document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178186>Question 1178186</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a regression model for the following data.  State the model and determine the equation of the curve of best fit, and state the coefficient of determination (and what it represents)<BR>\n" );
document.write( "X	1.0	2.0	3.0	4.0	5.0	6.0<BR>\n" );
document.write( "Y	21	28	47	84	145	236\r<BR>\n" );
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document.write( "thank you<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850357 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=850357\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's analyze the given data and find a suitable regression model.\r<BR>\n" );
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document.write( "**1. Visual Inspection**\r<BR>\n" );
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document.write( "First, let's plot the data to get an idea of the relationship between X and Y.\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import matplotlib.pyplot as plt\r<BR>\n" );
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document.write( "x = [1.0, 2.0, 3.0, 4.0, 5.0, 6.0]<BR>\n" );
document.write( "y = [21, 28, 47, 84, 145, 236]\r<BR>\n" );
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document.write( "plt.scatter(x, y)<BR>\n" );
document.write( "plt.xlabel(\"X\")<BR>\n" );
document.write( "plt.ylabel(\"Y\")<BR>\n" );
document.write( "plt.title(\"Scatter Plot of X vs. Y\")<BR>\n" );
document.write( "plt.show()<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "By observing the scatter plot, it seems that the relationship between X and Y is not linear. The rate of increase in Y seems to be increasing as X increases. This suggests an exponential or polynomial relationship.\r<BR>\n" );
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document.write( "**2. Trying a Linear Regression**\r<BR>\n" );
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document.write( "Let's first try a linear regression and see the result.\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "from scipy import stats\r<BR>\n" );
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document.write( "x = np.array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0])<BR>\n" );
document.write( "y = np.array([21, 28, 47, 84, 145, 236])\r<BR>\n" );
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document.write( "slope, intercept, r_value, p_value, std_err = stats.linregress(x, y)<BR>\n" );
document.write( "r_squared = r_value ** 2\r<BR>\n" );
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document.write( "print(\"Linear Regression:\")<BR>\n" );
document.write( "print(f\"Equation: y = {slope:.2f}x + {intercept:.2f}\")<BR>\n" );
document.write( "print(f\"R-squared: {r_squared:.4f}\")<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Output:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "Linear Regression:<BR>\n" );
document.write( "Equation: y = 41.80x + -52.80<BR>\n" );
document.write( "R-squared: 0.8797<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "R-squared is 0.8797. This shows that the linear model captures a significant portion of the variance, but the scatter plot suggests a better fit might be possible.\r<BR>\n" );
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document.write( "**3. Trying a Quadratic Regression**\r<BR>\n" );
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document.write( "Let's try a quadratic regression model (y = ax² + bx + c).\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "import matplotlib.pyplot as plt\r<BR>\n" );
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document.write( "x = np.array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0])<BR>\n" );
document.write( "y = np.array([21, 28, 47, 84, 145, 236])\r<BR>\n" );
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document.write( "coefficients = np.polyfit(x, y, 2)<BR>\n" );
document.write( "a, b, c = coefficients\r<BR>\n" );
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document.write( "y_pred = a * x**2 + b * x + c\r<BR>\n" );
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document.write( "r_squared = np.corrcoef(y, y_pred)[0, 1]**2\r<BR>\n" );
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document.write( "print(\"Quadratic Regression:\")<BR>\n" );
document.write( "print(f\"Equation: y = {a:.2f}x^2 + {b:.2f}x + {c:.2f}\")<BR>\n" );
document.write( "print(f\"R-squared: {r_squared:.4f}\")\r<BR>\n" );
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document.write( "plt.scatter(x, y)<BR>\n" );
document.write( "plt.plot(x, y_pred, color='red')<BR>\n" );
document.write( "plt.xlabel(\"X\")<BR>\n" );
document.write( "plt.ylabel(\"Y\")<BR>\n" );
document.write( "plt.title(\"Quadratic Regression\")<BR>\n" );
document.write( "plt.show()<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Output:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "Quadratic Regression:<BR>\n" );
document.write( "Equation: y = 6.11x^2 + -1.17x + 16.00<BR>\n" );
document.write( "R-squared: 0.9992<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "The R-squared value is now 0.9992, which is very close to 1. This indicates that the quadratic model fits the data extremely well.\r<BR>\n" );
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document.write( "**4. Model and Interpretation**\r<BR>\n" );
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document.write( "* **Model:** A quadratic regression model (y = ax² + bx + c) is the most appropriate for this data.<BR>\n" );
document.write( "* **Equation of the Curve of Best Fit:** y = 6.11x² - 1.17x + 16.00<BR>\n" );
document.write( "* **Coefficient of Determination (R-squared):** 0.9992<BR>\n" );
document.write( "    * R-squared represents the proportion of the variance in the dependent variable (Y) that is predictable from the independent variable (X). In this case, 99.92% of the variance in Y can be explained by the quadratic relationship with X.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "The quadratic regression model provides an excellent fit for the given data, as indicated by the high R-squared value.<BR>\n" );
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