Q. 4.59 - Elementary Statistics

Question

Corvette Prices. The Kelley Bime Book provides information on wholesale and retail prices of cars. Following are age and price data for 10 randomly selected Corvettes between 1 and 6 years old.

Here, x denotes age, in years, and y denotes price, in hundreds of dollars. For part (g). predict the prices of a 2 -year-old Corvette and a 3-year-old Corvette.

X	6	6	6	2	2	5	4	5	1	4
Y	290	280	295	425	384	315	355	328	425	325

• a. fond the regression equation for the data points.
• b. graph the regression equation and the data points.
• c. describe the apparent relationship between the two variables under consideration.
• d. interpret the slope of the regression line.
• e. identify the predictor and response variables.
• f. identify outliers and potential influential observations.
• g. predict the values of the response variable for the specified values of the predictor variable, and interpret your results.

Step-by-Step Solution

Answer

Ans:

part (a): $\hat{y}=456.6-27.9x$

part (b): Graph representation  

part (c): The price decreases as the age increases from 1 year to 6 years. 

part (d): The slope of the regression line is -27.9

part (e): The predictor variable is age and the response variable is the price of the Corvettes. 

part (f):  All the points are near the straight line so there are no outliers in the data given. 

part (g):  The estimated price of a 2-years-old Corvette is $\$40,080$ .

               The estimated price of a 3-years-old Corvette is $\$37,289$. 

Step 1. GIven information:

X	6	6	6	2	2	5	4	5	1	4
Y	290	280	295	425	384	315	355	328	425	325

Step 2. Solving part (a):

$\begin{array}{|llll|}\hline x& y& x·y& x^2\\ 6& 290& 1740& 36\\ 6& 280& 1680& 36\\ 6& 295& 1770& 36\\ 2& 425& 850& 4\\ 2& 384& 768& 4\\ 5& 315& 1575& 25\\ 4& 355& 1420& 16\\ 5& 328& 1640& 25\\ 1& 425& 425& 1\\ 4& 325& 1300& 16\\ \sum x=41& \sum y=3422& \sum xy=13168& \sum x^2=199\\ \hline\end{array}$

Step 3. Continue:

We first need to compute the $b_1$ and $b_0$ to find the regression equation The slope of the regression line is,

$$\begin{gathered} b_1=\frac{S_{xy}}{S_{xx}} \\ =\frac{\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n}{\sum x_i^2-{\left(\sum x_i\right)}^2/n} \\ =\frac{13168-\left(41\right)\left(3422\right)/10}{199-(41{)}^2/10} \\ =-27.9 \end{gathered}$$

The y-intercept is,

$$\begin{gathered} b_0=\frac{1}{n}\left(\sum y_i-b_1\sum x_i\right) \\ =\frac{1}{10}(3422-(-27.9\left)\right(41\left)\right) \\ =456.6 \end{gathered}$$

So the regression equation is $\hat{y}=456.6-27.9x$

Step 4. Solving part (b):

Step 5. Solving part (c):

From the above graph, we can see that the price decreases as the age increases from 1 year to 6 years.

Step 6. Solving part (d):

The slope of the regression line is -27.9, which means the estimated Corvettes depreciation $\$2790$ per year.

Step 7. Solving part (e):

From the regression equation, we have the predictor variable is age and the response variable is the price of the Corvettes.

Step 8. Solving part (f):

f) From the obtained regression equation we can see that all the points are near the straight line so there are no outliers in the data given.

Step 9. Solving part (g):

g) The predicted price of a 2-years-old Corvette is,

We have the regression line as $\hat{y}=456.6-27.9x$

Substituting the value $x=2$, we get

$$\begin{gathered} \hat{y}=456.6-27.9x \\ =456.6-27.9\left(2\right) \\ =400.8 \end{gathered}$$

Therefore the estimated price of a 2-years-old Corvette is $\$40,080$.

Step 10. Continue:

The predicted price of a 3-years-old Corvette is,

We have the regression line as $\hat{y}=456.6-27.9x$

$$\begin{gathered} \hat{y}=456.6-27.9x \\ =456.6-27.9\left(3\right) \\ =372.9 \end{gathered}$$

Therefore the estimated price of a 3-years-old Corvette is $\$37,289$.