You have spent the last nine lessons learning inference procedures one at a time: a confidence interval for a proportion here, a chi-square test there, a matched-pairs t-test last week. On a homework set, you always knew which one to use — it was the procedure from that chapter.
The AP exam is not so kind. A free-response question simply describes a study and asks, "Is there convincing evidence...?" Nobody tells you it is a two-sample t-test. Choosing the right procedure is the single highest-leverage skill on the exam, because if you name the wrong one, the rest of your work earns almost nothing — correct arithmetic on the wrong test is still the wrong test.
Quick gut-check before we build the framework. For each, just say "proportion" or "mean":
(Answers: 1 = proportions, 2 = mean, 3 = neither — that's a count/categorical situation. Hold that thought.)
Every inference procedure in this course answers one of two big questions: what is the parameter? (confidence interval) or is there convincing evidence about the parameter? (significance test). The trick is figuring out which parameter — and that depends entirely on the structure of the data. We decode that structure with four questions, asked in order.
Question 1 — Is the variable of interest CATEGORICAL or QUANTITATIVE?
This is the most important fork, and the one students blow most often.
Categorical → proportion or chi-square procedures (z and χ²). Quantitative → mean procedures (t). Get this fork wrong and nothing downstream can be right.
Question 2 — How many samples or groups: one, two, or several?
Count the populations being compared.
Question 3 — For two quantitative samples: INDEPENDENT or PAIRED?
This fork only matters once you are in mean-land with two sets of numbers.
Question 4 — Confidence INTERVAL or significance TEST?
[GRAPH: Decision flowchart for choosing an inference procedure.
START: "What kind of variable of interest?"
│
├── CATEGORICAL (proportion / counts)
│ │
│ ├── ONE sample, ONE categorical variable ──► ONE-PROPORTION z
│ │ (interval or test)
│ │
│ ├── TWO groups, compare a proportion ──────► TWO-PROPORTION z
│ │ (interval or test)
│ │
│ └── COUNTS in a two-way table ─────────────► CHI-SQUARE
│ ├── several groups, ONE response ──► χ² test for HOMOGENEITY
│ └── ONE sample, TWO variables ─────► χ² test for INDEPENDENCE
│
└── QUANTITATIVE (mean, measured units)
│
├── ONE sample / one mean ──────────────────► ONE-SAMPLE t
│ (interval or test)
│
└── TWO sets of numbers
├── INDEPENDENT groups ────────────► TWO-SAMPLE t
│ (interval or test)
└── PAIRED (one column of diffs) ──► MATCHED-PAIRS t
(one-sample t on differences)
At every leaf: ask "INTERVAL or TEST?" — that picks Int vs. Test on the menu.]| Scenario features | Procedure | Conditions | TI-84 |
|---|---|---|---|
| One categorical variable, one sample, one % | 1-proportion z | Random sample/assignment; 10% (n ≤ 0.10N) for independence; Large Counts — test: np₀ ≥ 10 and n(1−p₀) ≥ 10; interval: np̂ ≥ 10 and n(1−p̂) ≥ 10 | 1-PropZInt / 1-PropZTest |
| Compare a % between two groups | 2-proportion z | Random (both); 10% (both); Large Counts — all of n₁p̂₁, n₁(1−p̂₁), n₂p̂₂, n₂(1−p̂₂) ≥ 10 (test uses pooled p̂_c) | 2-PropZInt / 2-PropZTest |
| Several groups, one categorical response (counts) | χ² homogeneity | Random; 10% (each group); all expected counts ≥ 5 | χ²-Test (matrix) |
| One sample, two categorical variables (counts) | χ² independence | Random; 10%; all expected counts ≥ 5 | χ²-Test (matrix) |
| One quantitative variable, one mean | 1-sample t | Random; 10%; Normal/Large Sample — n ≥ 30, or stated normal, or graph of data shows no strong skew/outliers; df = n−1 | TInterval / T-Test |
| Two independent groups, compare means | 2-sample t | Random (both); 10% (both); Normal/Large Sample for each group (each n ≥ 30, or stated normal, or graphs ok) | 2-SampTInt / 2-SampTTest |
| Paired data → one column of differences | Matched-pairs t | Random (pairs/assignment); 10%; Normal/Large Sample on the differences (n_diff ≥ 30 or diffs roughly symmetric, no strong skew/outliers); df = n_pairs − 1 | TInterval / T-Test on the diff list |
Two things to burn in. First, every procedure needs Random — without it you cannot generalize, period. Second, the three "shapes" of the third condition track the procedure family: proportions use Large Counts (the np ≥ 10 rule), chi-square uses expected counts ≥ 5, and means use Normal/Large Sample (CLT). Naming the wrong condition shape is a tell that you named the wrong procedure.
Both use the exact same χ²-Test calculator routine and the same (r−1)(c−1) degrees of freedom. They differ only in how the data were collected, and that determines what you may conclude:
Same math, different design, different conclusion sentence. On the exam, the data-collection description is your only clue — read it carefully.
On the new point-based FRQ rubric, the naming step is usually its own scored component. If you write "two-sample z-test for proportions" when the variable is quantitative, you don't just lose the naming point — you forfeit the conditions, the mechanics, and the conclusion, because they're all evaluated against the procedure you named. Spending ten seconds on the four questions protects every other point in the problem.
For each: identify the procedure, justify it with the decision questions, and state conditions. We do not compute — selection is the skill here.
A nutritionist records the sodium content (in mg) of 40 randomly selected frozen dinners and wants to estimate the true mean sodium content.
Procedure: one-sample t-interval for a mean. (TI-84: TInterval.)
Conditions: Random (stated); 10% (40 ≤ 10% of all frozen dinners); Normal/Large Sample (n = 40 ≥ 30, so CLT applies). df = 39.
A polling firm asks a random sample of 600 voters whether they support a ballot measure. They want to know if there is convincing evidence that more than half support it.
H₀: p = 0.5.Procedure: one-proportion z-test. (TI-84: 1-PropZTest.)
Conditions: Random (stated); 10% (600 ≤ 10% of all voters); Large Counts using p₀ = 0.5: 600(0.5) = 300 ≥ 10 and 600(0.5) = 300 ≥ 10. ✓
A school nurse measures resting heart rate before and after an 8-week exercise program for the same 25 students. Is there convincing evidence the program lowered heart rate?
H₀: μ_diff = 0, Hₐ: μ_diff < 0 with diff = after − before).Procedure: matched-pairs t-test (one-sample t on the differences). (TI-84: T-Test on the difference list.)
Conditions: Random (treat as random/representative; the order of measurement is fixed, but inference is on the within-subject differences); 10% (25 ≤ 10% of comparable students); Normal/Large Sample on the differences — n_diff = 25 < 30, so graph the 25 differences and check for no strong skew or outliers. df = 24.
Researchers take separate random samples of 150 dog owners and 150 cat owners and record each person's preferred social media platform (Instagram, TikTok, or Facebook). Is there evidence the distribution of preferred platform differs between dog and cat owners?
Procedure: chi-square test for homogeneity. (TI-84: χ²-Test, after entering the 2×3 counts as a matrix.)
Conditions: Random (both samples); 10% (each owner type ≤ 10% of its population); all expected counts ≥ 5. df = (2−1)(3−1) = 2.
A company tests two website layouts. It randomly assigns 500 visitors to Layout A and 500 to Layout B, then records whether each visitor made a purchase. Estimate the difference in purchase rates between the layouts.
Procedure: two-proportion z-interval. (TI-84: 2-PropZInt.)
Conditions: Random — randomized experiment, so OK to compare; 10% — not strictly required for a randomized experiment, but note if treating as samples; Large Counts — need x_A, 500−x_A, x_B, 500−x_B all ≥ 10 (check the number of successes and failures in each group ≥ 10).
Note: because this is a randomized experiment rather than random sampling from a population, the 10% condition is about sampling and is often waived; the Random condition is met by random assignment, which licenses a cause-and-effect conclusion.
A fitness app reports daily step counts. A researcher takes a random sample of 50 users who use the app's reminder feature and a separate random sample of 50 who don't, and compares the mean daily steps.
Procedure: two-sample t-test (or interval) for means. (TI-84: 2-SampTTest / 2-SampTInt.)
Conditions: Random (both samples); 10% (each ≤ 10% of its population); Normal/Large Sample for each group (n = 50 ≥ 30 each, CLT ✓).
1. Proportion vs. mean confusion (the #1 error). Students latch onto a yes/no grouping variable and pick a proportion procedure, when the variable of interest being measured is a number. Ask: "What did they record on each individual — a category or a measured number?" In Example 6, the grouping is categorical, but the response (steps) is quantitative → it's a mean procedure.
2. Homogeneity vs. independence. These share calculator and df, so students treat them as interchangeable. They are distinguished only by design: several separate samples + one response = homogeneity; one sample + two variables recorded = independence. The wrong choice gives the wrong conclusion sentence ("distributions differ" vs. "variables are associated").
3. Independent vs. paired. The classic miss is running a two-sample t on before/after data from the same subjects. If each value in one column is naturally tied to exactly one value in the other (same person, twins, matched), it's paired — collapse to one column of differences and run a one-sample t. The tell: can you subtract row by row meaningfully?
4. Interval vs. test. "Estimate / how much / what is the difference" → interval. "Is there convincing evidence / is there a difference" → test. Running a test when asked to estimate (or vice versa) loses the naming and conclusion points even if the procedure family is right.
5. One- vs. two-sample. Count the populations. "Compared to the national rate of 0.20" is one sample tested against a known value p₀ = 0.20, not two samples. Two-sample procedures require two groups of data you actually collected.
For MC items, choose the best procedure. "ID" items ask you to name the procedure and conditions.
A random sample of 80 college students reports hours of sleep per night; you want to estimate the mean. Which procedure?
- (A) One-proportion z-interval
- (B) One-sample t-interval
- (C) Two-sample t-interval
- (D) Matched-pairs t-interval
You compare the proportion of defective items from two production lines using a random sample from each. Which procedure?
- (A) Two-sample t-test
- (B) χ² test for homogeneity
- (C) Two-proportion z-test
- (D) One-proportion z-test
Each of 30 volunteers is tested with their left hand and their right hand on a reaction-time task (ms). You test for a difference. Which procedure?
- (A) Two-sample t-test
- (B) Matched-pairs t-test
- (C) Two-proportion z-test
- (D) One-sample t-test on raw times
A single random sample of 400 adults is classified by both education level (3 categories) and whether they own a home (yes/no). You ask whether the two variables are associated. Which procedure?
- (A) χ² test for homogeneity
- (B) χ² test for independence
- (C) Two-proportion z-test
- (D) Two-sample t-test
A coffee shop claims 90% of customers are satisfied. In a random sample of 200, you test whether the true satisfaction proportion is less than 0.90. Which procedure?
- (A) One-proportion z-test
- (B) One-sample t-test
- (C) Two-proportion z-test
- (D) χ² test for independence
Separate random samples of patients at three hospitals are each classified as readmitted / not readmitted. You ask whether the readmission distribution is the same across hospitals. Which procedure?
- (A) χ² test for independence
- (B) χ² test for homogeneity
- (C) Two-proportion z-test
- (D) One-sample t-test
Which condition shape belongs to a two-sample t procedure?
- (A) np₀ ≥ 10 and n(1−p₀) ≥ 10
- (B) All expected counts ≥ 5
- (C) Normal/Large Sample checked for each group separately
- (D) np̂ ≥ 10 and n(1−p̂) ≥ 10
A researcher wants to estimate, with 95% confidence, the difference in mean test scores between students taught online vs. in person, using two independent random samples. Which procedure?
- (A) Matched-pairs t-interval
- (B) Two-sample t-interval
- (C) Two-proportion z-interval
- (D) One-sample t-interval
Which scenario calls for a one-proportion z-test (not two-sample)?
- (A) Comparing pass rates at two schools
- (B) Testing whether a single coin's heads-rate differs from 0.5
- (C) Estimating mean height from one sample
- (D) Comparing mean income across three cities
(In context) A gym surveys a random sample of 250 members, asking each their membership tier (Basic/Plus/Premium) and whether they attend group classes (yes/no), to see if tier and class attendance are related. ID: name the procedure, the TI-84 command, and the conditions.
(In context) A teacher records each of her 28 students' quiz scores (0–100) before and after a new study method and wants to know if scores improved. ID: name the procedure, the TI-84 command, and the conditions, including the correct df.
(In context) A public-health official takes a random sample of 1,000 residents and finds 230 are uninsured. She wants a 95% confidence interval for the true proportion uninsured. ID: name the procedure, the TI-84 command, and the conditions.
A study draws independent random samples of 45 urban and 45 rural commuters and compares their mean commute times (minutes). Which procedure?
- (A) Two-proportion z-test
- (B) Two-sample t-test
- (C) Matched-pairs t-test
- (D) χ² test for homogeneity
Which of the following always requires the Random condition?
- (A) Only proportion procedures
- (B) Only test procedures (not intervals)
- (C) Every inference procedure in this course
- (D) Only chi-square procedures
(In context) Four different fertilizers are each applied to a separate random sample of 50 plots; each plot is later classified as high-yield / not high-yield. Researchers ask whether the proportion of high-yield plots is the same across fertilizers. ID: name the procedure, the TI-84 command, and the conditions, including df.
Statistical Practice 1 — Formulate Questions
A statistics teacher is reviewing four different studies with her class. For each study below, the teacher asks students to (i) identify the appropriate inference procedure by name and (ii) state the conditions required to use it. (No computation is required.)
Study A. A consumer group takes a random sample of 60 family sedans and records each car's fuel economy (in miles per gallon). They want to estimate the true mean fuel economy of all such sedans.
Study B. A school district takes separate random samples of 120 elementary, 120 middle, and 120 high school teachers and records each teacher's primary mode of commuting (car, public transit, walk/bike). The district asks whether the distribution of commuting mode is the same across the three school levels.
Study C. A trainer measures each of 22 randomly selected athletes' vertical jump height (in inches) before and after a 6-week plyometric program. The trainer wants convincing evidence that the program increased jump height.
Study D. In a randomized experiment, 300 patients with chronic headaches are randomly assigned, 150 to a new drug and 150 to a placebo, and each is recorded as "improved" or "not improved." Researchers want to estimate the difference in improvement rates.
(a) For Study A, name the procedure and state its conditions. (2 pts)
(b) For Study B, name the procedure and state its conditions. (3 pts)
(c) For Study C, name the procedure and state its conditions. (2 pts)
(d) For Study D, name the procedure and state its conditions. (3 pts)
(a) Study A — One-sample t-interval for a mean.
Fuel economy (mpg) is a quantitative variable measured on a single sample, and we want to estimate a mean → one-sample t-interval. Conditions:
- Random: the 60 sedans are a random sample. ✓
- 10%: 60 ≤ 10% of all family sedans. ✓
- Normal/Large Sample: n = 60 ≥ 30, so the sampling distribution of x̄ is approximately Normal by the CLT. ✓ (df = 59)
(b) Study B — Chi-square (χ²) test for homogeneity.
The response (commuting mode) is categorical with three categories, recorded across three separately sampled groups, and we ask whether the distribution is the same → homogeneity. Conditions:
- Random: separate random samples from each school level. ✓
- 10%: each sample (120) ≤ 10% of teachers at that level. ✓
- Large/Expected Counts: all expected counts ≥ 5. ✓ (df = (3−1)(3−1) = 4)
(c) Study C — Matched-pairs t-test (one-sample t-test on the differences).
Jump height is quantitative, and the same 22 athletes are measured before and after, so the data pair naturally into one column of differences; we want evidence of an increase → matched-pairs t-test, H₀: μ_diff = 0 vs. Hₐ: μ_diff > 0 (diff = after − before). Conditions:
- Random: athletes randomly selected. ✓
- 10%: 22 ≤ 10% of comparable athletes. ✓
- Normal/Large Sample on the differences: n_diff = 22 < 30, so graph the 22 differences and confirm no strong skew or outliers. (df = 21)
(d) Study D — Two-proportion z-interval.
"Improved/not improved" is categorical → a proportion, compared between two groups (drug vs. placebo), and we want to estimate the difference → two-proportion z-interval. Conditions:
- Random: patients were randomly assigned to the two treatments. ✓
- 10%: because this is a randomized experiment (not sampling from a population), the 10% condition is not required.
- Large Counts: the number of "improved" and "not improved" in each group must each be ≥ 10. ✓
| Part | Point | Earned for |
|---|---|---|
| (a) | 1 | Correctly names one-sample t-interval (for a mean). |
| (a) | 1 | States all three conditions (Random, 10%, Normal/Large Sample via n ≥ 30). |
| (b) | 1 | Correctly names χ² test for homogeneity (must be homogeneity, not independence). |
| (b) | 1 | Justifies homogeneity via design (separate samples / compare distributions) or correct df = 4. |
| (b) | 1 | States conditions: Random, 10%, expected counts ≥ 5. |
| (c) | 1 | Correctly names matched-pairs t-test (or one-sample t on differences). |
| (c) | 1 | Conditions on the differences, noting n < 30 → graph differences for skew/outliers. |
| (d) | 1 | Correctly names two-proportion z-interval. |
| (d) | 1 | States Large Counts (≥ 10 successes and failures per group). |
| (d) | 1 | Addresses Random via random assignment (and/or correctly handles 10% for an experiment). |
Scoring guide: 9–10 = Complete (E); 7–8 = Substantial (P); 5–6 = Developing (D); 3–4 = Minimal (M); 0–2 = Incorrect (I).
- Part (b): writing "test for independence" instead of "homogeneity." Same math, but the design (three separate samples) makes it homogeneity. This is the most-missed point on the whole problem.
- Part (c): treating it as a two-sample t (forgetting the same athletes are measured twice), or checking Normality on the raw scores instead of the differences.
- Part (d): calling it a proportion test when the question says "estimate the difference" (it's an interval), or mechanically asserting the 10% condition for a randomized experiment.
- Throughout: naming the right family but the wrong member (e.g., "z-test for proportions" in Study A where the variable is quantitative). Naming the wrong procedure forfeits both that naming point and the linked conditions point.
1. (B) One-sample t-interval. Sleep hours = quantitative (mean), one sample, "estimate" → interval. Distractors: (A) proportion — wrong variable type; (C) only one group; (D) no pairing.
2. (C) Two-proportion z-test. Defective/not = categorical (proportion), two groups, "compare" with evidence → test. Distractors: (A) means, not proportions; (B) χ² works for 2×2 but the standard named procedure for comparing two proportions is the two-proportion z-test; (D) two groups, not one.
3. (B) Matched-pairs t-test. Same volunteers, left vs. right hand → naturally paired → one column of differences. Distractors: (A) ignores the pairing; (C) reaction time is quantitative, not a proportion; (D) "one-sample on raw times" misses that the analysis is on differences.
4. (B) χ² test for independence. One sample, two categorical variables recorded per person, asking about association → independence. Distractors: (A) homogeneity needs several separate samples; (C)/(D) not the structure (one sample, two categorical variables = chi-square).
5. (A) One-proportion z-test. Satisfied/not = proportion, one sample tested against a known p₀ = 0.90 → one-proportion z-test. Distractors: (B) satisfaction here is categorical, not a measured mean; (C) only one sample; (D) only one categorical variable, no two-way table.
6. (B) χ² test for homogeneity. Separate samples from three hospitals, one categorical response, comparing distributions → homogeneity. Distractor (A): independence requires a single sample with two variables.
7. (C) Normal/Large Sample for each group. That is the mean-procedure condition shape, applied per group for two-sample t. Distractors: (A)/(D) are proportion (Large Counts) shapes; (B) is the chi-square shape — each signals a different procedure family.
8. (B) Two-sample t-interval. Test scores = quantitative (mean), two independent groups, "estimate the difference, 95% confidence" → interval. Distractors: (A) no pairing — separate samples; (C) means, not proportions; (D) two groups, not one.
9. (B) Testing a single coin's heads-rate vs. 0.5. One sample, one proportion, compared to a known value → one-proportion z-test. Distractors: (A) two groups; (C) a mean; (D) means across several groups.
10. χ² test for independence. One sample of 250 members, two categorical variables (tier, class attendance) recorded per person, asking if they're related. TI-84: χ²-Test (enter the 3×2 counts as Matrix [A]). Conditions: Random sample; 10% (250 ≤ 10% of all members); all expected counts ≥ 5. df = (3−1)(2−1) = 2.
11. Matched-pairs t-test (one-sample t on differences). Same 28 students, before vs. after → one column of differences; "improved" → one-sided, Hₐ: μ_diff > 0 (after − before). TI-84: T-Test on the list of differences. Conditions: Random/representative; 10% (28 ≤ 10% of comparable students); Normal/Large Sample on the differences — n_diff = 28 < 30, so graph the differences and check for no strong skew/outliers. df = 27.
12. One-proportion z-interval. Uninsured/not = proportion, one sample, "confidence interval" → interval. TI-84: 1-PropZInt (x = 230, n = 1000, C-Level = .95). Conditions: Random (stated); 10% (1,000 ≤ 10% of all residents); Large Counts using p̂ = 0.23: np̂ = 230 ≥ 10 and n(1−p̂) = 770 ≥ 10. ✓
13. (B) Two-sample t-test. Commute time = quantitative (mean), two independent samples, "compare means" → two-sample t. Distractors: (A) means, not proportions; (C) no pairing between an urban and a rural commuter; (D) chi-square is for categorical counts.
14. (C) Every inference procedure in this course. Without random sampling or random assignment you cannot generalize or claim cause — Random is universal. Distractors: (A)/(B)/(D) wrongly restrict it; the condition applies across all families and to both intervals and tests.
15. χ² test for homogeneity. Four separate random samples (one per fertilizer), one categorical response (high-yield / not), comparing the proportion/distribution across groups → homogeneity. TI-84: χ²-Test (enter the 4×2 counts as a matrix). Conditions: Random (each sample); 10% (each 50 ≤ 10% of comparable plots — often waived if experimental); all expected counts ≥ 5. df = (4−1)(2−1) = 3.
(Note on #15: with only two response categories across several groups, this is equivalently a comparison of several proportions; the named chi-square procedure for comparing the response distribution across several separate samples is the test for homogeneity.)
StatsIQ · Lesson 28 of 30 · Units 3–4 · Phase 5 — Choosing the Right Inference Procedure
This lesson is exam-preparation material aligned to the 2026–27 AP Statistics Course and Exam Description. AP® is a trademark registered by the College Board, which is not affiliated with and does not endorse this product. Built and reviewed by a retired actuary for statistical accuracy. Statistical content reviewed for accuracy; if you find an error, please report it for correction.