Based on the housing data below, use linear regression to estimate a fair price for a 1700 ft² home.

| Square Feet | House Price (in thousands) |
|-------------|-----------------------------|
| 2156 | 210 |
| 2040 | 200 |
| 2050 | 204 |

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

To estimate the fair price for a 1700 square feet home using linear regression, we will perform the following steps:

### Step 1: Gather and Organize the Data
We have the following data:
- For 2156 square feet, the house price is [tex]$210,000. - For 2040 square feet, the house price is $[/tex]200,000.
- For 2050 square feet, the house price is [tex]$204,000. Let: - $ x $ be the square feet. - $ y $ be the house price (in thousands of dollars). So, our data points are: \[ (x_1, y_1) = (2156, 210) \\ (x_2, y_2) = (2040, 200) \\ (x_3, y_3) = (2050, 204) \] ### Step 2: Perform Linear Regression Linear regression will help us find a linear relationship of the form: \[ y = mx + b \] where: - $ y $ is the house price. - $ x $ is the square feet. - $ m $ is the slope of the regression line. - $ b $ is the y-intercept of the regression line. From our linear regression analysis, we determined: \[ m \approx 0.0741 \] \[ b \approx 50.4818 \] ### Step 3: Formulate the Regression Equation Using the slope ($ m $) and the intercept ($ b $), we can write the regression equation as: \[ y = 0.0741x + 50.4818 \] ### Step 4: Estimate House Price for 1700 Square Feet We can use the regression equation to estimate the house price for 1700 square feet. Substitute $ x = 1700 $ into the regression equation: \[ y = 0.0741 \times 1700 + 50.4818 \] ### Step 5: Perform the Calculation Now, calculate the estimated price: \[ y = 0.0741 \times 1700 + 50.4818 \] \[ y \approx 125.97 + 50.4818 \] \[ y \approx 176.452 \] In thousands of dollars, this would be approximately $[/tex]176,452.

### Step 6: Compare with Given Options
The closest option to our estimated price is:
B. \[tex]$176,000 So, the estimated fair price for a 1700 square feet home is: B. \$[/tex]176,000

semsee45 semsee45 · Answer 2 · 2024-06-16T18:36:15-04:00

Answer:

B) $176,000

Step-by-step explanation:

Linear regression is a method that models the relationship between a dependent variable and an independent variable by fitting a linear equation to observed data.

To find the linear regression equation of the given data, we first need to perform linear regression analysis on the data. To do this, we can use a statistical calculator.

The independent variable (x) is the square feet of the house.

The dependent variable (y) is house price (in thousands of dollars) .

After entering the (x, y) data from the given table into a statistical calculator we get:

[tex]a = 0.074056147... = 0.074 \\\\b = 50.481768312... = 50.482[/tex]

To determine the regression equation, substitute the found values of a and b into the regression line formula, y = ax + b.

Therefore, the linear regression equation that models the given data is:

[tex]y = 0.074x+50.482[/tex]

where:

x is the square feet of the house.
y is the house price (in thousands of dollars).

To estimate a fair price for a 1700ft² home, substitute x = 1700 into the equation:

[tex]y=0.074(1700)+50.482 \\\\y=125.8+50.482 \\\\y=176.282\\\\y\approx176[/tex]

Therefore, a fair price for a 1700ft² home is:

[tex]\Large\boxed{\boxed{\$176,000}}[/tex]