We appreciate your visit to The following table gives retail values of a 2020 Corvette for various odometer readings tex begin tabular c c hline Odometer Reading Retail Value hline. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
We are given the following data for odometer readings ([tex]$x$[/tex]) and retail values ([tex]$y$[/tex]):
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Odometer Reading } (x) & \text{Retail Value } (y) \\
\hline
13000 & 52525 \\
\hline
18000 & 51625 \\
\hline
20000 & 51350 \\
\hline
25000 & 50325 \\
\hline
29000 & 49875 \\
\hline
32000 & 49225 \\
\hline
\end{array}
\][/tex]
We now explain the steps for finding the answers.
──────────────────────────────
Step (a): Least-Squares Line
To find the least-squares line of the form
[tex]\[
\hat{y} = a + b x,
\][/tex]
we compute the slope [tex]$b$[/tex] and intercept [tex]$a$[/tex] using the standard formulas:
[tex]\[
b = \frac{n\sum (xy) - \left(\sum x\right)\left(\sum y\right)}{n\sum{x^2} - \left(\sum x\right)^2}, \quad
a = \overline{y} - b \, \overline{x},
\][/tex]
where [tex]$n$[/tex] is the number of data points.
After carrying out the computations and rounding the values to two decimal places, the resulting equation is
[tex]\[
\hat{y} = 54734.53 - 0.17 x.
\][/tex]
Since this is a linear equation, the degree of the polynomial is 1.
──────────────────────────────
Step (b): Predicting the Retail Value at 30000 Miles
To predict the retail value when the odometer reading is [tex]$x = 30000$[/tex], substitute into the equation:
[tex]\[
\hat{y} = 54734.53 - 0.17(30000).
\][/tex]
Performing the multiplication:
[tex]\[
0.17 \times 30000 = 5100,
\][/tex]
so
[tex]\[
\hat{y} \approx 54734.53 - 5100 = 49634.53.
\][/tex]
Rounding to the nearest \[tex]$100 gives a predicted value of
\[
\$[/tex]49600.
\]
──────────────────────────────
Step (c): Linear Correlation Coefficient
The linear correlation coefficient [tex]$r$[/tex] is calculated by
[tex]\[
r = \frac{\sum \left(x - \overline{x}\right)\left(y - \overline{y}\right)}{\sqrt{\sum \left(x - \overline{x}\right)^2 \sum \left(y - \overline{y}\right)^2}}.
\][/tex]
After doing the appropriate computations and rounding to four decimal places, we obtain
[tex]\[
r = -0.9979.
\][/tex]
──────────────────────────────
Step (d): Interpretation of the Negative Correlation
A negative linear correlation coefficient indicates that as the odometer reading increases, the retail value of the vehicle tends to decrease. In other words, higher mileage is associated with a lower retail value.
──────────────────────────────
Summary of Answers
(a) The equation of the least-squares line is
[tex]\[
\hat{y} = 54734.53 - 0.17x \quad (\text{with a polynomial degree of } 1).
\][/tex]
(b) For an odometer reading of [tex]$30000$[/tex], the predicted retail value is approximately
[tex]\[
\$49600.
\][/tex]
(c) The linear correlation coefficient is
[tex]\[
r = -0.9979.
\][/tex]
(d) The negative correlation coefficient means that as the odometer reading increases, the retail value decreases.
These are the detailed steps and results for this problem.
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Odometer Reading } (x) & \text{Retail Value } (y) \\
\hline
13000 & 52525 \\
\hline
18000 & 51625 \\
\hline
20000 & 51350 \\
\hline
25000 & 50325 \\
\hline
29000 & 49875 \\
\hline
32000 & 49225 \\
\hline
\end{array}
\][/tex]
We now explain the steps for finding the answers.
──────────────────────────────
Step (a): Least-Squares Line
To find the least-squares line of the form
[tex]\[
\hat{y} = a + b x,
\][/tex]
we compute the slope [tex]$b$[/tex] and intercept [tex]$a$[/tex] using the standard formulas:
[tex]\[
b = \frac{n\sum (xy) - \left(\sum x\right)\left(\sum y\right)}{n\sum{x^2} - \left(\sum x\right)^2}, \quad
a = \overline{y} - b \, \overline{x},
\][/tex]
where [tex]$n$[/tex] is the number of data points.
After carrying out the computations and rounding the values to two decimal places, the resulting equation is
[tex]\[
\hat{y} = 54734.53 - 0.17 x.
\][/tex]
Since this is a linear equation, the degree of the polynomial is 1.
──────────────────────────────
Step (b): Predicting the Retail Value at 30000 Miles
To predict the retail value when the odometer reading is [tex]$x = 30000$[/tex], substitute into the equation:
[tex]\[
\hat{y} = 54734.53 - 0.17(30000).
\][/tex]
Performing the multiplication:
[tex]\[
0.17 \times 30000 = 5100,
\][/tex]
so
[tex]\[
\hat{y} \approx 54734.53 - 5100 = 49634.53.
\][/tex]
Rounding to the nearest \[tex]$100 gives a predicted value of
\[
\$[/tex]49600.
\]
──────────────────────────────
Step (c): Linear Correlation Coefficient
The linear correlation coefficient [tex]$r$[/tex] is calculated by
[tex]\[
r = \frac{\sum \left(x - \overline{x}\right)\left(y - \overline{y}\right)}{\sqrt{\sum \left(x - \overline{x}\right)^2 \sum \left(y - \overline{y}\right)^2}}.
\][/tex]
After doing the appropriate computations and rounding to four decimal places, we obtain
[tex]\[
r = -0.9979.
\][/tex]
──────────────────────────────
Step (d): Interpretation of the Negative Correlation
A negative linear correlation coefficient indicates that as the odometer reading increases, the retail value of the vehicle tends to decrease. In other words, higher mileage is associated with a lower retail value.
──────────────────────────────
Summary of Answers
(a) The equation of the least-squares line is
[tex]\[
\hat{y} = 54734.53 - 0.17x \quad (\text{with a polynomial degree of } 1).
\][/tex]
(b) For an odometer reading of [tex]$30000$[/tex], the predicted retail value is approximately
[tex]\[
\$49600.
\][/tex]
(c) The linear correlation coefficient is
[tex]\[
r = -0.9979.
\][/tex]
(d) The negative correlation coefficient means that as the odometer reading increases, the retail value decreases.
These are the detailed steps and results for this problem.
Thanks for taking the time to read The following table gives retail values of a 2020 Corvette for various odometer readings tex begin tabular c c hline Odometer Reading Retail Value hline. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
