Difficulty: Intermediate | Prerequisites: Familiarity with all inference procedures covered in the course (z-tests, t-tests, chi-squared tests, regression inference).
Big picture: One of the hardest parts of inference is not performing the test, it is knowing which test to use. This guide walks through the decision process. On an exam, the scenario description contains all the clues you need. Your job is to identify the type of data, the number of samples, and what the question is asking, then match those to the correct procedure. If you can do this reliably, you can handle any inference question the course throws at you.
Start with the type of data. Categorical data uses z (for proportions) or χ² (for distributions and associations). Quantitative data uses t (for means) or the regression t (for slopes). Then count the samples and read what the question is asking: estimate (confidence interval) or test a claim (hypothesis test).
Categorical (yes/no, success/failure, categories like colour or party affiliation) → go to Step 2A
Quantitative (numbers you can average, like height, score, time, weight) → go to Step 2B
Bivariate quantitative (two quantitative variables, looking at a relationship) → go to the Regression section
How many variables?
One categorical variable, compared to a hypothesised distribution → Chi-squared goodness-of-fit test (χ²GOF-Test)
One categorical variable, but you care about a single proportion (success/failure):
One sample → 1-PropZInt or 1-PropZTest
Two samples → 2-PropZInt or 2-PropZTest
Two categorical variables → ask: how many samples?
One sample, classified by both variables → Chi-squared test for independence (χ²-Test)
Two or more separate samples, compared on one variable → Chi-squared test for homogeneity (χ²-Test)
How many samples?
One sample (or paired/matched data, where you work with the differences) → TInterval or T-Test
Two independent samples → 2-SampTInt or 2-SampTTest
Watch for paired data. If the two groups are linked (before/after, twins, left/right), compute the differences and use a one-sample t-procedure on those differences. Do not use a two-sample t-test.
Two quantitative variables, testing for a linear relationship → LinRegTTest
Two quantitative variables, estimating the true slope → LinRegTInt
The question asks you to estimate a parameter or find a range of plausible values → Confidence interval
The question asks you to test a claim, gives you a hypothesised value, or asks whether there is "evidence that..." → Hypothesis test
Scenario | Interval | Test |
|---|---|---|
One proportion | 1-PropZInt | 1-PropZTest |
Two proportions | 2-PropZInt | 2-PropZTest |
One mean | TInterval | T-Test |
Two means (independent) | 2-SampTInt | 2-SampTTest |
Paired means | TInterval (on differences) | T-Test (on differences) |
One categorical variable vs. expected distribution | n/a | χ²GOF-Test |
Two categorical variables (one sample) | n/a | χ²-Test (independence) |
One categorical variable across populations | n/a | χ²-Test (homogeneity) |
Slope of regression line | LinRegTInt | LinRegTTest |
"Two groups always means a two-sample test." Not if the groups are paired. Before/after data on the same subjects requires a one-sample t-procedure on the differences.
"Chi-squared and the two-proportion z-test never overlap." A two-proportion z-test on a 2 × 2 table gives the same p-value as a chi-squared test for homogeneity (or independence) on that table. The z-test gives you a direction (one proportion is higher), while chi-squared only tells you the distributions differ.
"I need to memorise which calculator function matches which test." Understanding the logic is more important, but you should be comfortable enough with the names that you do not waste exam time searching menus. The naming convention is consistent: the function name describes the procedure.
⚠️ Many exams include questions specifically designed to test whether you can identify the correct procedure. Read the problem carefully before touching your calculator.
⚠️ The words in the problem are your clues: "proportion," "percentage," "fraction" → z-procedure. "Mean," "average" → t-procedure. "Distribution," "breakdown," "association," "relationship between categorical variables" → chi-squared. "Slope," "regression," "linear relationship" → LinReg.
⚠️ If the problem gives you raw data in two columns (x and y), it is probably regression. If it gives you counts in a table, it is probably chi-squared.
A researcher measures the blood pressure of 50 patients before and after a medication. Which procedure?
A survey asks 300 people for their favourite season (spring, summer, autumn, winter). The researcher wants to test whether preferences are equally distributed. Which procedure?
A study compares the pass rate of students in online vs. in-person classes. Which procedure?
A dataset has hours studied (x) and exam score (y) for 40 students. The researcher wants to know if more studying is associated with higher scores. Which procedure?
A company samples customers from three regions and records whether each customer chose Plan A, Plan B, or Plan C. Which procedure?
Answers: 1. Paired (one-sample) t-test on the differences. 2. χ² goodness-of-fit test. 3. Two-proportion z-test (or χ² test for homogeneity on a 2 × 2 table). 4. LinRegTTest. 5. χ² test for homogeneity.
This decision framework ties together every inference procedure in the course. Once you have the logic down, new procedures (like ANOVA, which extends the two-sample t-test to three or more groups) follow the same pattern: identify the data type, count the groups, choose the procedure. The structure of "hypotheses → conditions → test statistic → p-value → conclusion" remains the same throughout.
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