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 carsAge 8 3 6 9 2 5 6 3Price 18 94 50 21 145 42 36 99(a) Find the regression line with price as a dependent variable and age as an independent variable.(b) Predict the price of a 7 year old car of this model.(c) Estimate the price of an 18 year old car of this model.

Answer 1

Answer:

a)

Price=150.414-16.626 Age

b)

$34.032

c)

We can't estimate the price of 18 year old car using the above regression equation because the price of 18 year old car falls outside the scope of the model

Step-by-step explanation:

Age(X)   Price(Y)

8              18

3              94

6              50

9              21

2              145

5              42

6              36

3              99

a)

The regression equation can be written as

y=a+bx

$b=\frac{sum(x-xbar)(y-ybar)}{sum(x-xbar)^{2} }$

a=ybar-bxbar

xbar=sumx/n

xbar=(8+3+6+9+2+5+6+3)/8=5.25

ybar=sumy/n

ybar=(18+94+50+21+145+42+36+99)/8=63.125

x y x-xbar y-ybar (x-xbar)^2 (x-xbar)(y-ybar)

8 18 2.75  -45.125 7.5625        -124.09375

3 94 -2.25 30.875 5.0625        -69.46875

6 50 0.75 -13.125 0.5625        -9.84375

9 21 3.75  -42.125 14.0625        -157.96875

2 145 -3.25 81.875 10.5625        -266.09375

5 42 -0.25 -21.125 0.0625         5.28125

6 36 0.75 -27.125 0.5625        -20.34375

3 99 -2.25 35.875 5.0625        -80.71875

Total             43.5        -723.25

$b=\frac{sum(x-xbar)(y-ybar)}{sum(x-xbar)^{2} }$

b=-723.25/43.5

b=-16.626

a=ybar-bxbar

a=63.125-(16.626)*5.25

a=150.414

Thus, the required regression equation is

y=150.414-16.626 x

Price=150.414-16.626 Age

b)

For predicting price of 7 years old car we put x=7 in the estimated regression equation

y=150.414-16.626 x

y=150.414-16.626(7)

y=150.414-116.382

y=34.032

The predicted price of a 7 year old car is $34.032.

c)

We can't estimate the price of 18 year old car using the above regression equation because the price of 18 year old car falls outside the scope of the model. The estimate of price is only valid for 2 to 9 year old car.