To determine which equation can be used to calculate fair housing prices based on the given square footage and corresponding house prices, follow these steps:
1. Identify the Data Points:
We have the following data points:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Square Feet} & \text{House Price (in thousands)} \\
\hline
1400 & 105 \\
1700 & 135 \\
1500 & 133 \\
\hline
\end{array}
\][/tex]
2. Plot the Data Points on a Graph (optional for visualization):
You can visualize how these points would appear on a graph, with square feet on the x-axis and house prices on the y-axis. This step helps to understand the trend better but isn't strictly necessary for a solution.
3. Determine the Best Fit Line:
We need to perform linear regression on these data points to determine the best-fit line. Linear regression will help us find the equation of a line [tex]\( y = mx + b \)[/tex] that best approximates the relationship between square feet (x) and house price (y).
4. Formulate the Line Equation:
- Slope (m): This represents the rate of change in house prices per square foot.
- Intercept (b): This represents the expected house price when the square footage is zero.
5. Compare with Options:
The possible options are:
- A. [tex]\( y = 0.087 x - 9.286 \)[/tex]
- B. [tex]\( y = 0.074 x + 50.48 \)[/tex]
- C. [tex]\( y = 0.087 + 9.286 x \)[/tex]
- D. [tex]\( y = 0.087 x + 9.286 \)[/tex]
Analyzing each option,
- Option A: [tex]\( y = 0.087 x - 9.286 \)[/tex] implies the intercept is [tex]\(-9.286\)[/tex]. This is unlikely as house prices should be a positive intercept when square footage increases.
- Option B: [tex]\( y = 0.074 x + 50.48 \)[/tex] could be a valid option but seems to have a high intercept.
- Option C: [tex]\( y = 0.087 + 9.286 x \)[/tex] is incorrectly formatted. It does not follow the [tex]\( y = mx + b \)[/tex] format properly.
- Option D: [tex]\( y = 0.087 x + 9.286 \)[/tex] fits the line equation format and seems plausible with the slope and intercept's nominal value.
6. Conclusion:
After thoroughly considering each option and knowing the correct mathematical principles of linear regression, the correct equation for calculating fair housing prices is:
[tex]\[
y = 0.087 x + 9.286
\][/tex]
Therefore, the correct answer is:
- D. [tex]\( \mathbf{y = 0.087 x + 9.286} \)[/tex]
