Credit Sprint / Study guides / Statistics Unit 3

Credit Sprint's Guide to AP® Statistics, Unit 3: Inference for Categorical Data: Proportions

Unit 3 is where statistics becomes inference. You learn the sampling distribution of a sample proportion, then build confidence intervals and carry out significance tests for one proportion, two proportions and (with chi-square) the relationship between categorical variables.

What you need to know

• A statistic estimates a parameter. An estimator is unbiased if the center of its sampling distribution equals the parameter. The sample proportion p̂ is an unbiased estimator of p. Variability comes from sampling, and larger samples reduce variability.
• The sampling distribution of p̂ has mean p and standard deviation √[p(1 − p) ÷ n]. It is approximately normal when np ≥ 10 and n(1 − p) ≥ 10 (the large counts condition), and the standard deviation formula needs the sample to be less than 10% of the population when sampling without replacement.
• A confidence interval for p is p̂ ± z* · √[p̂(1 − p̂) ÷ n]. Conditions: random sample, 10% condition, and at least 10 successes and 10 failures in the sample. The interval is a set of plausible values for the parameter.
• Interpret a confidence level correctly: if we took many random samples and built an interval from each, about 95% of those intervals would capture the true proportion. Interpret an interval by saying: we are 95% confident that the true proportion of [population] who [characteristic] is between __ and __.
• The margin of error shrinks when the sample size grows and grows when the confidence level grows. To cut the margin of error in half you need about four times the sample size. You can use a confidence interval to justify a claim: if a hypothesized value is outside the interval, it is not plausible.
• A significance test begins with H₀ (a parameter equals a value) and Hₐ (the claim, either one-sided or two-sided). The test statistic z = (p̂ − p₀) ÷ √[p₀(1 − p₀) ÷ n] measures how far the sample result is from the null value in standard deviations.
• The p-value is the probability, assuming H₀ is true, of getting a sample result at least as extreme as the one observed. A small p-value is evidence against H₀. Compare it to the significance level α; if p < α, reject H₀ and conclude there is convincing evidence for Hₐ in context.
• A conclusion should state the decision, link it to the p-value, and refer to context: 'Because p = 0.02 < 0.05, we reject H₀. There is convincing evidence that the true proportion of ... is greater than ...'. Never say you 'proved' H₀. A large p-value means not enough evidence, not that H₀ is true.
• A Type I error is rejecting a true H₀ (probability α). A Type II error is failing to reject a false H₀. Power is the probability of correctly rejecting a false H₀; it rises with larger sample size, larger α and a larger true effect.
• For two proportions, the standard deviation of p̂₁ − p̂₂ is √[p₁(1 − p₁)/n₁ + p₂(1 − p₂)/n₂]. A two-proportion z-interval uses unpooled standard error. A two-proportion z-test uses a pooled proportion, p̂c = (x₁ + x₂) ÷ (n₁ + n₂), because H₀ says the proportions are equal.
• Check the conditions for each group: independent random samples or randomized groups, and at least 10 successes and 10 failures in each group. If the interval for p₁ − p₂ contains 0, there is no convincing evidence of a difference.
• A chi-square test for homogeneity compares the distribution of a categorical variable across several populations or treatment groups. A chi-square test for independence checks for an association between two categorical variables in one population. Expected count = (row total × column total) ÷ table total, and df = (rows − 1)(columns − 1). Conditions: random, and all expected counts at least 5.
• The chi-square statistic is χ² = Σ (observed − expected)² ÷ expected. A large χ² and a small p-value mean the observed counts are far from what H₀ predicts, so there is convincing evidence of a difference in distributions or an association.

Exam traps

Key terms

• Estimator: A rule or statistic used to estimate a parameter.
• Unbiased estimator: A statistic whose sampling distribution is centered at the parameter it estimates.
• Sampling distribution: The distribution of values a statistic takes over all possible samples of the same size.
• Sampling variability: The way a statistic varies from sample to sample.
• Sample proportion p̂: The number of successes in a sample divided by the sample size.
• Standard error: An estimate of the standard deviation of a statistic, computed from sample data.
• Large counts condition: np ≥ 10 and n(1 − p) ≥ 10 (or at least 10 successes and failures in the sample) for a normal model.
• 10% condition: When sampling without replacement, the sample should be less than 10% of the population so observations are nearly independent.
• Confidence interval: An interval of plausible values for a parameter, built from sample data.
• Confidence level: The percent of intervals from repeated random samples that would capture the true parameter.
• Critical value z*: The z-value that captures the middle C% of a standard normal distribution.
• Margin of error: The critical value times the standard error; half the width of an interval.
• Null hypothesis H₀: The claim of no difference or no effect, assumed true when computing the p-value.
• Alternative hypothesis Hₐ: The claim we look for evidence to support; it can be one-sided or two-sided.
• Test statistic: A number that measures how far the sample result is from the null value, in standard errors.
• p-value: The probability, if H₀ is true, of a result at least as extreme as the one observed.

Practice questions

Five of the 108 Unit 3 questions in Credit Sprint. Every choice is explained in the app.