Q. 4.59 - Elementary Statistics

Question

Corvette Prices. The Kelley Blue 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.

1. find the regression equation for the data points.
2. graph the regression equation and the data points.
3. describe the apparent relationship between the two variables under consideration.
4. interpret the slope of the regression line.
5. identify the predictor and response variables.
6. identify outliers and potential influential observations.
7. predict the values of the response variable for the specified values of the predictor variable, and interpret your results.

Step-by-Step Solution

Answer

1. The following is the regression equation for predicting price $\left(y\right)$ from age $\left(x\right)$: $\hat{y}=456.6019-27.9029x.$
   
2. 
   

 c. The regression line slopes downward.

 d. If a Corvette's age grows by one year, the price drops by an average of $27.9029$ hundred dollars.

e. Variable to predict: age. Price is a response variable.

f. There are no outliers or significant observations. 

g. ${\hat{y}}_2=400.7961$ and ${\hat{y}}_3=372.8932$

Part (a) Step 1: Given Information

Given in the question that, 

For the data points, we must find the regression equation.

Part(a) Step 2: Explanation

Let's use the regression beta coefficients for the calculation. 

$\hat{y}=b_0+b_{1}x$

According to the information,

We have to find the necessary sum as below: 

$\sum x_i=6+6+\dots +4=41$

$\sum x_i^2=6^2+6^2+\dots +4^2=199$

$\sum y_i=290+280+\dots +325=3422$

$\sum x_i{y}_i=\left(6\right)\left(290\right)+\left(6\right)\left(280\right)+\dots +\left(4\right)\left(325\right)=13168$

Let's find $s_{xy}$ as follow:

$s_{xy}=\sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)=\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n$

$$\begin{gathered} s_{xy}=13168-\frac{\left(4\mathrm{I}\right)\left(3442\right)}{10} \\ =-862.2 \end{gathered}$$

Then, find $s_{xx}$ as follow:

$s_{xx}=\sum {\left(x_i-\overline{x}\right)}^2=\sum x_i^2-{\left(\sum x_i\right)}^2/n$

$$\begin{gathered} s_{xx}=199-\frac{\left(41\right)\left(41\right)}{10} \\ =30.9 \end{gathered}$$

Part(a) Step 3: Calculate the parameters and averages

We can find the averages by using the given formula: 

$\overline{x}=\frac{\sum x_i}{n}$

$\overline{y}=\frac{\sum y_i}{n}$

Here, the value of  $\overline{x}$ is:

$\overline{x}=\frac{41}{10}=4.1$

Then, find the value of $\overline{y}$ is:

$\overline{y}=\frac{3422}{10}=342.2$

Therefore, the parameters are:

$$\begin{gathered} b_1=\frac{s_{xy}}{s_{xx}} \\ {b}_1=\frac{-862.2}{30.9} \\ =-27.9029 \end{gathered}$$

$$\begin{gathered} b_0=\overline{y}-b_1\overline{x} \\ {b}_0=342.2-(-27.9029)\times 4.1 \\ =456.6019 \end{gathered}$$

Part (b) Step 1: Given Information

Given in the question that,

We have to graph the regression equation and the data points. 

Part (b) Step 2: Explanation

When you're six years old, the price is:

${\hat{y}}_6=456.6019-27.9029\left(6\right)$

    $=289.1845$

The anticipated values for the provided data are also presented in the table below:

$\begin{array}{|lll|}\hline x& y& \hat{y}\\ 6& 290& 289.1845\\ 6& 280& 289.1845\\ 6& 295& 289.1845\\ 2& 425& 400.7961\\ 2& 384& 400.7961\\ 5& 315& 317.0874\\ 4& 355& 344.9903\\ 5& 328& 317.0874\\ 1& 425& 428.699\\ 4& 325& 344.9903\\ \hline\end{array}$

The given points and the fitted regression line are represented in the graph below. 

Part (c) Step 1: Given Information

The apparent relationship between the two variables under investigation must be described. 

Part (c) Step 2: Explanation

It's worth noting that the regression line slopes downwards, implying that the price $y$ falls as age $x$ rises.

Part (d) Step 1: Given Information

The apparent relationship between the two variables under investigation must be described. 

Part (d) Step 2: Explanation

The average growth in $y$ per increment in $x$ is represented by the slope.

The slope was estimated as $-27.9029$ in section (a).

Part (e) Step 1: Given Information

The predictor and response variables must be identified. 

Part (e) Step 2: Explanation

The response variable seems to be the one that will be measured.

The predictive variable is a variable that is used to estimate the output of the response variable.

The value of a Corvette is determined by its age in this country. As a result, the price is the response variable, whereas the age is the predictor variable.

Part (f) Step 1: Given Information

Outliers and potentially influential observations must be identified. 

Part (f) Step 2: Explanation

An outlier is just a data point that is far off the regression line.

An impactful observation is one where the removal of a point causes a significant change in the regression equation. That instance, removing a point creates a significant shift in the regression line's direction.

All of the points in component (b) are closed to the regression line, indicating that there are no outliers in the dataset. 

There are no substantial changes in the orientation of the regression line when a point is removed, hence there are no influencing observations.

Part (g) Step 1: Given Information

We must forecast the response variable's values based on the values of the predictor variable and evaluate the results. 

Part (g) Step 2: Explanation

It was determined that, 

$\hat{y}=456.6019-27.9029x$

When you're two years old, a Corvette costs:

$$\begin{gathered} {\hat{y}}_2=456.6019-27.9029\left(2\right) \\ =400.7961 \end{gathered}$$

When you're three years old, a Corvette costs:

$$\begin{gathered} {\hat{y}}_2=456.6019-27.9029\left(3\right) \\ =372.8932 \end{gathered}$$

A two-year-old Corvette is expected to cost $400.7961$ dollars, while a three-year-old Corvette is expected to cost $372.8932$ dollars. As a result, the value of a Corvette reduces as it gets older.