--- title: "Regression of maintenance cost on car age" subject: "Probability and Statistics" chapter: 5 key: "semester_5/probability_and_statistics:car-age-maintenance-regression" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/car-age-maintenance-regression site: BCT Engineering --- # Regression of maintenance cost on car age Probability and Statistics · Chapter 5: Correlation and regression - Asked in: 2068 Chaitra, 5 marks - Video: [Find regression equation for cost related to age.also predict the maintenance cost,if x=12](https://www.youtube.com/watch?v=M9qkU184ho8) — Statistics is Fun A.H - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/car-age-maintenance-regression - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5 ## Question The following table gives the age of the cars of a certain company and annual maintenance costs: Age of cars (Years): 2 4 6 8 10 Maintenance costs (Rs.000): 10 15 22 32 46 Obtain the regression equation for cost related to age and also estimate the cost of maintenance for 10 yrs old car. ## Short answer Regression equation: ŷ = −1.70 + 4.45X (cost in Rs.000). For a 10-year-old car, estimated maintenance cost = 42.80 (Rs.000). ## Answer Let X = age of car (years) and Y = annual maintenance cost (Rs. 000). Regression of cost on age: Ŷ = a + bX. Sums (n = 5): ```text x y xy x² y² 2 10 20 4 100 4 15 60 16 225 6 22 132 36 484 8 32 256 64 1024 10 46 460 100 2116 --------------------------------------------- 30 125 928 220 3949 ``` Σx = 30, Σy = 125, Σxy = 928, Σx² = 220, Σy² = 3949 Means: x̄ = 6.0000, ȳ = 25.0000 Sxx = Σx² − (Σx)²/n = 40.0000 Sxy = Σxy − (Σx)(Σy)/n = 178.0000 Syy = Σy² − (Σy)²/n = 824.0000 Slope b = Sxy/Sxx = 178.0000/40.0000 = 4.4500 Intercept a = ȳ − bx̄ = 25.0000 − (4.4500)(6.0000) = -1.7000 **Regression equation: Ŷ = -1.70 + 4.45 X** (r = 0.980) #### Cost for a 10-year-old car Ŷ = -1.7000 + 4.4500 × 10 = **42.80** (Rs. 000) **Answer: Ŷ = -1.70 + 4.45X; maintenance cost for a 10-year-old car ≈ Rs. 42800 (the fitted value; the actual observed cost is Rs. 46,000).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which is the dependent variable? A. Age of car B. Annual maintenance cost C. Number of cars D. Year Answer: B. Cost is regressed on age, so cost is Y. (Key point: Dependent variable) 2. What is the slope? A. 178/40 = 4.45 B. 40/178 C. 824/40 D. 25/6 Answer: A. b = Sxy/Sxx = 178/40 = 4.45. (Key point: Slope) 3. What is the fitted equation? A. Ŷ = 1.70 + 4.45X B. Ŷ = -1.70 + 4.45X C. Ŷ = 25 + 6X D. Ŷ = -4.45 + 1.70X Answer: B. a = 25 − 4.45(6) = -1.70. (Key point: Equation) 4. Estimated cost for a 10-year-old car (fitted value)? A. Rs. 46,000 B. Rs. 42,800 C. Rs. 44,500 D. Rs. 4,280 Answer: B. Ŷ = -1.7 + 44.5 = 42.8 (Rs. 000); 46 is the observed value. 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