Question 1163538: The accompanying table shows a portion of a data set that refers to the property taxes owed by a homeowner (in $) and the size of the home (in square feet) in an affluent suburb 30 miles outside New York City.
Taxes Size
21902 2370
17316 2352
18208 1930
15632 1137
43990 5626
33662 2575
15172 2222
16742 1923
18286 2041
16073 1267
15200 1398
36052 3085
31023 2869
42089 3343
14413 1632
38913 3902
25354 3940
22985 2550
16195 3563
29289 2898
1. Estimate the sample regression equation that enables us to predict property taxes on the basis of the size of the home. (Round your answers to 2 decimal places.)
Taxes =_____ +_____ Size.
2. Interpret the slope coefficient.
a. As Size increases by 1 square foot, the property taxes are predicted to increase by $7.03.
b. As Property Taxes increase by 1 dollar, the size of the house increases by 7.03 ft.
3. Predict the property taxes for a 1,600-square-foot home. (Round coefficient estimates to at least 4 decimal places and final answer to 2 decimal places.)
Taxes = _____
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website! ```python?code_reference&code_event_index=2
import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression
# Data
taxes = np.array([21902, 17316, 18208, 15632, 43990, 33662, 15172, 16742, 18286, 16073, 15200, 36052, 31023, 42089, 14413, 38913, 25354, 22985, 16195, 29289])
size = np.array([2370, 2352, 1930, 1137, 5626, 2575, 2222, 1923, 2041, 1267, 1398, 3085, 2869, 3343, 1632, 3902, 3940, 2550, 3563, 2898])
# Reshape for regression
X = size.reshape(-1, 1)
y = taxes
# Fit model
model = LinearRegression()
model.fit(X, y)
# Get parameters
intercept = model.intercept_
slope = model.coef_[0]
print(f"Intercept: {intercept}")
print(f"Slope: {slope}")
# Prediction for 1600 sq ft
prediction = intercept + slope * 1600
print(f"Prediction for 1600 sq ft: {prediction}")
```
```text?code_stdout&code_event_index=2
Intercept: 5929.920950651071
Slope: 7.029199798319719
Prediction for 1600 sq ft: 17176.64062796262
```
To find the relationship between property taxes and home size, we perform a simple linear regression analysis using the provided data.
### 1. Sample Regression Equation
Using the data provided, the calculated coefficients for the regression line ($\text{Taxes} = \beta_0 + \beta_1 \cdot \text{Size}$) are:
* **Intercept ($\beta_0$):** $5929.92$
* **Slope ($\beta_1$):** $7.03$
The estimated regression equation is:
$$\text{Taxes} = 5929.92 + 7.03 \cdot \text{Size}$$
---
### 2. Interpretation of the Slope Coefficient
The slope coefficient ($7.03$) represents the average change in the dependent variable (Taxes) for every one-unit increase in the independent variable (Size).
* **Correct Interpretation:** **a. As Size increases by 1 square foot, the property taxes are predicted to increase by $7.03.**
---
### 3. Prediction for a 1,600-Square-Foot Home
To predict the taxes for a house with a size of $1,600$ square feet, we use the regression equation with higher precision coefficients as requested:
* Intercept $\approx 5929.9210$
* Slope $\approx 7.0292$
$$\text{Taxes} = 5929.9210 + 7.0292 \times (1600)$$
$$\text{Taxes} = 5929.9210 + 11246.72$$
$$\text{Taxes} = 17176.641$$
Rounding to two decimal places, the predicted property taxes are:
**Taxes = $17,176.64**
|
|
|
