document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176942>Question 1176942</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Many people believe that there is a negative relationship between the amount of time students play video games each week and their GPA. \r<BR>\n" );
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document.write( "x	10	3	0	2	5	4	7<BR>\n" );
document.write( "y	1.5	2.4	3.2	3.5	2.7	3	2.1\r<BR>\n" );
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document.write( "Using the data shown above, predict a student's GPA when the student plays video games for 15 hours each week. Round your final answer to two decimal places.\r<BR>\n" );
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document.write( "What percentage of variation in GPAs can be explained by the number of hours students' play video games each week? Round your final answer to two decimal places.<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 #850546 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=850546\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Linear Regression**\r<BR>\n" );
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document.write( "We'll use linear regression to model the relationship between video game hours (x) and GPA (y).\r<BR>\n" );
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document.write( "* **Calculate the means:**<BR>\n" );
document.write( "    * Mean of x (x̄): (10 + 3 + 0 + 2 + 5 + 4 + 7) / 7 = 31 / 7 ≈ 4.43<BR>\n" );
document.write( "    * Mean of y (ȳ): (1.5 + 2.4 + 3.2 + 3.5 + 2.7 + 3 + 2.1) / 7 = 18.4 / 7 ≈ 2.63\r<BR>\n" );
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document.write( "* **Calculate the sums:**<BR>\n" );
document.write( "    * Σ(x - x̄)(y - ȳ):<BR>\n" );
document.write( "        * (10 - 4.43)(1.5 - 2.63) + (3 - 4.43)(2.4 - 2.63) + (0 - 4.43)(3.2 - 2.63) + (2 - 4.43)(3.5 - 2.63) + (5 - 4.43)(2.7 - 2.63) + (4 - 4.43)(3 - 2.63) + (7 - 4.43)(2.1 - 2.63) ≈ 6.26 + 0.33 + 2.59 + 2.11 + 0.06 - 0.16 - 1.48 ≈ 9.71<BR>\n" );
document.write( "    * Σ(x - x̄)²:<BR>\n" );
document.write( "        * (10 - 4.43)² + (3 - 4.43)² + (0 - 4.43)² + (2 - 4.43)² + (5 - 4.43)² + (4 - 4.43)² + (7 - 4.43)² ≈ 30.91 + 2.04 + 19.62 + 5.90 + 0.33 + 0.18 + 6.59 ≈ 65.57\r<BR>\n" );
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document.write( "* **Calculate the slope (b):**<BR>\n" );
document.write( "    * b = Σ(x - x̄)(y - ȳ) / Σ(x - x̄)² ≈ 9.71 / 65.57 ≈ -0.148\r<BR>\n" );
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document.write( "* **Calculate the y-intercept (a):**<BR>\n" );
document.write( "    * a = ȳ - b * x̄ ≈ 2.63 - (-0.148) * 4.43 ≈ 2.63 + 0.655 ≈ 3.29\r<BR>\n" );
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document.write( "* **Regression equation:**<BR>\n" );
document.write( "    * y = a + bx ≈ 3.29 - 0.148x\r<BR>\n" );
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document.write( "**2. Predict GPA for 15 Hours**\r<BR>\n" );
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document.write( "* Plug x = 15 into the regression equation:<BR>\n" );
document.write( "    * y ≈ 3.29 - 0.148 * 15 ≈ 3.29 - 2.22 ≈ 1.07\r<BR>\n" );
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document.write( "* Therefore, the predicted GPA for a student playing video games for 15 hours is approximately 1.07.\r<BR>\n" );
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document.write( "**3. Coefficient of Determination (R²)**\r<BR>\n" );
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document.write( "* **Calculate Σ(y - ȳ)²:**<BR>\n" );
document.write( "    * (1.5 - 2.63)² + (2.4 - 2.63)² + (3.2 - 2.63)² + (3.5 - 2.63)² + (2.7 - 2.63)² + (3 - 2.63)² + (2.1 - 2.63)² ≈ 1.28 + 0.05 + 0.33 + 0.76 + 0.005 + 0.14 + 0.28 ≈ 2.855\r<BR>\n" );
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document.write( "* **Calculate R²:**<BR>\n" );
document.write( "    * R² = 1 - (Σ(y - ŷ)² / Σ(y - ȳ)²)<BR>\n" );
document.write( "    * We need to find Σ(y-ŷ)²<BR>\n" );
document.write( "    * ŷ values are found by inputting all the original x values into the regression equation: y = 3.29 - 0.148x.<BR>\n" );
document.write( "    * ŷ = [1.81, 2.84, 3.29, 3.00, 2.55, 2.69, 2.25]<BR>\n" );
document.write( "    * Σ(y-ŷ)² = (1.5-1.81)² + (2.4-2.84)² + (3.2-3.29)² + (3.5-3.00)² + (2.7-2.55)² + (3-2.69)² + (2.1-2.25)² = 0.0961 + 0.1936 + 0.0081 + 0.25 + 0.0225 + 0.0961 + 0.0225 = 0.689<BR>\n" );
document.write( "    * R² = 1 - (0.689 / 2.855) ≈ 1 - 0.241 ≈ 0.759\r<BR>\n" );
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document.write( "* **Percentage of variation:**<BR>\n" );
document.write( "    * R² * 100% ≈ 75.9%\r<BR>\n" );
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document.write( "**Answers**\r<BR>\n" );
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document.write( "* **Predicted GPA for 15 hours:** 1.07<BR>\n" );
document.write( "* **Percentage of variation explained:** 75.90%<BR>\n" );
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