80% value.
- Confusing MOE and width: width = 2 × MOE.
- Rounding n down in a sample-size question.
- Rounding p̂ early, which shifts the endpoints.
Quick self-check
- In a random sample of 150 adults, 60 say yes. Find the 95% interval. (p̂ = 0.4, SE = 0.04, MOE = 1.960 × 0.04 = 0.078, interval (0.322, 0.478))
- A 90% interval is (0.55, 0.63). What are p̂ and the margin of error? (p̂ = 0.59, MOE = 0.04)
- What sample size gives a 95% margin of error of at most 0.04 with no prior estimate? (n = (1.960/0.04)² × 0.25 = 600.25, so 601)
Next: practice questions.