8.7 Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data

Use a two-proportion z-test when you’re comparing exactly two proportions (like treatment vs control) and the sampling distribution of the difference in sample proportions is approximately normal. Conditions: independent random samples, and "large counts"—each sample should have at least about 10 successes and 10 failures (so the standard error formula is reliable). If you’re testing equality of two proportions, compute the pooled proportion for the SE. Use a chi-square test when you have categorical data with more than two groups or more than two categories per variable, or when you want a single test that handles an r × c contingency table. Types: goodness-of-fit (one categorical variable, compare observed to expected), test of independence or homogeneity (two categorical variables). Conditions: independent random sampling and expected cell counts generally ≥ 5. Degrees of freedom = (rows−1)(cols−1). For tiny samples or when expected counts <5, use Fisher’s exact test or a randomization test instead. For AP alignment and practice, review Topic 8.7 (selecting procedures) on Fiveable (study guide: https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl; unit overview: https://library.fiveable.me/ap-statistics/unit-8) and try practice problems (https://library.fiveable.me/practice/ap-statistics).

Goodness-of-fit and test for independence both use χ² but answer different questions. - Goodness-of-fit: asks whether one categorical variable’s observed distribution matches a specified single distribution (e.g., fair die or given percentages). You use one-way counts, expected counts from the claimed proportions, χ² = Σ(Observed–Expected)²/Expected, df = k − 1 (k = number of categories). Check large-counts condition (expected ≥ 5 or use simulation/Fisher’s ideas if small). - Test for independence: asks whether two categorical variables are associated (no relationship) in a contingency table. Observed are the table counts; expected = (row total × column total)/grand total; χ² uses those expected counts and df = (r − 1)(c − 1). Also check large-counts condition; if many small expected counts, consider Fisher’s exact test or a permutation/randomization approach. Both are in Unit 8 (chi-square inference). For quick review, see the Topic 8.7 study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl), the unit overview (https://library.fiveable.me/ap-statistics/unit-8), and lots of practice problems at (https://library.fiveable.me/practice/ap-statistics).

Short cheat sheet so you’ll know which test to pick: - One-proportion z-test: use when you’re testing a single population proportion (H0: p = something). Need a random sample and large counts (np̂ and n(1−p̂) both ≳ 10 on AP problems; use the formula and z-critical values). - Two-proportion z-test (unpooled vs pooled): use to compare p1 and p2. For hypothesis tests that assume p1 = p2 under H0, use the pooled proportion to compute the SE. Also requires large counts in each group (expected successes/failures ≳ 5). - Chi-square goodness-of-fit: use when you test whether a single categorical distribution matches specified proportions across 3+ categories. Work with counts, compute expected counts, df = (k−1). - Chi-square test of independence / homogeneity: use for relationships between two categorical variables in an r×c table. Conditions: random sample, expected counts generally ≥5, df = (r−1)(c−1). Remember chi-square uses counts (not proportions). - Fisher’s exact test or permutation/randomization: use when sample sizes or expected counts are too small for chi-square (especially 2×2 tables). AP accepts randomization/permutation approaches too. Extras: Yates’ continuity correction is rarely required on the AP. For practice on selecting methods, see the Topic 8.7 study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl), the Unit 8 overview (https://library.fiveable.me/ap-statistics/unit-8), and >1000 practice problems (https://library.fiveable.me/practice/ap-statistics).

Step-by-step for a chi-square goodness-of-fit test (AP style): 1. State hypotheses in context: H0 = the categorical distribution matches the claimed proportions; Ha = it does not. 2. Check conditions: data from a random sample/experiment and the large counts rule (all expected counts ≥ 5). If not met, consider simulation/Fisher’s exact (when small). 3. Compute expected counts: expected = (total sample size) × (claimed proportion) for each category. 4. Calculate the chi-square statistic: χ² = Σ[(observed − expected)² / expected] across categories. 5. Degrees of freedom: df = k − 1 (k = number of categories). 6. Find the p-value: use χ² table, calculator, or software (AP students may use table on the formula sheet). Compare p to α. 7. Conclusion in context: if p ≤ α reject H0 (evidence the distribution differs); if p > α fail to reject H0 (no convincing evidence). 8. Always state limitations and connect conclusion to the real-world context. Review Topic 8.2–8.3 for examples and walk-throughs (study guide: https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl). For unit review and extra practice, see https://library.fiveable.me/ap-statistics/unit-8 and thousands of problems at https://library.fiveable.me/practice/ap-statistics.

The chi-square test statistic is χ² = Σ (Observed − Expected)² / Expected You compute this sum over all cells in your table. “Observed” are the counts you actually saw; “Expected” are the counts you’d expect under H₀ (for goodness-of-fit: expected = n × hypothesized proportion for each category; for independence/homogeneity: expected = (row total × column total) / grand total). Degrees of freedom: goodness-of-fit df = k − 1; independence/homogeneity df = (rows − 1)(cols − 1). Check the large-counts condition (usually expected counts ≥ 5) before using χ²; otherwise use exact or simulation methods. The AP formula sheet gives the same χ² formula—practice applying it on the Topic 8.7 study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl). For more review and practice, see the Unit 8 overview (https://library.fiveable.me/ap-statistics/unit-8) and Fiveable practice problems (https://library.fiveable.me/practice/ap-statistics).

Use a one-sample z-test for a proportion when you’re asking about a single population proportion (H0: p = p0). Example: “Is the proportion of students who bike to school equal to 0.30?” Conditions: random sample, independence (10% rule), and large counts under H0 (np0 and n(1−p0) ≥ 10). Use a two-sample z-test for proportions when you’re comparing two independent groups (H0: p1 = p2). Example: “Are freshmen and seniors equally likely to prefer online classes?” Conditions: two independent random samples, each satisfies large-counts (n1 p̂1, n1(1−p̂1), n2 p̂2, n2(1−p̂2) ≥ 10). For the test statistic, if H0 assumes p1 = p2 you use the pooled proportion p̂c = (x1 + x2)/(n1 + n2) to compute the standard error. AP exam keywords to check: pooled proportion, large counts condition, independence, two-proportion z-test (see Topic 8.7 study guide for decision rules and examples: https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl). For more review and practice, see the Unit 8 overview (https://library.fiveable.me/ap-statistics/unit-8) and practice problems (https://library.fiveable.me/practice/ap-statistics).

Quick checklist you can memorize: - Data type and design: counts (not means or percentiles) from a random sample or randomized experiment; categories are mutually exclusive. This satisfies the independence assumption (observations independent). - Expected counts (large counts rule): every expected cell count should be ≥ 5 for the chi-square approximation to be reliable. If many cells are <5, use Fisher’s exact test (small samples) or a simulation/randomization test. - Degrees of freedom: goodness-of-fit df = k − 1 (k = # categories); for two-way tables df = (rows − 1)(cols − 1). Use the chi-square statistic χ² = Σ((obs − exp)²/exp) and compare to the χ² table (AP formula sheet/Table C). - Extras to know for the exam: for 2×2 some texts mention Yates’ continuity correction, but AP expects you to check expected counts and switch to Fisher’s or simulation if counts are too small. You can also use a permutation/randomization test for independence/homogeneity when conditions fail. For a short AP-focused review, see the Topic 8.7 study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl) and practice questions (https://library.fiveable.me/practice/ap-statistics).

Pick the tail based on the research question—not the sample result. Write H0 as the “=” statement and Ha to match the claim’s direction: - If the claim says “greater than” (p > p0), do a right-tailed (one-tailed) test. - If it says “less than” (p < p0), do a left-tailed (one-tailed) test. - If it asks “different from” or makes no direction (p ≠ p0), do a two-tailed test. On the AP exam you must state H0 and Ha correctly (Skill 1). Also check conditions: random sample, large counts (np and n(1−p) ≥ 10 for one-sample z), and whether you need a pooled proportion for two-sample z tests (CED keywords: pooled proportion, large counts). Use the tail to find critical z or to interpret the p-value: compare p-value to α in context. For more on choosing procedures for categorical data, see the Topic 8.7 study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl) and the Unit 8 overview (https://library.fiveable.me/ap-statistics/unit-8). For extra practice, try problems at (https://library.fiveable.me/practice/ap-statistics).

Testing for homogeneity vs. testing for independence are very similar mathematically but answer different questions: - Test of homogeneity: you have separate groups (different populations or treatments) and you ask whether the distribution of a categorical variable is the same across those groups. Null: the category proportions are the same for every group. Data typically come from multiple independent samples. - Test of independence: you have one sample and two categorical variables measured on each unit; you ask whether the variables are associated. Null: the two variables are independent (no association) in the population. Both use chi-square on an r×c contingency table with expected = (row total × column total)/grand total and df = (r−1)(c−1), and both require random sampling and large expected counts (≈≥5) or else use Fisher’s exact or a randomization test. For a 2×2 you might instead see a two-proportion z-test (uses a pooled p under H0) but AP usually wants you to pick the correct categorical procedure (chi-square independence vs. homogeneity)—practice choosing these on the Unit 8 study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl) and more problems at (https://library.fiveable.me/practice/ap-statistics).

Use a z (normal) test when you’re comparing one or two proportions and the sampling distribution of the sample proportion(s) is approximately normal. That means: - The question is about a single proportion or difference of two proportions (one-sample z or two-sample z). - The independence/randomness condition holds. - The success–failure (large counts) condition is met (commonly np and n(1−p) ≳ 10 for z-tests). For a two-sample z-test you may use a pooled p̂ when H0 assumes equal proportions. Use a chi-square test when you have counts in ≥2 categories and you’re testing overall fit, homogeneity, or independence: - Chi-square goodness-of-fit for one categorical variable with many levels. - Chi-square test of independence / homogeneity for contingency tables (use df = (rows−1)(cols−1)). - Expected counts in every cell should be large enough (AP uses the “large counts” rule; a common classroom rule is expected count ≥ 5). - If cell counts are too small use Fisher’s exact test or a randomization/permutation approach. For AP-aligned review/examples, see the Topic 8.7 study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl) and try practice problems (https://library.fiveable.me/practice/ap-statistics).

For a chi-square test you calculate expected counts differently depending on the setup: - Goodness-of-fit (one categorical variable): expected count for category i = (total sample size) × (hypothesized proportion for category i). - Independence / homogeneity (two-way table): expected count for cell (row i, column j) = (row i total × column j total) / grand total. You use those expected counts in the chi-square statistic: χ² = Σ (observed − expected)² / expected. Expected counts matter because the χ² approximation to the sampling distribution assumes cells aren’t too small—AP’s large-counts rule is that all expected counts should be ≥ 5 (or at least most ≥ 5); if not, use Fisher’s exact test or a simulation/randomization test. Also compute degrees of freedom (goodness-of-fit: k−1; two-way: (r−1)(c−1)) to get p-values from the χ² table. For quick review and examples tied to Topic 8.7, see the Fiveable study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl). For more practice, try the AP Stats practice problems (https://library.fiveable.me/practice/ap-statistics).

If you’re comparing two proportions (two groups, yes/no outcome), start with the two-proportion z-test. Use it when: - You want to test p1 vs p2 (or build a CI for p1−p2). - The groups are independent (random samples or randomized experiment). - Large counts condition holds: each group has at least about 10 successes and 10 failures (so sampling distribution ≈ normal). - If H0: p1 = p2, use the pooled proportion p̂c = (x1+x2)/(n1+n2) to compute the SE and z statistic. If the sample sizes or expected counts are small (cells <10), use Fisher’s exact test (for 2×2) or a randomization/permutation test instead. If your data are a contingency table with >2 categories per variable, use a chi-square test of independence or homogeneity (check expected counts and df). AP CED keywords and conditions are covered in the Topic 8.7 study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl). For extra practice, try problems at (https://library.fiveable.me/practice/ap-statistics).

When you do a two-proportion z-test, follow these steps: 1. State parameters & hypotheses: p1, p2 and H0: p1 = p2 (or p1 − p2 = 0). Choose Ha (≠, >, or <) in context. 2. Check conditions (AP/CED): random samples or randomized experiment; independence (each sample ≤10% of population); large counts—use pooled counts for the test: X1, X2 so that np̂c and n(1−p̂c) are ≥10 for each sample when H0 assumes equal proportions. 3. Compute pooled proportion p̂c = (X1 + X2)/(n1 + n2). 4. Compute standard error under H0: SE = sqrt[p̂c(1−p̂c)(1/n1 + 1/n2)]. 5. Compute z = (p̂1 − p̂2)/SE and find p-value from normal model (use z-table or calculator). 6. Draw conclusion: compare p-value to α, state decision in context and interpret effect size/ practical significance. 7. (Optional) For interval estimation use unpooled SE and build a CI for p1 − p2. For a quick topic review, see the Fiveable study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl) and more practice (https://library.fiveable.me/practice/ap-statistics).

Your calculator can give different p-values for the “same” categorical test because different procedures or adjustments are being used. Common reasons: - Different tests: chi-square (approximate), Fisher’s exact (exact for 2×2), and two-proportion z-tests all target similar questions but give different p-values—use Fisher’s for small counts or exact answers. - Yates’ continuity correction: some calculator/computer functions apply Yates’ correction for 2×2 chi-square, which makes p-values a bit larger (more conservative). - Condition violations: the chi-square approximation needs expected counts “large enough” (usually all ≥1 and most ≥5). If counts are small, software may switch to exact or simulation methods (giving different p). - Different implementations: menu choices on your calculator (chi-square test vs inputting table into regression/STAT commands) can call different algorithms. For AP work, check the test you intend to use (chi-square goodness-of-fit, test of independence/homogeneity, Fisher’s exact, or two-proportion z), verify expected counts and independence/randomness, and pick the correct method. Want a quick walkthrough for your specific table? Share the observed counts and I’ll tell you which procedure the AP CED expects (see the Topic 8.7 study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl) and unit overview (https://library.fiveable.me/ap-statistics/unit-8)). For extra practice, Fiveable has lots of AP-style problems (https://library.fiveable.me/practice/ap-statistics).

Quick roadmap—which test to use for categorical data: - One proportion (z) test / CI: use when you have one sample and want to test or estimate p. Conditions: random, large counts (np̂ and n(1−p̂) ≳ 10 for AP-style), use normal approx. - Two-proportion z-test / CI: comparing two independent proportions (p1 vs p2). Conditions: random, large counts in all four cells; if H0 assumes p1=p2 you use the pooled proportion for the SE. - Chi-square goodness-of-fit: one categorical variable with >2 categories; test whether observed counts match expected distribution. Conditions: random, expected counts ≳ 5, df = k−1. - Chi-square test of independence / homogeneity (two-way table): use for association (independence) or comparing distributions across groups (homogeneity). Conditions: random, expected counts ≳ 5 in each cell, df = (r−1)(c−1). - Fisher’s exact test: use instead of chi-square when sample is small and expected counts <5 (especially 2×2). - Randomization/permutation test: alternative when assumptions fail (nonrandomization, small n); uses simulation to get p-value. - Continuity correction (Yates’) and other tweaks: rarely required on AP; mention only if asked. For AP exam practice and quick refresh see the Topic 8.7 study guide (https://library.fiveable.me/ap-statistics/unit-8/selecting-an-appropriate-inference-procedure-for-categorical-data/study-guide/0hC6NNjpHXs0x44bc2Fl), the Unit 8 overview (https://library.fiveable.me/ap-statistics/unit-8), and lots of practice problems (https://library.fiveable.me/practice/ap-statistics).