Revision and practice with answers
On this page 3 headings
Try these questions without looking at the answers. Then open each explanation and check your method, units and denominator.
One-page method guide
| Pair or question | Method |
|---|---|
| Two categorical variables | Contingency table; compare conditional shares. |
| Two numerical variables | Scatterplot before covariance or correlation. |
| Direction without units | Sign of r. |
| Strength of linear association | Magnitude of r, alongside the plot. |
| Predict y from x | Fitted line; inspect residuals and range. |
| Two groups and numerical observations | Compare distributions; point-biserial for an explicit 0/1 code. |
Practice with explained answers
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1. A two-way row has Yes=12 and No=8. Find its within-row Yes share.
Show explained answer
12/(12+8) = 12/20 = 60%.
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2. If all rows have the same conditional percentages, what does that say about this table?
Show explained answer
The empirical category distributions are identical across rows, so knowing the row category gives no extra information about the column category.
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3. In the phone survey, distinguish 34/44 and 34/76.
Show explained answer
34/44 is ownership among female participants. 34/76 is the female share among owners. They condition on different groups.
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4. What should you describe before computing correlation?
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Direction, linearity or curvature, variation around the pattern, and unusual points.
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5. For (1,3),(2,2),(3,1), find sample covariance and correlation.
Show explained answer
Means are 2 and 2. Products are −1,0,−1. Sample covariance = −2/2 = −1. Each sample variance is 1, so r = −1.
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6. If one variable is converted from meters to centimeters, what happens to covariance and correlation?
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Covariance is multiplied by 100. Correlation is unchanged because standardization cancels the positive scale change.
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7. What if r=0 but the scatterplot is a clear U-shape?
Show explained answer
There is a nonlinear association. Pearson r only summarizes a linear relationship.
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8. Predicted y = 10+3x. At x=4, actual y=20. Find predicted y and residual.
Show explained answer
Predicted y = 22. Residual = observed−predicted = 20−22 = −2.
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9. A simple fitted line with intercept has r=−0.6. Find R² and interpret it.
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R² = 0.36. The line accounts for 36% of observed response variation relative to the mean-only prediction. The direction is negative.
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10. Does R²=0.9 mean that 90% of individual predictions are exactly correct?
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No. It describes variation accounted for by the fitted model, not the percentage of correct predictions.
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11. With F=0 and M=1, point-biserial r is +0.1881. What happens if the codes are reversed?
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The value becomes −0.1881. Coding changes the sign; the underlying group difference is unchanged.
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12. What happens if every x value is identical?
Show explained answer
SD of x is zero, so correlation is undefined and a unique regression slope cannot be calculated.
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13. Why can the largest ownership count belong to a group without the largest ownership rate?
Show explained answer
The group may simply be larger. Compare within-group shares using each group’s own total.
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14. Can a fitted line prove causation or guarantee a prediction far outside the data?
Show explained answer
No. A fitted association does neither. Inspect context, possible other variables and the observed range.
Before moving on
Explain one result in a complete sentence with the group, unit and method. If your result disagrees with a spreadsheet, check the definition, sample/population denominator and percentile convention before changing your answer.