Apply — Unit Review
Unit 4 extends inference to richer data: categorical associations, comparing many groups, and relationships between numerical variables. The procedures are new, but the habit is the same — read the variables (how many, what type, how many groups) and let that pick the method. This review drills exactly that match.
The big picture
Three families of questions, each tied to a data shape:
- Categorical data → chi-square. Does one categorical variable match an expected distribution (goodness of fit)? Are two categorical variables associated (test of independence)?
- One numerical outcome across several groups → ANOVA. Do three or more group means differ (ANOVA)? If so, which pairs differ (multiple comparisons)?
- Two numerical variables → correlation and regression. How strong and what direction is the linear association (correlation)? What line predicts one from the other, and is the slope real (linear regression)? How well does the model fit, and how far can you trust it (prediction & model fit)?
Which method?
Read the variables; the row names the method:
| Variables in the question | Method | Statistic |
|---|---|---|
| One categorical vs. a claimed distribution | chi-square goodness of fit | \(\chi^2\) |
| Two categorical (associated?) | chi-square test of independence | \(\chi^2\) |
| One numerical outcome across 3+ groups | ANOVA (then post-hoc comparisons) | F |
| Two numerical (strength/direction) | correlation | \(r\) |
| Two numerical (predict / is the slope real?) | linear regression | \(t\) on the slope |
ANOVA answers “are any of the group means different?” — not which ones. A significant F says at least one pair differs; you need multiple-comparison procedures (which guard against the inflated error of many pairwise tests) to say where the difference is.
A quick decision guide
- What types are the variables?
- All categorical → chi-square. One variable vs. an expected split → goodness of fit; two variables → independence.
- A numerical outcome split by a categorical group → two means? use Unit 3 t; three or more groups? → ANOVA.
- Two numerical variables → correlation (how strong) and/or regression (predict, test the slope).
- Check the method’s conditions (e.g., chi-square expected counts; regression’s linear, roughly-constant-spread residuals).
Key ideas to carry forward
Goodness of fit vs. independence · expected counts · \(\chi^2\) statistic · ANOVA · F-statistic · between- vs. within-group variation · post-hoc / multiple comparisons · correlation \(r\) · least-squares line · slope inference · \(R^2\) · residual · extrapolation.
Common pitfalls
- Chi-square needs adequate expected counts (commonly each ≥ 5) — not observed counts.
- A significant ANOVA doesn’t say which groups differ — that’s what post-hoc comparisons are for, and running many naive t-tests inflates the Type I error.
- Correlation captures only linear association — a strong curve can have \(r \approx 0\).
- \(R^2\) is the fraction of variation explained, not the slope and not a probability.
- Don’t extrapolate — a regression line is only trustworthy within the range of the observed data.
- A real (discernible) slope is not necessarily a large or useful one — check the size, not just the p-value.
Exercises
The Unit 4 skill is reading the variables and choosing the right method. Answers are provided so you can check your work.
Which method? For each scenario, name the procedure (chi-square goodness of fit, chi-square independence, ANOVA, correlation, or linear regression).
- Is eye color associated with whether a person is left- or right-handed?
- Do mean test scores differ among students taught by four different methods?
- Does a die’s roll distribution match the expected uniform 1/6 for each face?
- How strongly, and in what direction, are height and arm span related?
- Can we predict a house’s price from its square footage, and is the relationship real?
Show answer
a. chi-square test of independence (two categorical). b. ANOVA (one numerical outcome, four groups). c. chi-square goodness of fit (one categorical vs. a claimed distribution). d. correlation (two numerical, strength/direction). e. linear regression (predict + test the slope).
Two groups or many? A study compares a numerical outcome (recovery time) across treatment groups. When does this call for a two-sample t (Unit 3) and when for ANOVA (Unit 4)?
Show answer
Two groups → two-sample t. Three or more groups → ANOVA (the F-test), because running all pairwise t-tests would inflate the overall Type I error. A significant ANOVA is then followed by multiple-comparison procedures to locate the differences.
After a significant F. An ANOVA on five diets yields a small p-value. A student concludes “diet 1 and diet 5 differ.” Is that conclusion justified by the ANOVA alone?
Show answer
No. A significant F only says at least one pair of diet means differs — not which. To claim diet 1 vs. diet 5 specifically, use a multiple-comparison procedure that controls the family-wise error across all the pairs.
Correlation gotcha. Two numerical variables show a strong U-shaped relationship, yet \(r \approx 0\). Is “no relationship” the right read?
Show answer
No. \(r\) measures only linear association. A strong curved pattern can produce \(r\) near 0 — always look at the scatterplot, not just the correlation.
Don’t over-trust the line. A regression of crop yield on rainfall is fit from seasons with 10–30 inches of rain. A planner uses it to predict yield at 60 inches. What’s the problem?
Show answer
Extrapolation. 60 inches is far outside the observed 10–30 inch range, where the linear pattern may not hold at all (too much rain could reduce yield). Predictions are only trustworthy within the range of the data used to fit the line.