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Based on the housing data below, which equation can be used to calculate fair housing prices?

| Square Feet | House Price (in thousands) |
|-------------|-----------------------------|
| 1900 | 196 |
| 2000 | 205 |
| 2200 | 225 |

A. [tex]\( y = 0.097x + 11.142 \)[/tex]

B. [tex]\( y = 0.087x - 9.286 \)[/tex]

C. [tex]\( y = 0.087x + 9.286 \)[/tex]

D. [tex]\( y = 0.074x - 50.48 \)[/tex]

Sagot :

GinnyAnswer
To determine which equation best matches the given housing data, we can proceed by finding the linear regression line based on the provided data points and comparing it with the given options.

First, let's understand the provided data:

| Square Feet | House Price (in thousands) |
|-------------|-----------------------------|
| 1900 | 196 |
| 2000 | 205 |
| 2200 | 225 |

We perform a linear regression to find the best-fit line [tex]\( y = mx + b \)[/tex] that minimizes the distance between the actual prices and those predicted by the equation. The result of this linear regression gives:
- Slope [tex]\(m\)[/tex]
- Intercept [tex]\(b\)[/tex]

The derived regression equation is used to calculate the house price for a house with 2000 square feet. This calculated house price is then compared with the predicted prices from the given options to find the closest.

1. Calculate the house price predicted by the derived regression equation for 2000 square feet:
- The calculated house price is approximately [tex]\( 205.428 \)[/tex] thousand dollars.

2. Evaluate each given option by plugging in 2000 square feet:

- Option A: [tex]\( y = 0.097 \times 2000 + 11.142 \)[/tex]
- [tex]\( y = 194 + 11.142 = 205.142 \)[/tex]

- Option B: [tex]\( y = 0.087 \times 2000 - 9.286 \)[/tex]
- [tex]\( y = 174 - 9.286 = 164.714 \)[/tex]

- Option C: [tex]\( y = 0.087 \times 2000 + 9.286 \)[/tex]
- [tex]\( y = 174 + 9.286 = 183.286 \)[/tex]

- Option D: [tex]\( y = 0.074 \times 2000 - 50.48 \)[/tex]
- [tex]\( y = 148 - 50.48 = 97.52 \)[/tex]

3. Calculate the differences between the predicted price (from regression) and each option:

- Difference for Option A: [tex]\( |205.428 - 205.142| \approx 0.286 \)[/tex]
- Difference for Option B: [tex]\( |205.428 - 164.714| \approx 40.714 \)[/tex]
- Difference for Option C: [tex]\( |205.428 - 183.286| \approx 22.143 \)[/tex]
- Difference for Option D: [tex]\( |205.428 - 97.52| \approx 107.908 \)[/tex]

4. Determine which option is the closest to the calculated regression price of 205.428:
- The smallest difference is for Option A, which is approximately [tex]\( 0.286 \)[/tex].

Therefore, the equation that can be used to calculate fair housing prices for the given data is:

A. [tex]\( y = 0.097x + 11.142 \)[/tex].
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