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The following table gives retail values of a 2020 Corvette for various odometer readings.

[tex]
\[
\begin{tabular}{|c|c|}
\hline
\text{Odometer Reading} & \text{Retail Value (\$)} \\
\hline
13,000 & 52,525 \\
\hline
18,000 & 51,625 \\
\hline
20,000 & 51,350 \\
\hline
25,000 & 50,325 \\
\hline
29,000 & 49,875 \\
\hline
32,000 & 49,225 \\
\hline
\end{tabular}
\]
[/tex]

(a) Find the equation of the least-squares line for the data, where the odometer reading is the independent variable [tex]$x$[/tex], and retail value is the dependent variable. Round your numerical values to two decimal places.

[tex]\hat{y} = \square[/tex]

(b) Use the equation from part (a) to predict the retail price of a 2020 Corvette with an odometer reading of 30,000. Round to the nearest [tex]\$100[/tex].

[tex]\$ \square[/tex]

(c) Find the linear correlation coefficient for these data. Round your answer to four decimal places.

[tex]r = \square[/tex]

(d) What is the significance of the fact that the linear correlation coefficient is negative for these data?

The variables are negatively correlated, which means that as the odometer reading goes up, the retail value goes down.

Answer :

Let's work through the problem step-by-step.

(a) Find the equation of the least-squares line

  1. Organize the given data:

    • Odometer Reading (x): 13000, 18000, 20000, 25000, 29000, 32000
    • Retail Value (y): 52525, 51625, 51350, 50325, 49875, 49225

  2. Calculate the means of x and y:

    • Mean of x: [tex]\bar{x} = \frac{13000 + 18000 + 20000 + 25000 + 29000 + 32000}{6} = 22833.33[/tex]
    • Mean of y: [tex]\bar{y} = \frac{52525 + 51625 + 51350 + 50325 + 49875 + 49225}{6} = 50870.83[/tex]

  3. Calculate the slope (b) of the least-squares line:

    • [tex]b = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}[/tex]
    • After calculating, we find [tex]b = -0.3179[/tex]

  4. Calculate the y-intercept (a) of the least-squares line:

    • [tex]a = \bar{y} - b\bar{x}[/tex]
    • Using the calculated slope, [tex]a = 58110.11[/tex]

  5. Equation of the least-squares line:

    • [tex]\hat{y} = -0.32x + 58110.11[/tex] (rounded to two decimal places)

(b) Predict the retail price of a 2020 Corvette with 30,000 miles

Use the equation from part (a):

  • [tex]\hat{y} = -0.32 \times 30000 + 58110.11[/tex]
  • [tex]\hat{y} = 48470.11[/tex]
  • Rounding to the nearest hundred dollars, the predicted retail price is [tex]\$48,500[/tex].

(c) Find the linear correlation coefficient (r)

  • Formula:
    [tex]r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \cdot \sum{(y_i - \bar{y})^2}}}[/tex]
  • After plugging in the values, we find [tex]r = -0.9953[/tex].

(d) What is the significance of the fact that the linear correlation coefficient is negative?

A negative correlation coefficient indicates a negative linear relationship between the two variables in this data set. This means that as the odometer reading increases, the retail value of the Corvette decreases, which is expected for used vehicles as more mileage typically reduces value.

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