Question

An auto manufacturing company wanted to investigate how the price of one of its car models depreciates with age. The research department at the company took a sample of eight cars of this model and collected the following information on the ages (in years) and prices (in hundreds of dollars) of these cars.

Age

8

3

6

9

2

5

6

3

Price

45

210

100

33

267

134

109

235

 

1. What is the independent variable?
2. What is the dependent variable?
3. Develop the regression model
4. Predict the price of a 7-year-old car of this model.
5. Estimate the price of an 18-year-old car of this model. Comment on this finding.

                                   

Answer 1

1. Age, $X$ is the independent variable

$$8, 3, 6, 9, 2, 5, 6, 3$$

2. Price, $Y$ the dependent variable

$$45, 210, 100, 33, 267, 134, 109, 235$$

3.

$$\bar{X}=\dfrac{1}{n}\sum _iX_i=\dfrac{42}{8}=5.25$$ $$\bar{Y}=\dfrac{1}{n}\sum _iY_i=\dfrac{1133}{8}=141.625$$ $$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum _iX_i)^2$$ $$=264-\dfrac{(42)^2}{8}=43.5$$ $$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum _iY_i)^2$$ $$=213565-\dfrac{(1133)^2}{8}=53103.875$$ $$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum _iX_i)(\sum _iY_i)$$ $$=4450−\dfrac{42(1133)}{8}=-1498.25$$ $$\beta_1=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{-1498.25}{43.5}=-34.442529$$ $$\beta_0=\bar{Y}-\beta_1\cdot\bar{X}=141.625+\dfrac{1498.25}{43.5}(5.25)$$ $$=322.448276$$

The regression equation is:

$$Y=322.448276-34.442529X$$

4.

$$Y(7)=322.448276-34.442529(7)=81.35$$

The price of a 7-year-old car of this mode will be 81.35.

5.

$$Y(18)=322.448276-34.442529(18)$$

$$=-297.52<0$$

A 18-year-old car of this mode does not exist.

$$322.448276-34.442529X\geq0$$ $$X\leq9.36$$

The price of a 9-year-old car of this mode will be 12.47.

A 10-year-old car of this mode does not exist.