High School

Based on the housing data below, use linear regression to estimate a fair price for a [tex]$1700 \, \text{ft}^2$[/tex] home.

[tex]
\[
\begin{tabular}{|c|c|}
\hline
Square Feet & \text{House Price (in thousands)} \\
\hline
2156 & 210 \\
\hline
2040 & 200 \\
\hline
2050 & 204 \\
\hline
\end{tabular}
\]
[/tex]

A. [tex]$\$ 156,000$[/tex]
B. [tex]$\$ 176,000$[/tex]
C. [tex]$\$ 190,000$[/tex]
D. [tex]$\$ 180,000$[/tex]

Answer :

To estimate a fair price for a 1700 ft² home using linear regression based on the given housing data, we can follow these steps:

1. Understand the Data: We have data for three homes with their respective square footage and prices (in thousands of dollars):
- 2156 square feet: [tex]$210,000
- 2040 square feet: $[/tex]200,000
- 2050 square feet: [tex]$204,000

2. Set Up the Equation: Linear regression helps us find a straight line (equation) that best fits our data. The equation of a line is generally given by:
\[
\text{Price} = (\text{slope} \times \text{Square Feet}) + \text{intercept}
\]

3. Calculate the Slope and Intercept: For the data above, when calculating using linear regression methods, we determine:
- Slope: Approximately \(0.0741\)
- Intercept: Approximately \(50.48\)

4. Estimate the Price for 1700 ft²: With the slope and intercept values, we substitute into the linear equation to find the estimated price for a 1700 ft² home:
\[
\text{Estimated Price} = (0.0741 \times 1700) + 50.48
\]

5. Perform the Calculation:
- Multiply the slope (\(0.0741\)) by 1700 to get approximately \(125.97\).
- Add the intercept (\(50.48\)) to \(125.97\), giving an estimated price of approximately \(176.45\).

6. Convert the Price to Thousands: Since the prices in the data are in thousands, the estimated price in thousands is approximately \(176\).

7. Find the Closest Option: Among the given options, the closest estimated price for a 1700 ft² home is option B, which is \(\$[/tex]176,000\).

Conclusion: The estimated fair price for a 1700 ft² home is approximately [tex]$176,000. So, the correct answer is B. \$[/tex]176,000.