Selecting a Categorical Procedure
Which categorical procedure?
- For categorical data, choose among proportion $z$-procedures and chi-square tests.
- The decision hinges on how many variables and how many samples.
- Get those right and the procedure follows.
- This skills lesson ties Units 6 and 8 together.
Count the variables and samples
- One variable, one sample, matched against a claimed distribution → goodness-of-fit.
- One variable, several separate samples → homogeneity.
- Two variables, one sample → independence.
- A single proportion (two categories, one sample) can also be a one-proportion $z$-test.
Chi-square vs. two-proportion z
- Comparing two groups on a two-category variable? Two valid options overlap here.
- A two-proportion $z$-test works and gives a direction (one-sided possible).
- A chi-square test of homogeneity also works (a $2\times2$ table) but is only non-directional.
- For a directional two-group proportion question, prefer the $z$-test.
Implement and communicate
- State hypotheses, check conditions, compute the statistic and p-value, conclude in context.
- Name the procedure explicitly and justify why it fits the data.
- Word the conclusion to match the test (fit / difference / association).
- Clear communication is graded as heavily as the arithmetic.
Two questions pick the categorical procedure: how many VARIABLES, and how many SAMPLES. One variable vs. a claimed distribution → goodness-of-fit; one variable across several samples → homogeneity; two variables in one sample → independence. Chi-square is never directional — if you need a one-sided claim about two proportions, use the two-proportion $z$-test instead.
"Do three schools have the same distribution of favorite sport?"
- Variables: one (favorite sport). Samples: three (separate schools).
- One variable, several samples → chi-square test of homogeneity.
- (Had it been "is sport related to gender in one survey?", → independence.)
Pick a categorical procedure by number of variables and number of samples: goodness-of-fit (one variable vs. a distribution), homogeneity (one variable, several samples), independence (two variables, one sample), or a proportion $z$-procedure. Implement fully and communicate the reasoning and conclusion in context.
Categorical data to analyze
Count the variables and samples to pick the right test.
One categorical variable checked against a claimed distribution (one sample) calls for a...
One variable vs. a distribution → goodness-of-fit.
Three separately-sampled schools compared on one variable calls for a...
One variable, several samples → homogeneity.
Two variables recorded in ONE sample calls for a...
Two variables, one sample → independence.
A chi-square test can give a one-sided (directional) conclusion.
Chi-square is never directional — use a two-proportion z-test for that.
Order the two questions that pick a categorical procedure, then the action.
Variables, then samples, then run the procedure.