Choosing an inference procedure happens before any formula is used, and the study design usually determines the choice. A useful sequence is to identify the response type, identify the groups and their relationship, and then decide whether the goal is estimation or testing.

Question 1, the kind of response recorded

If each individual contributes a category, the parameter usually involves a proportion or a categorical distribution. Examples include:

  • yes or no
  • success or failure
  • preferred category

If each individual contributes a numerical measurement, the parameter usually involves a mean. Examples include:

  • height
  • wait time
  • blood pressure
  • test score

This first distinction separates proportion procedures from mean procedures.

Question 2, the number of groups

One sample gives a one-sample procedure, two independently collected groups give a two-sample procedure, and several categorical groups summarized in a two-way table may call for chi-square. The number of columns in a dataset is not enough, and the way the observations were collected matters.

Question 3, whether numerical observations are paired

Two measurements can look like two samples while actually forming pairs. Examples include:

  • before and after measurements on the same people
  • twins or deliberately matched subjects
  • two measurements on the same experimental unit

For paired data, compute one difference for each pair, and the analysis becomes a one-sample \(t\)-procedure on those differences. The parameter is \(\mu_d\), the population mean difference, and it is not \(\mu_1-\mu_2\) for two independent populations.

Question 4, interval or test

A confidence interval estimates a parameter or difference, and a hypothesis test evaluates evidence about a specific claim. Wording such as “estimate,” “find a plausible range,” or “by how much” usually points toward an interval. Wording such as “is there convincing evidence,” “test the claim,” or “do the data support” usually points toward a test. The purpose and the procedure family are separate decisions.

Classify the design

Twelve scenarios, cycling. Both rows have to be answered before the verdict appears, because the two decisions are independent: the family comes from how the data were collected and the purpose comes from what the question asks for. The feedback never names the arithmetic. It names the feature of the design that settled the choice, which is the only part that transfers to a scenario you have not seen. Four of the twelve are built as confusable pairs and are worth returning to.

The drill presents scenarios without arithmetic, and that is intentional. The point is to identify which feature of the design determines the method, and a correct choice should be explainable in words before any calculator command is entered.

Two chi-square designs

A two-way table can arise in two ways. One sample classified according to two categorical variables calls for a test of independence, and several separate groups compared on one categorical response call for a test of homogeneity. The tables and test statistic can be identical. The data collection distinguishes the procedures.

A compact map

Design Typical procedure
One categorical sample One-proportion \(z\)
Two independent categorical groups Two-proportion \(z\)
One numerical sample One-sample \(t\)
Two independent numerical groups Two-sample \(t\)
Paired numerical data One-sample \(t\) on differences
One sample, two categorical variables Chi-square test for independence
Several groups, one categorical variable Chi-square test for homogeneity

Then decide whether the question asks for an interval or a test. Chi-square appears as a test only.

The fastest way to become fluent is to practice classification separately from computation.