document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163538>Question 1163538</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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.<BR>\n" );
document.write( "Taxes	Size<BR>\n" );
document.write( "21902	2370<BR>\n" );
document.write( "17316	2352<BR>\n" );
document.write( "18208	1930<BR>\n" );
document.write( "15632	1137<BR>\n" );
document.write( "43990	5626<BR>\n" );
document.write( "33662	2575<BR>\n" );
document.write( "15172	2222<BR>\n" );
document.write( "16742	1923<BR>\n" );
document.write( "18286	2041<BR>\n" );
document.write( "16073	1267<BR>\n" );
document.write( "15200	1398<BR>\n" );
document.write( "36052	3085<BR>\n" );
document.write( "31023	2869<BR>\n" );
document.write( "42089	3343<BR>\n" );
document.write( "14413	1632<BR>\n" );
document.write( "38913	3902<BR>\n" );
document.write( "25354	3940<BR>\n" );
document.write( "22985	2550<BR>\n" );
document.write( "16195	3563<BR>\n" );
document.write( "29289	2898\r<BR>\n" );
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document.write( "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.)\r<BR>\n" );
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document.write( " Taxes =_____ +_____ Size.\r<BR>\n" );
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document.write( "2. Interpret the slope coefficient.<BR>\n" );
document.write( " <BR>\n" );
document.write( "  a. As Size increases by 1 square foot, the property taxes are predicted to increase by $7.03.<BR>\n" );
document.write( "  b. As Property Taxes increase by 1 dollar, the size of the house increases by 7.03 ft.\r<BR>\n" );
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document.write( "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.)<BR>\n" );
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document.write( "  Taxes = _____<BR>\n" );
document.write( "              <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854374 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854374\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "import pandas as pd<BR>\n" );
document.write( "from sklearn.linear_model import LinearRegression\r<BR>\n" );
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document.write( "# Data<BR>\n" );
document.write( "taxes = np.array([21902, 17316, 18208, 15632, 43990, 33662, 15172, 16742, 18286, 16073, 15200, 36052, 31023, 42089, 14413, 38913, 25354, 22985, 16195, 29289])<BR>\n" );
document.write( "size = np.array([2370, 2352, 1930, 1137, 5626, 2575, 2222, 1923, 2041, 1267, 1398, 3085, 2869, 3343, 1632, 3902, 3940, 2550, 3563, 2898])\r<BR>\n" );
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document.write( "# Reshape for regression<BR>\n" );
document.write( "X = size.reshape(-1, 1)<BR>\n" );
document.write( "y = taxes\r<BR>\n" );
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document.write( "# Fit model<BR>\n" );
document.write( "model = LinearRegression()<BR>\n" );
document.write( "model.fit(X, y)\r<BR>\n" );
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document.write( "# Get parameters<BR>\n" );
document.write( "intercept = model.intercept_<BR>\n" );
document.write( "slope = model.coef_[0]\r<BR>\n" );
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document.write( "print(f\"Intercept: {intercept}\")<BR>\n" );
document.write( "print(f\"Slope: {slope}\")\r<BR>\n" );
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document.write( "# Prediction for 1600 sq ft<BR>\n" );
document.write( "prediction = intercept + slope * 1600<BR>\n" );
document.write( "print(f\"Prediction for 1600 sq ft: {prediction}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Intercept: 5929.920950651071<BR>\n" );
document.write( "Slope: 7.029199798319719<BR>\n" );
document.write( "Prediction for 1600 sq ft: 17176.64062796262\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To find the relationship between property taxes and home size, we perform a simple linear regression analysis using the provided data.\r<BR>\n" );
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document.write( "### 1. Sample Regression Equation<BR>\n" );
document.write( "Using the data provided, the calculated coefficients for the regression line ($\text{Taxes} = \beta_0 + \beta_1 \cdot \text{Size}$) are:<BR>\n" );
document.write( "* **Intercept ($\beta_0$):** $5929.92$<BR>\n" );
document.write( "* **Slope ($\beta_1$):** $7.03$\r<BR>\n" );
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document.write( "The estimated regression equation is:<BR>\n" );
document.write( "$$\text{Taxes} = 5929.92 + 7.03 \cdot \text{Size}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 2. Interpretation of the Slope Coefficient<BR>\n" );
document.write( "The slope coefficient ($7.03$) represents the average change in the dependent variable (Taxes) for every one-unit increase in the independent variable (Size).\r<BR>\n" );
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document.write( "* **Correct Interpretation:** **a. As Size increases by 1 square foot, the property taxes are predicted to increase by $7.03.**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 3. Prediction for a 1,600-Square-Foot Home<BR>\n" );
document.write( "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:<BR>\n" );
document.write( "* Intercept $\approx 5929.9210$<BR>\n" );
document.write( "* Slope $\approx 7.0292$\r<BR>\n" );
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document.write( "$$\text{Taxes} = 5929.9210 + 7.0292 \times (1600)$$<BR>\n" );
document.write( "$$\text{Taxes} = 5929.9210 + 11246.72$$<BR>\n" );
document.write( "$$\text{Taxes} = 17176.641$$\r<BR>\n" );
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document.write( "Rounding to two decimal places, the predicted property taxes are:<BR>\n" );
document.write( "**Taxes = $17,176.64** </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );