--- title: "Correlation vs regression; stopping distance line" subject: "Probability and Statistics" chapter: 5 key: "semester_5/probability_and_statistics:correlation-regression-plot-stopping-distance" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/correlation-regression-plot-stopping-distance site: BCT Engineering --- # Correlation vs regression; stopping distance line Probability and Statistics · Chapter 5: Correlation and regression - Asked in: 2070 Asar, 2+4 marks - Video: [Plot the regression line of Y on X | given, Speed limit(x) and Stopping distance(y) | bcs 40 dec 23](https://www.youtube.com/watch?v=fQdAnI_BCas) — IGNOUverse - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/correlation-regression-plot-stopping-distance - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5 ## Question 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 | ## 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. ## Answer #### 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): ```text 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. ```text 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) ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which statement best distinguishes correlation from regression? A. Correlation gives an equation; regression gives a single number B. Correlation gives a single number r; regression gives an equation to estimate one variable from the other C. Both treat variables identically and give lines D. Regression is symmetric in X and Y Answer: B. Correlation measures strength/direction symmetrically; regression gives a line with one variable dependent. (Key point: Correlation vs regression) 2. Which is symmetric? A. Regression of Y on X B. r_xy = r_yx C. Slope b D. Regression lines of Y on X and X on Y Answer: B. Correlation is symmetric, while the two regression lines generally differ. (Key point: Symmetry) 3. What is the regression line of stopping distance on speed? A. ŷ = -104.5 + 8.75x B. ŷ = 104.5 + 8.75x C. ŷ = -104.5 + 0.114x D. ŷ = 8.75 − 104.5x Answer: A. b = 8750/1000 = 8.75 and a = 333 − 8.75(50) = -104.5. (Key point: Fitted line) 4. Through which point must the regression line pass? A. (0, 0) B. (x̄, ȳ) = (50, 333) C. (30, 160) D. (70, 500) Answer: B. The least-squares line always passes through the means (x̄, ȳ). 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