Inference for Proportions

Unit 6 of AP Statistics, Inference for Categorical Data: Proportions, is where the course turns from describing data to drawing conclusions about a population. The same structure returns in every later inference unit, so learning it well here pays off four times.

The two tools

• Confidence interval: estimates the population proportion p.

> p̂ ± z* √( p̂(1 − p̂) / n )

• Significance test: asks whether the data give convincing evidence against a claimed value p₀.

> z = (p̂ − p₀) / √( p₀(1 − p₀) / n )

Notice the difference: the interval uses p̂ in the standard error, the test uses p₀. The test assumes the null hypothesis is true, so it must use the null value.

The three conditions

1. Random: the data come from a random sample or a randomized experiment.
2. 10% condition: when sampling without replacement, n is at most 10% of the population.
3. Large Counts: for an interval, np̂ ≥ 10 and n(1 − p̂) ≥ 10; for a test, np₀ ≥ 10 and n(1 − p₀) ≥ 10.

Name each condition and show the numbers. Writing "conditions are met" with no check earns nothing.

Worked example

A random sample of 200 students finds that 124 use a planner. A school claims 55% do. Is there convincing evidence that the true proportion is higher? Use α = 0.05.

1. Hypotheses: H₀: p = 0.55, Hₐ: p > 0.55, where p is the proportion of all students at the school who use a planner.
2. Conditions: random sample stated; 200 is less than 10% of all students; 200 × 0.55 = 110 and 200 × 0.45 = 90, both at least 10.
3. Statistic: p̂ = 124/200 = 0.62. Standard error √(0.55 × 0.45 / 200) ≈ 0.0352, so z = 0.07 / 0.0352 ≈ 1.99.
4. P-value: about 0.023.
5. Conclusion: because 0.023 < 0.05, reject H₀. There is convincing evidence that more than 55% of students at the school use a planner.

The 95% interval for the same data: 0.62 ± 1.96 × √(0.62 × 0.38 / 200) ≈ 0.62 ± 0.067, so about 0.553 to 0.687.

Interpreting what you found

• Confidence interval: "We are 95% confident that the interval from 0.553 to 0.687 captures the true proportion of students who use a planner."
• Confidence level: in repeated random sampling, about 95% of intervals built this way capture the true proportion. It is a statement about the method, not about one interval.
• Margin of error: it shrinks with √n, so four times the sample size halves it.
• Errors: a Type I error rejects a true H₀; a Type II error fails to reject a false H₀. Always describe the error in the context of the problem.

The free response approach

State the procedure by name, define the parameter in words, check the conditions with numbers, show the calculation, and link the conclusion to the P-value and to the context. Graders look for all five.