Chapter 3 — Regression Analysis
Key Concepts Quick Recap
| Term | Formula | Meaning |
|---|---|---|
| Regression of Y on X | Ŷ = a + byx·X | Predict Y from X |
| Regression coefficient byx | r(σy/σx) | Change in Y per unit change in X |
| Regression of X on Y | X̂ = c + bxy·Y | Predict X from Y |
| Regression coefficient bxy | r(σx/σy) | Change in X per unit change in Y |
| Relation | r = ±√(byx · bxy) | r = geometric mean of regression coefficients |
| Means | Both lines pass through (X̄, Ȳ) | Point of intersection |
I. Choose the Best Answer (MCQs)
1. The regression line Y on X is used to predict: (a) X for a given Y (b) Y for a given X (c) Both X and Y (d) Neither
2. If byx = 0.6 and bxy = 0.4, then r is: (a) 0.24 (b) 0.49 (c) 0.49 (d) ±0.49
3. The two lines of regression intersect at: (a) Origin (b) (σx, σy) (c) (X̄, Ȳ) (d) (r, 0)
4. If r = 0, the regression lines are: (a) Parallel (b) Coincident (c) Perpendicular to each other (d) At 45° to each other
5. If r = ±1, the two regression lines: (a) Are perpendicular (b) Coincide (become one line) (c) Are parallel (d) Make 45° angle
| When | r | = 1, perfect correlation → all points lie on a straight line → both regression lines are the same. |
|---|
6. The regression coefficient byx is always: (a) Greater than 1 (b) Less than 1 (c) Equal to r (d) Depends on the ratio σy/σx
7. If the regression equation of Y on X is Y = 2 + 0.5X, then when X = 10, Y is: (a) 5 (b) 7 (c) 10 (d) 12
II. Short Answer Questions
Common Mistakes
- Using Y-on-X equation to predict X (should use X-on-Y equation instead).
- Forgetting r = ±√(byx × bxy) — the sign of r must match the signs of byx and bxy.
- Forgetting both lines always pass through (X̄, Ȳ).
Exam Checklist
- Two regression lines: Y on X (predict Y) and X on Y (predict X).
- byx = r(σy/σx); bxy = r(σx/σy).
- r = ±√(byx × bxy); if both b's positive → r positive.
- Both lines through (X̄, Ȳ).
- r = 0 → lines perpendicular; r = ±1 → lines coincide.
- Normal equations for least squares fitting.