College

A real estate agent made a scatter plot with homeowners' property taxes on the [tex]y[/tex]-axis and the sizes of their homes, in square feet, on the [tex]x[/tex]-axis. The variables have a strong linear correlation, and the equation for the least squares regression line is [tex]\hat{y} = 0.392x + 638.118[/tex].

Based on the equation, what should the agent expect the property taxes to be for a 2400-square-foot home in the area?

A. \[tex]$940.80

B. \$[/tex]1190.94
C. \[tex]$1578.92

D. \$[/tex]2646.12

Answer :

To determine the expected property taxes for a 2400-square-foot home using the given equation for the least squares regression line, we follow these steps:

1. Understand the Equation: The equation provided is [tex]\(\hat{y} = 0.392x + 638.118\)[/tex]. This equation represents the relationship between the size of the home (in square feet) and the expected property taxes. In this equation:
- [tex]\(\hat{y}\)[/tex] is the predicted property taxes.
- [tex]\(x\)[/tex] represents the size of the home in square feet.

2. Identify the Given Value: We need to find the property taxes when the size of the home, [tex]\(x\)[/tex], is 2400 square feet.

3. Substitute the Value into the Equation: Replace [tex]\(x\)[/tex] in the equation with 2400:
[tex]\[
\hat{y} = 0.392 \times 2400 + 638.118
\][/tex]

4. Perform the Calculation: Calculate each part of the equation:
- Multiply [tex]\(0.392\)[/tex] by [tex]\(2400\)[/tex]:
[tex]\[
0.392 \times 2400 = 940.8
\][/tex]
- Add the result to the constant term [tex]\(638.118\)[/tex]:
[tex]\[
940.8 + 638.118 = 1578.918
\][/tex]

5. Round the Result: If necessary, round to the required decimal places. Here, the calculation yields:
[tex]\[
1578.92
\][/tex]

Thus, the expected property taxes for a 2400-square-foot home are approximately \[tex]$1578.92. Therefore, the correct answer is \(\$[/tex]1578.92\).

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