9.6 Skills Focus: Selecting an Appropriate Inference Procedure

Use a z-test for a mean only when the population standard deviation σ is known (rare in practice) and the sampling distribution of the mean is approx. normal. If σ is unknown (the usual case) you must use a t-test, which replaces σ with the sample SD s and uses a t-distribution with df = n−1 (one sample) or the Welch df for two samples (pooled t only when you can assume equal variances). Quick decision flow: - One-sample or two-sample mean, σ unknown → use t-test (or paired t for matched pairs). - σ known → z-test for mean (textbook case only). - For large n (CLT) the t and z give similar results, but AP expects t when σ is unknown. Check normality (histogram/CLT) and independence (random sample, 10% rule). Use Table B (t critical values) on the AP formula sheet. Want targeted practice on picking procedures? See the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ) and try more problems (https://library.fiveable.me/practice/ap-statistics).

One-sample vs two-sample tests—short version: a one-sample test compares a sample statistic to a single population value (e.g., test whether μ = 10 or p = 0.5). A two-sample test compares the difference between two independent groups (e.g., μ1 − μ2 or p1 − p2) to see if that difference is zero or some other value. Key AP details you should check before picking a test (Skill 1 in the CED): - Are you comparing one group to a known value? → one-sample t or z for a mean/proportion. - Are you comparing two independent groups? → two-sample t (pooled rarely used) or two-proportion z. - Are the two measurements matched/paired (same subjects before/after)? → use paired (matched pairs) t (that’s effectively a one-sample test on differences). - Conditions: CLT/normality for means (or use t), success–failure for proportions, equal-variance assumption when considering pooled vs unpooled t, independence/random sampling. For more examples and a decision flow for choosing procedures, check the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ). For extra practice, Fiveable has tons of practice problems (https://library.fiveable.me/practice/ap-statistics).

Use a chi-square test when your data are categorical; use a t-test when your data are quantitative (means). Quick checklist: - Chi-square tests (goodness-of-fit or test of independence): both variables are categorical. Use χ² = Σ[(obs − exp)²/exp]. Conditions: counts are from a random sample, expected counts (usually) ≥ 5 for each cell, and degrees of freedom = (rows−1)(cols−1) for independence or k−1 for goodness-of-fit. AP covers these in Unit 8/Topic 9.6 choices about inference procedures. See the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ). - t-tests (one-sample, two-sample, paired): response is quantitative and you’re testing a mean (μ or μ1−μ2). Conditions: random sample or experiment, approximately normal sampling distribution (CLT helps if n is large), and consider equal-variances vs unpooled two-sample t and correct df. Use paired t for matched pairs. Also note: for proportions use z-tests/CI for p (success–failure condition). For more practice problems, go to Fiveable practice (https://library.fiveable.me/practice/ap-statistics).

Use a paired (matched-pairs) t when your data are two measurements on the same unit (before/after, left/right, matched subjects). You analyze the n differences with a one-sample t on those differences: check independence of pairs, approximate normality of the differences (or large n), and use t(df = n−1). Use a two-sample t when you have two independent groups (different people in each group). Then compare group means with a t for (x̄1 − x̄2). Check: independent samples, each group’s distribution approx. normal (or large n), and 10% condition if sampling without replacement. Decide pooled vs unpooled: only pool variances when you have good reason to assume equal variances (homoscedasticity); otherwise use the unpooled Welch t (more common). Degrees of freedom for Welch t are adjusted (calculator/computer gives df). On the AP exam you must “select statistical methods” (Skill 1) and verify conditions—practice choosing between paired vs two-sample problems (see the Topic 9.6 study guide: https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ). For extra practice, try problems at (https://library.fiveable.me/practice/ap-statistics).

Decide by the research question—the alternative hypothesis (Ha) determines one- vs two-tailed. If you want to detect any difference (not specified direction), use a two-tailed test: Ha: parameter ≠ value (rejecting H0 means “different”). If you’re specifically testing for an increase or a decrease, use a one-tailed test: Ha: parameter > value (right-tailed) or Ha: parameter < value (left-tailed). Key rules: 1) Pick the tail BEFORE looking at the data. 2) Your p-value is computed according to that tail: two-tailed doubles the one-sided tail area for symmetric tests. 3) Be explicit in context (e.g., Ha: μ > 50 means you’re testing for an increase in the population mean). On the AP exam you’ll be asked to identify Ha and the correct tail, compare p to α, and state conclusions in context (Topic 9.6 skills). For more guidance and practice, see the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ) and the Unit 9 overview (https://library.fiveable.me/ap-statistics/unit-9). Practice problems are at (https://library.fiveable.me/practice/ap-statistics).

Before you pick an inference test check these conditions (quick checklist by test): General first: randomness (random sample or randomized experiment) and independence (sample ≤ 10% of population when sampling without replacement). Proportions (z-test / CI): success–failure: np ≥ 10 and n(1–p) ≥ 10 (or use bootstrap/permutation if not). Use pooled p̂ for two-proportion tests when H0 assumes equal p. Means (one/two-sample t, paired t): roughly normal sampling distribution—either population approximately normal or n ≥ 30 (CLT). For two-sample t check equal-variances if you plan to pool; otherwise use unpooled (Welch). For paired use differences and treat as one-sample t on differences. Regression slope (t for β): linear relationship, residuals ~ Normal for each x, constant variance (homoscedasticity), independence of observations, and sufficient sample size (df = n−2). Check residual plot. Chi-square tests: expected count ≥ 5 (or mostly ≥5) in each cell for goodness-of-fit / independence; independence/random sample. ANOVA (one-way): independent groups, roughly normal groups (or large n), equal variances. If conditions fail, use bootstrap CIs or permutation/randomization tests. For more AP-aligned practice and summaries see the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ) and unit overview (https://library.fiveable.me/ap-statistics/unit-9). For extra practice problems go to (https://library.fiveable.me/practice/ap-statistics).

Use a proportion test when your response is categorical (yes/no, success/failure) and you’re estimating or testing a population proportion p (one-sample z for p, two-sample z for p1−p2). Use a mean test when your response is quantitative and you’re estimating or testing a population mean μ (one-sample t, two-sample t, paired t for matched data, or t for slope in regression). Key checklist from the CED: - Data type: categorical → proportion test; quantitative → mean test. - Sample size/conditions: for proportions check the success–failure rule (np and n(1−p) large) so CLT gives approx normal; for means check normality of the population or that n is large (CLT) or use t if σ unknown. - If two related measurements on same unit → paired t. If two independent samples → two-sample t (decide pooled vs unpooled if variances assumed equal). On the AP exam you’ll pick the test, state H0/Ha, verify conditions, compute statistic and p-value (tables/calculator allowed). For more practice and a quick decision guide see the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ) and lots of problems at (https://library.fiveable.me/practice/ap-statistics).

Use a confidence interval when your goal is estimation—you want a plausible range for a population parameter (mean, slope, difference, proportion). Use a hypothesis test when your goal is decision—you’re testing a specific claim (H0 vs Ha) and want a p-value or reject/do-not-reject conclusion. Quick checklist (AP-style): - Type of parameter: mean → t (or z if σ known), slope → t for regression, proportion → z (use success–failure), categorical association → χ², more than 2 means → ANOVA. - Paired data → paired t; two independent samples → two-sample t (check equal-variances rule for pooled vs unpooled). - Conditions: CLT/normality for means or large n, success–failure for proportions, expected cell counts for χ², linearity and equal variance for regression slope. If conditions fail, use bootstrap CIs or permutation tests (AP recognizes these). - On the exam: show hypotheses or CI formula, check conditions, use correct t/z/χ²/ANOVA table or calculator—you’ll get formula sheets and tables (CED). For more targeted choosing practice and examples, see the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ) and the Unit 9 overview (https://library.fiveable.me/ap-statistics/unit-9). Need practice? Thousands of problems are at (https://library.fiveable.me/practice/ap-statistics).

Quick step-by-step you can use on the exam: 1) Identify the parameter and data type: mean (quantitative), proportion (categorical), difference of means/proportions, slope, or association (chi-square/ANOVA). 2) Design and pairing: Is it one sample, two independent samples, or paired/matched? Paired → paired t. Independent → one- or two-sample procedures. 3) Pick the method: one-sample t / two-sample t (or pooled t if equal-variances and justified) for means; z-test/CI for proportions; t for slope (regression); chi-square GOF or independence for categorical tables; ANOVA for >2 group means; bootstrap/permutation when conditions fail or for simulation-based inference. 4) Check conditions: randomization/independence, CLT/normality (or look at skew and n), success–failure for proportions, equal-variances for pooled t, expected counts for chi-square. Use t df or z as required. 5) If any condition fails, switch to bootstrap or permutation/randomization methods. Want a checklist and examples? See the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ) and try practice problems (https://library.fiveable.me/practice/ap-statistics).

Use a t-test when you’re comparing means for two groups; use one-way ANOVA when you’re comparing means for three or more groups. Quick rules tied to the CED: - One-sample t: compare one mean to a value. - Two-sample t (independent): compare means of two independent groups. Choose pooled t if equal-variances (homoscedasticity) is plausible; otherwise use unpooled (Welch) t. - Paired t: compare two related measurements (matched pairs). - One-way ANOVA: test H0: all group means are equal vs Ha: at least one differs—appropriate when k ≥ 3 groups. The ANOVA F statistic follows an F distribution and assumes independent samples, approximate normality within groups, and equal variances across groups (check those conditions). On the AP exam you must identify the right test from context (Skill 1) and verify conditions (normality, CLT, equal-variances when required). For practice picking procedures and checking conditions, see the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ) and hundreds of practice questions (https://library.fiveable.me/practice/ap-statistics).

Start with a short checklist every time—it’ll make the choice automatic. 1. What parameter are you testing? mean, proportion, slope (regression), or association (chi-square/ANOVA)? 2. How many groups/samples? one-sample, two independent samples, or paired/matched pairs? 3. Data type: quantitative → t-tests/ANOVA/regression; categorical → z-test for proportions, chi-square goodness-of-fit or independence. 4. Conditions: For proportions check success–failure (np ≥ 10). For means check normality or CLT (n≥30) and whether variances are equal (pooled vs. unpooled t). For slope use t-test for b with linearity, normal residuals, equal variance. If conditions fail, use bootstrap or permutation/randomization tests. 5. If >2 groups for means, use one-way ANOVA; if >2 categorical levels and testing independence, use χ2 test. On the AP exam you’re being scored on Skill 1 (selecting methods), so explicitly state these decisions and which conditions you checked. For practice and guided examples see the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ) and the Unit 9 overview (https://library.fiveable.me/ap-statistics/unit-9). For extra practice try the 1000+ AP practice problems (https://library.fiveable.me/practice/ap-statistics).

Use a chi-square goodness-of-fit when you have one categorical variable and you want to test whether the observed counts match a specific expected distribution (e.g., "are birthdays equally likely across months?" or "do voters follow the party proportions the poll claims?"). Goodness-of-fit compares observed vs expected counts; df ≈ k − 1 (k = # categories). Use a chi-square test of independence when you have two categorical variables and you want to test whether they’re associated (e.g., "is smoking status related to exercise level?"). That uses a contingency table and df = (rows − 1)(cols − 1). Common AP/CEF checks for both: data are counts from random sample, expected count in each cell/category ≳ 5 (if not, consider combining categories or using an exact/simulation method), and you compute χ² = Σ (obs − exp)²/exp. On the AP exam you’ll be expected to pick the right test, state H0/Ha in context, check conditions, and interpret p-value. For a quick refresher, see the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ); for more practice problems go to (https://library.fiveable.me/practice/ap-statistics).

Quick guide—when to use which t-test and why: - One-sample t-test: use when you compare a sample mean to a known population value (H0: μ = value). Conditions: random, independent (10% rule), sample distribution ≈ normal or n big enough (CLT). Uses t with df = n−1. - Two-sample (independent) t-test: compare means from two independent groups (H0: μ1 = μ2). Use unpooled (Welch) t by default unless you can justify equal variances (homoscedasticity). SE = sqrt(s1^2/n1 + s2^2/n2); df depends on variances/sample sizes. - Paired (matched pairs) t-test: use when observations are paired (before/after, matched subjects). Convert to one-sample test on differences; df = n_pairs − 1. Key AP stuff to remember: check normality (plots/outliers), independence, and success-failure only for proportions (z-tests). Use pooled vs. unpooled only if equal-variance condition is justified. If assumptions fail or samples are small, use bootstrap CI or permutation/randomization tests (also tested on the AP). For categorical comparisons use z for proportions or chi-square tests and ANOVA for >2 group mean comparisons. For more AP-aligned practice and quick review see the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ), the Unit 9 overview (https://library.fiveable.me/ap-statistics/unit-9), and tons of practice questions (https://library.fiveable.me/practice/ap-statistics).

Quick flowchart you can keep in your head: 1) What’s your response? Categorical (counts/proportions) or quantitative (means/slopes)? - If categorical → consider z-tests for proportions (one- or two-sample) or chi-square (goodness-of-fit or independence for tables). - Check success–failure: np and n(1−p) ~≥10 for z-approx. - For tables use expected counts ≥5 (chi-square condition). - If quantitative → are you comparing means or modeling relationship? - Comparing two related measurements → matched-pairs t (paired). - Comparing two independent groups → two-sample t (check normality or large n; check equal-variances for pooled vs unpooled). - One group vs a value → one-sample t. - More than two groups → ANOVA (one-way). - Regression slope → t-test for slope (check linearity, normal residuals, constant variance). 2) Are model assumptions violated or sample small? → use permutation/randomization tests or bootstrap CIs. 3) Always state H0/Ha, check CLT/normality, and report df/SE/z or t and p-value. For AP-aligned practice and a tidy flowchart, see the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ) and Unit 9 overview (https://library.fiveable.me/ap-statistics/unit-9). For lots of practice problems go to (https://library.fiveable.me/practice/ap-statistics).

Your calculator gives different p-values because each test uses a different sampling distribution, standard error, and degrees of freedom—so the same data plugged into different procedures yields different null-model probabilities. Quick checklist of why p-values change: - Test choice changes the statistic and SE: one-sample t uses s/√n; two-sample (pooled vs unpooled) uses √(s1²/n1 + s2²/n2); z for proportions uses a pooled p̂ in the SE when H0 assumes equality. Those different SEs change the z/t value and p. - Paired test = one-sample test on differences (usually smaller SE → different p). - t vs z: t has thicker tails and df matters, so p depends on df. - Conditions matter: CLT/normality and success–failure affect whether z, t, or a simulation (bootstrap/permutation) is appropriate—simulated/randomization tests give p-values that can differ from parametric tests. - Also check calculator settings: order of samples, direction of Ha (one- vs two-tailed), and rounding. On the AP exam you must pick the correct procedure (paired vs two-sample, pooled vs unpooled, z vs t, chi-square, ANOVA, or a simulation-based test) and justify conditions (CLT, normality, success–failure). For a focused review, see the Topic 9.6 study guide (https://library.fiveable.me/ap-statistics/unit-9/selecting-an-appropriate-inference-procedure/study-guide/TYrF9PsFWB1lTadg0ljQ) and practice problems (https://library.fiveable.me/practice/ap-statistics).