Inference for Proportions: every key term you need (+ practice quiz)

More AP Statistics guides

• Designing Studies
• Exploring Bivariate Data
• Probability
• Random Variables & Distributions
• Sampling Distributions

One-Proportion Confidence Interval

p̂ ± z*√(p̂(1-p̂)/n). Estimate population proportion p. Requires: np̂ ≥ 10 and n(1-p̂) ≥ 10 for normal approximation.

Sample Size for Proportions

n = (z*/ME)²·p(1-p). Use p=0.5 if unknown (most conservative). Larger ME = smaller n needed. Costly: doubling precision requires 4x sample.

Two-Proportion CI

(p̂₁ - p̂₂) ± z*√(p̂₁(1-p̂₁)/n₁ + p̂₂(1-p̂₂)/n₂). Compares two populations. If CI doesn't contain 0, proportions differ significantly.

Hypothesis Test for Proportion

H₀: p = p₀. Test statistic: z = (p̂ - p₀)/√(p₀(1-p₀)/n). Use p₀ (hypothesized) in denominator, not p̂.

Two-Proportion Z-test

H₀: p₁ = p₂ vs Hₐ: p₁ ≠ p₂. Test statistic: z = (p̂₁ - p̂₂)/√(p̂(1-p̂)(1/n₁ + 1/n₂)) where p̂ = combined proportion.

Chi-Square Test

Tests association between two categorical variables. X² = Σ(observed - expected)²/expected. Higher values = stronger association.

Expected Frequencies

For independence: expected = (row total × column total)/total. Chi-square test valid if all expected ≥ 5.

Conditions for Chi-Square

Random sample, independence, expected frequencies ≥ 5 in each cell. If not met, may combine categories or use exact test.

Effect Size for Proportions

Sample size alone doesn't indicate importance; consider practical significance. Difference of 0.3% vs 30% in proportions very different.

Unit 7 Summary

Confidence intervals and tests for proportions use normal approximation. Chi-square test analyzes categorical association. Always check conditions.

One-Sample z Interval for p

p̂ ± z* · sqrt(p̂(1-p̂)/n). Conditions: random, 10%, Large Counts using p̂: n·p̂ ≥ 10 and n(1-p̂) ≥ 10.

One-Sample z Test for p

z = (p̂ - p0)/sqrt(p0(1-p0)/n). Large Counts uses p0 (the hypothesized value), not p̂, because we assume H0 is true.

Interval vs Test Standard Error

Interval: SE uses p̂ (we don't assume any p). Test: SE uses p0 (we assume H0). This is why the two can occasionally disagree slightly.

Sample Size for a Proportion

n ≥ (z*/ME)^2 · p*(1-p*), with p* a guess or 0.5 for the most conservative (largest) n. Round up.

Two-Sample z Interval for p1 - p2

(p̂1 - p̂2) ± z* · sqrt(p̂1(1-p̂1)/n1 + p̂2(1-p̂2)/n2). Large Counts: all four observed counts ≥ 10 (some texts use 5).

Pooled Proportion

p̂_c = (X1 + X2)/(n1 + n2). Used ONLY in the two-sample z TEST because H0 says p1 = p2, so we combine to estimate the common p.

Two-Sample z Test

z = (p̂1 - p̂2)/sqrt(p̂_c(1-p̂_c)(1/n1 + 1/n2)). Large Counts checked with the pooled proportion times each n.

Margin of Error for p

ME = z*·sqrt(p̂(1-p̂)/n). Largest when p̂ = 0.5; that's why 0.5 is the conservative guess.

Critical z Values

90%: 1.645; 95%: 1.960; 99%: 2.576. Higher confidence means wider intervals.

Plus-Four Interval

Add 2 successes and 2 failures before computing p̂ to improve coverage when counts are small. Not required on the AP exam but sometimes referenced.

Interpretation Template (proportion)

'We are 95% confident that the interval from a to b captures the true proportion of [population] who [characteristic].'

Direction of Ha

Choose Ha before seeing data based on the research question. Using the data to pick a direction inflates Type I error.

Statistical vs Practical Significance

With huge n, a difference of 0.5 percentage points may be significant yet meaningless. Report the interval to convey the size of the effect.

Two-Sample Interval Interpretation

'We are 95% confident that the true difference p1 - p2 lies between a and b.' If the interval includes 0, no convincing evidence of a difference.

Experiment vs Survey Conditions

For a randomized experiment comparing two proportions, the Random condition is random assignment; the 10% condition does not apply.

Effect of Confidence Level on Width

Moving from 95% to 99% raises z* from 1.96 to 2.576, widening the interval by about 31% for the same data.

Effect of n on Width

Width is proportional to 1/sqrt(n). Doubling n reduces width by a factor of about 0.71, not half.

Sign of z and Ha

For Ha: p > p0, P-value is the area to the right of z; for Ha: p < p0, the area to the left; two-sided doubles the tail.

Rejecting with Small Effect

Rejecting H0 says the difference is unlikely due to chance alone; it does not mean the difference is large.

Nonresponse and Intervals

A confidence interval's margin of error covers only random sampling variability. Bias from nonresponse or wording is not included.