Question

Tom is considering buying a used car but he is concerned about the repair costs. To investigate how repair cost is related to the age of a car, he has collected the following data:

Repairs (y)   Age (x)

($) (months)

327.67 110

376.68 113

392.52 114

443.14 134

342.62 93

476.16 141

324.74 115

338.98 115

433.45 115

526.37 142

362.42 96

448.76 139

335.27 89

350.94 93

291.81 91

467.80 109

474.48 138

354.15 83

420.11 100

416.04 137

Note: Round all of your numerical answers below to three decimal places.

(a) Fill out the blanks in the regression below that Tom has estimated based on the least squares approach:

ŷ = _______________ + _______________ x Age

(b) Interpret the economic implication of the estimated coefficient on “Age” in Part (a).

(c) The standard error of the estimate of the simple regression model is __________.

(d) The regression result indicates that age of the car ____________ (has or does not have) explanatory power for repair cost at the 5% significance level because the p-value of the coefficient is ___________ (less or greater) than 5%.

(e) The regression result suggests that the model can explain __________% of the changes in repair cost.

Answer 1

> Repairs=c(327.67,376.68,392.52,443.14,342.62,476.16,324.74,338.98,433.45,526.37,362.42,448.76,335.27,350.94,291.81,467.80,474.48,354.15,420.11,416.04)
> Age=c(110,113,114,134,93,141,115,115,115,142,96,139,89,93,91,109,138,83,100,137)
> model=lm(Repairs~Age)
> summary(model)

Call:
lm(formula = Repairs ~ Age)

Residuals:
Min 1Q Median 3Q Max
-74.547 -22.660 -0.981 22.233 83.354

Coefficients:

	Estimate	Std. error	t value	Pr(>|t|)
Intercept	114.8525	58.6854	1.957	0.06603
Age	2.4733	0.5106	4.8844	0.00013

Residual standard error: 43.32 on 18 degrees of freedom
Multiple R-squared: 0.5659,   Adjusted R-squared: 0.5417
F-statistic: 23.46 on 1 and 18 DF, p-value: 0.0001304
(#blue color shows the r-code)

a) The regression that Tom has estimated based on the least squares approach:

Repairs =114.853 + 2.477 * Age

b) Here estimated coefficient on “Age”, i.e., slope= 2.4773, Interpreted as the unit change in Age then the expected Repairs cost will be increase by 2.477.

c) The standard error of the estimate of the simple regression model is 43.32

d) The regression result indicates that age of the car has explanatory power for repair cost at the 5% significance level because the p-value of the coefficient is less than 5%.

d) correlation coeffeicient = sqrt(R^2) = 0.7523 , The r>0 , indicated that there is positive assoction between Repairs and Age. as age increases the repairs also get increases.

(e) The regression result suggests that the model can explain 56.59% of the changes in repair cost.

(#If need calculations by hand the please comment)