- 2070 Asar · 2+4 marks
Correlation vs regression; stopping distance line
Short answer
Correlation measures strength and direction of linear relationship (r, -1 to +1); regression gives an equation (ŷ = -104.50 + 8.75x) for prediction. To plot, mark the five data points and draw the line through (50,333) with slope 8.75.
What is the difference between correlation and regression? Plot the sample regression line of Y on X.
| Speed x | 30 | 40 | 50 | 60 | 70 |
|---|---|---|---|---|---|
| Stopping distance y | 160 | 240 | 330 | 435 | 500 |
Difference between correlation and regression
| Basis | Correlation | Regression |
|---|---|---|
| Aim | Measures strength and direction of linear association | Gives an equation to estimate one variable from the other |
| Variables | Both treated symmetrically | One dependent, one independent |
| Result | Single number r, −1 ≤ r ≤ +1 | Line Y = a + bX with units |
| Symmetry | r_xy = r_yx | Y on X and X on Y lines differ |
Regression line of Y (stopping distance) on X (speed)
Sums (n = 5):
x y xy x² y²
30 160 4800 900 25600
40 240 9600 1600 57600
50 330 16500 2500 108900
60 435 26100 3600 189225
70 500 35000 4900 250000
---------------------------------------------
250 1665 92000 13500 631325
Σx = 250, Σy = 1665, Σxy = 92000, Σx² = 13500, Σy² = 631325
Means: x̄ = 50.0000, ȳ = 333.0000
Sxx = Σx² − (Σx)²/n = 1000.0000 Sxy = Σxy − (Σx)(Σy)/n = 8750.0000 Syy = Σy² − (Σy)²/n = 76880.0000
Slope b = Sxy/Sxx = 8750.0000/1000.0000 = 8.7500 Intercept a = ȳ − bx̄ = 333.0000 − (8.7500)(50.0000) = -104.5000
r = 0.9979 (very strong positive correlation).
Regression line: ŷ = -104.50 + 8.75 x
Fitted values:
| x | 30 | 40 | 50 | 60 | 70 |
|---|---|---|---|---|---|
| y (observed) | 160 | 240 | 330 | 435 | 500 |
| ŷ (line) | 158.0 | 245.5 | 333.0 | 420.5 | 508.0 |
To plot: mark the five points (x, y) on a scatter diagram and draw the line through (x̄, ȳ) = (50, 333), e.g. through (30, 158.0) and (70, 508.0). The points lie very close to the line.
Stopping distance y (o = data, . = fitted line)
550 | ...
500 | ...o.
450 | o......
400 | ......
350 | o....
300 | ......
250 | ..o...
200 | ......
150 | ...o.
100 |.
+-------------------------------------------------
30 40 50 60 70 x (speed)
✓ Answer checked by Claude Sonnet 5.5
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