Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Sections 4.1–4.2
Chapter Summary
This chapter has two halves that share a theme: what to do when relationships aren’t simple averages. The first half is correlation and regression — describing how two quantitative variables move together. A scatterplot shows the pattern, the correlation coefficient r measures the strength and direction of the linear relationship (−1 to +1), and the least-squares regression line gives the best-fitting straight line for making predictions. Realistic example: for 30 used cars, mileage and price have r ≈ −0.85 — a strong negative linear relationship — and the regression line ŷ = 24,000 − 0.12x predicts a car with 60,000 miles sells for about $16,800. The coefficient of determination, r², tells you what fraction of the variation in price is explained by mileage (here about 72%).
Three warnings come with this power: correlation is not causation, predictions outside the data’s range (extrapolation) are unreliable, and a single unusual point can distort everything. The second half introduces nonparametric tests — the backup plan when the normality requirements of t-tests and ANOVA fail. Instead of means, these tests work with signs and ranks: the sign test for a median, the Wilcoxon signed-rank test for paired data, the Mann-Whitney test for two independent samples, and the Kruskal-Wallis test for three or more groups. They make fewer assumptions, at the cost of slightly less power when parametric conditions actually hold.
Study Guide
Key terms and definitions
- Scatterplot — graph of ordered pairs (x, y); reveals form, direction, and strength of a relationship.
- Linear correlation coefficient (r) — between −1 and 1; sign = direction, magnitude = strength of linear association. Near 0 means no linear relationship (a curved one may still exist).
- Least-squares regression line — ŷ = b₁x + b₀; the line minimizing the sum of squared residuals.
- Residual — observed y minus predicted ŷ; leftover variation the line doesn’t explain.
- Coefficient of determination (r²) — proportion of variation in y explained by the linear relationship with x.
- Extrapolation — predicting for x-values outside the observed range; unreliable.
- Influential observation — a point whose removal markedly changes the regression results.
- Nonparametric test — a hypothesis test not requiring a normal population; typically based on signs or ranks.
- Sign test — tests a claim about a population median using the binomial distribution on +/− signs (p = 0.5).
- Wilcoxon signed-rank test — nonparametric counterpart of the paired t-test.
- Mann-Whitney test — nonparametric counterpart of the two-independent-samples t-test.
- Kruskal-Wallis test — nonparametric counterpart of one-way ANOVA; test statistic H is approximately chi-square with df = k − 1.
Essential formulas
- Slope: b₁ = r · (s_y ÷ s_x). Intercept: b₀ = ȳ − b₁x̄.
- Regression line: ŷ = b₁x + b₀.
- Residual: e = y − ŷ.
- Coefficient of determination: r² (interpret as a percent: “r² × 100% of the variation in y is explained by x”).
- Kruskal-Wallis df: k − 1, where k = number of groups.
Step-by-step procedures
- Analyze a bivariate relationship: (1) draw the scatterplot and describe form/direction/strength; (2) compute r and interpret it; (3) find the regression line ŷ = b₁x + b₀; (4) use it to predict within the data range; (5) compute r² and interpret; (6) check residuals/outliers for influential points.
- Choose a nonparametric test: (1) confirm parametric requirements fail (small sample, skewed, outliers); (2) match the design: one median → sign test; paired data → Wilcoxon signed-rank; two independent samples → Mann-Whitney; three or more groups → Kruskal-Wallis; (3) run the test and conclude as usual with the P-value.
Common mistakes to avoid
- Saying correlation proves causation — it doesn’t; lurking variables are always suspects.
- Extrapolating beyond the data (predicting a car’s price at 500,000 miles from data up to 120,000).
- Interpreting r² as “percent correct” — it’s the percent of variation explained, not prediction accuracy.
- Using parametric tests on small, badly skewed samples when a nonparametric alternative exists — and the reverse error: using nonparametric tests (lower power) when conditions are actually met.
Self-check: I can…
- [ ] Draw and interpret a scatterplot.
- [ ] Compute and interpret the correlation coefficient r.
- [ ] Find the least-squares regression line and use it for predictions.
- [ ] Interpret r² and residuals in context.
- [ ] Explain the dangers of extrapolation and influential points.
- [ ] Choose the correct nonparametric test for a given design (sign, Wilcoxon signed-rank, Mann-Whitney, Kruskal-Wallis).
StatCrunch How-To
- Scatterplot: Graph > Scatterplot — select X and Y columns.
- Correlation: Stat > Summary Stats > Correlation — select the two columns > Compute.
- Regression line: Stat > Regression > Simple Linear — select X and Y > Compute. Output includes the equation, r, r², and residual plots. Use “Predict Y” options for predictions.
- Residual plot check: in the Simple Linear output, examine the residuals vs. fitted plot for patterns (a funnel or curve means the linear model is shaky).
- Nonparametric tests: Stat > Nonparametrics > choose Sign Test, Mann-Whitney, Wilcoxon Signed Ranks, or Kruskal-Wallis — select the data column(s), enter the hypothesized median where asked, and Compute.
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