Inference for Categorical Data: Proportions
Unit 3 of AP Statistics, worth 15–25% of the exam. 13 questions below, each with the working. Every answer was checked by a second pass before it was published.
Confidence intervals and significance tests for one and two proportions, errors and power, chi-square tests.
How this unit is tested
Start every problem by identifying what kind of categorical inference is being asked for: one proportion or two, a confidence interval or a significance test, or a chi-square procedure (goodness of fit, independence, or homogeneity). The clue is almost always in how the data were collected — one sample with two outcomes, two samples compared, one sample classified by two variables, or several samples compared on one variable.
Once you know the procedure, check conditions before computing anything: Random (random sample or random assignment), 10% (sample size no more than 10% of the population when sampling without replacement, so trials are close enough to independent), and Large Counts (expected successes and failures, or expected cell counts, are large enough for the normal or chi-square approximation to hold — usually at least 10 for proportions, at least 5 for chi-square). Skipping this step is the single most common way points are lost, even when the arithmetic is right.
For confidence intervals, compute a point estimate plus or minus a critical value times a standard error, then interpret the interval and the confidence level in context — these are two different sentences and the exam grades them separately. For significance tests, state hypotheses in terms of population parameters, compute a test statistic and p-value, compare to alpha, and state a conclusion in context that never claims the null is 'true' or 'proven.'
For error and power questions, always write Type I and Type II errors in the context of the problem, not as abstract definitions — 'rejecting H0 when it's actually true' becomes a concrete false alarm about a specific claim. For chi-square problems, get degrees of freedom right (k−1 for goodness of fit, (r−1)(c−1) for two-way tables) and be able to distinguish independence (one sample, two variables) from homogeneity (multiple samples, one variable).
What you have to know
One-proportion z confidence interval
A confidence interval for a population proportion p is $\hat p \pm z^*\sqrt{\dfrac{\hat p(1-\hat p)}{n}}$, valid when Random, 10%, and Large Counts ($n\hat p\ge 10$, $n(1-\hat p)\ge 10$) conditions hold.
One-proportion z test statistic
To test $H_0: p=p_0$, compute $z=\dfrac{\hat p - p_0}{\sqrt{p_0(1-p_0)/n}}$ and find the p-value from the standard normal distribution.
Two-proportion test statistic (pooled)
To test $H_0: p_1=p_2$, use the pooled proportion $\hat p_c=\dfrac{x_1+x_2}{n_1+n_2}$ and compute $z=\dfrac{\hat p_1-\hat p_2}{\sqrt{\hat p_c(1-\hat p_c)\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}$.
Two-proportion confidence interval (unpooled)
A confidence interval for $p_1-p_2$ is $(\hat p_1-\hat p_2)\pm z^*\sqrt{\dfrac{\hat p_1(1-\hat p_1)}{n_1}+\dfrac{\hat p_2(1-\hat p_2)}{n_2}}$; the pooled proportion is used only for the significance test, never for the interval.
Chi-square statistic and degrees of freedom
$\chi^2=\sum \dfrac{(O-E)^2}{E}$ over all cells. Degrees of freedom equal $k-1$ for a goodness-of-fit test with $k$ categories, and $(r-1)(c-1)$ for a two-way table with $r$ rows and $c$ columns (independence or homogeneity).
Power and Type II error
Power is the probability of correctly rejecting a false null hypothesis: $\text{Power}=1-\beta$, where $\beta = P(\text{Type II error})$. Power increases with larger sample size, larger true effect size, and larger significance level $\alpha$.
13 practice questions
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What three conditions must be verified before constructing a confidence interval for a single proportion?
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Answer. Random, 10%, and Large Counts.
Random ensures no bias in how data were collected; the 10% condition (n ≤ 0.10N) keeps trials approximately independent when sampling without replacement; Large Counts requires $n\hat p \ge 10$ and $n(1-\hat p)\ge 10$ so the normal approximation is valid. -
In a class of 30 students, 18 support extending the lunch period. Construct a 95% confidence interval for the true proportion of all students who would support this.
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Answer. Approximately (0.425, 0.775).
$\hat p = 18/30 = 0.6$. SE $=\sqrt{0.6(0.4)/30}=0.0894$. Margin of error $=1.96(0.0894)=0.175$. Interval: $0.6\pm0.175 = (0.425, 0.775)$. -
A 95% confidence interval for the proportion of adults who favor a new policy is (0.42, 0.58). Interpret this interval.
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Answer. We are 95% confident that the true proportion of all adults who favor the policy is between 0.42 and 0.58.
The interpretation must reference the population parameter and the specific context, not just repeat the numbers. -
What does '95% confidence' mean in the context of repeated sampling?
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Answer. If many random samples of the same size were taken and a confidence interval computed from each, about 95% of those intervals would contain the true population proportion.
Confidence level describes the long-run success rate of the method, not the probability that one specific interval contains the parameter. -
A cereal company claims 20% of boxes contain a prize. In a random sample of 150 boxes, 21 contain a prize. Test at α = 0.05 whether the true proportion differs from 20%.
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Answer. z ≈ −1.84, p-value ≈ 0.066; fail to reject H0 — insufficient evidence the true proportion differs from 20%.
$\hat p = 21/150 = 0.14$. $z=\dfrac{0.14-0.20}{\sqrt{0.20(0.80)/150}}=\dfrac{-0.06}{0.0327}\approx -1.84$. Two-sided p-value ≈ 0.066 > 0.05, so H0 is not rejected. -
When testing H0: p1 = p2 for two proportions, what value should be used in the standard error, and why?
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Answer. The pooled proportion $\hat p_c = (x_1+x_2)/(n_1+n_2)$.
Under H0 both samples are assumed to come from a population with the same proportion, so combining the successes and sample sizes gives the best single estimate of that common proportion for the standard error. -
Sample 1: n₁ = 80, x₁ = 32. Sample 2: n₂ = 100, x₂ = 25. Construct a 90% confidence interval for p1 − p2.
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Answer. Approximately (0.035, 0.265).
$\hat p_1=0.40$, $\hat p_2=0.25$, difference = 0.15. SE $=\sqrt{0.4(0.6)/80+0.25(0.75)/100}=\sqrt{0.003+0.001875}=0.0698$. Margin $=1.645(0.0698)=0.115$. Interval: $0.15\pm0.115=(0.035,0.265)$. -
A quality control test uses H0: defect rate = 2% vs Ha: defect rate > 2%. Describe a Type I error in this context and one consequence.
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Answer. Concluding the defect rate is greater than 2% when it is actually still 2%.
A Type I error means rejecting a true H0. Here the consequence would be unnecessarily halting or adjusting a production process that was actually working fine, wasting time and money. -
For the same defect-rate test, describe a Type II error and explain how it relates to power.
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Answer. Concluding the defect rate is still 2% (failing to reject H0) when it has actually increased.
A Type II error is failing to detect a real effect. Power is the probability of avoiding this error — that is, correctly detecting the increased defect rate — and equals $1-\beta$. -
A die is rolled 60 times to test whether it is fair. Observed counts: 8, 12, 7, 15, 9, 9 for faces 1–6. Compute the chi-square statistic and state a conclusion at α = 0.05 (critical value 11.07, df = 5).
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Answer. χ² = 4.4; fail to reject H0 — no evidence the die is unfair.
Expected count for each face is 60/6 = 10. Sum of $(O-E)^2/E$: 0.4+0.4+0.9+2.5+0.1+0.1 = 4.4, with df = 6−1 = 5. Since 4.4 < 11.07, the result is not significant. -
A researcher takes one random sample of 200 people and records both their age group and their voting preference. Which chi-square test is appropriate, and why?
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Answer. A chi-square test of independence.
A single sample classified by two categorical variables tests whether those variables are associated. A test of homogeneity, by contrast, compares the distribution of one variable across two or more separately sampled populations. -
What condition on expected cell counts is required before using any chi-square test?
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Answer. All expected counts must be at least 5.
This Large Counts condition, along with Random sampling and the 10% condition (when sampling without replacement), ensures the chi-square statistic approximately follows a chi-square distribution. -
A two-way table has 4 rows and 3 columns. What are the degrees of freedom for a chi-square test of independence on this table?
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Answer. df = 6.
For a two-way table, degrees of freedom equal $(r-1)(c-1) = (4-1)(3-1) = 3 \times 2 = 6$.
What people get wrong
- Using the pooled proportion when constructing a two-proportion confidence interval. The pooled proportion is only for the significance test of H0: p1 = p2; confidence intervals must use each sample's own $\hat p_1$ and $\hat p_2$ in the standard error.
- Checking the Large Counts condition using n alone instead of the actual expected successes and failures. Always compute $n\hat p$, $n(1-\hat p)$ (or expected counts in a table) and confirm each is at least 10 for proportions or 5 for chi-square before proceeding.
- Interpreting a confidence interval as 'there is a 95% probability the true proportion is in this interval.' The parameter is fixed, not random — the correct interpretation is about the long-run percentage of intervals from repeated sampling that would capture the true value.
- Mixing up degrees of freedom formulas — using (r−1)(c−1) for a goodness-of-fit test or k−1 for a two-way table. Match the formula to the test: k−1 for goodness of fit, (r−1)(c−1) for independence or homogeneity.
- Treating the p-value as 'the probability the null hypothesis is true.' It is instead the probability of observing a result as extreme or more extreme than what was found, assuming H0 is true — describe it that way in conclusions.
- Describing a Type I or Type II error in the abstract ('rejecting a true null') instead of in the context of the specific claim being tested. Exam graders require the error stated in terms of the actual scenario.
Drill this unit until it sticks
These questions come back on a schedule built from what you get wrong, alongside the rest of AP Statistics. Free, and no account needed to start.