- Asked 2/17
- 2072 Chaitra · 5 marks
- 2072 Kartik · 5 marks
Regression of wear on oil viscosity
Short answer
Least squares gives ŷ = 234.071 − 3.509x. At x = 30, predicted wear volume is 128.8 × 10⁻⁴ mm³.
An article in Wear (Vol.152, 1992, pp. 171-181) presents data on the fretting wear of mild steel and oil viscosity. Representative data follow, with x = oil viscosity and y = wear volume (10^-4 cubic millimeters).
| y | 240 | 181 | 193 | 155 | 172 | 110 | 113 | 75 | 94 |
|---|---|---|---|---|---|---|---|---|---|
| x | 1.6 | 9.4 | 15.5 | 20.0 | 22.0 | 35.5 | 43.0 | 40.5 | 33.0 |
| i) Fit the sample linear regression model using least squares. | |||||||||
| ii) Predict fretting wear when viscosity x = 30. |
Take x = oil viscosity (independent) and y = wear volume (dependent). Model: ŷ = a + bx, with least squares estimates
b = Sxy/Sxx, a = ȳ − b·x̄
i) Fitting the regression line
Sums (n = 9):
x y xy x² y²
1.6 240 384 2.56 57600
9.4 181 1701.4 88.36 32761
15.5 193 2991.5 240.25 37249
20 155 3100 400 24025
22 172 3784 484 29584
35.5 110 3905 1260.25 12100
43 113 4859 1849 12769
40.5 75 3037.5 1640.25 5625
33 94 3102 1089 8836
---------------------------------------------
220.5 1333 26864.4 7053.67 220549
Σx = 220.5, Σy = 1333, Σxy = 26864.4, Σx² = 7053.67, Σy² = 220549
Means: x̄ = 24.5000, ȳ = 148.1111
Sxx = Σx² − (Σx)²/n = 1651.4200 Sxy = Σxy − (Σx)(Σy)/n = -5794.1000 Syy = Σy² − (Σy)²/n = 23116.8889
Slope b = Sxy/Sxx = -5794.1000/1651.4200 = -3.5086 Intercept a = ȳ − bx̄ = 148.1111 − (-3.5086)(24.5000) = 234.0707
Fitted line: ŷ = 234.071 − 3.509 x
(Correlation r = -0.938, strong negative: wear falls as viscosity rises.)
ii) Prediction at x = 30
ŷ = 234.0707 + (-3.5086)(30) = 234.0707 − 105.2567 = 128.81
Answer: ŷ = 234.07 − 3.509x; predicted wear at x = 30 is about 128.8 × 10⁻⁴ mm³.
✓ Answer checked by Claude Sonnet 5.5
Related questions
- Regression of permeability on compressive strength · Probability and Statistics, chapter 5
- Correlation vs regression; stopping distance line · Probability and Statistics, chapter 5
- Regression of rice yield on tillers · Probability and Statistics, chapter 5
- Twists regression; CI for slope · Probability and Statistics, chapter 5 · asked 3×
- Twists regression; predict at 20% · Probability and Statistics, chapter 5 · asked 3×
- Regression: operator experience and performance · Probability and Statistics, chapter 5
- Viscosity of oil between parallel plates · Hydro Power, chapter 1
- Viscosity of oil on inclined plane · Hydro Power, chapter 1
- Viscosity of oil from laminar pipe flow · Hydraulics, chapter 1
- Relative permeability and inductance of mild steel ring · Electrical Machines I, chapter 1