document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176842>Question 1176842</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( " Write the regression equation below. Round all numbers to four decimal places.\r<BR>\n" );
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document.write( "y hat = \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 #850561 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=850561\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this step-by-step.\r<BR>\n" );
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document.write( "**1. Calculate the Regression Equation**\r<BR>\n" );
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document.write( "We need to find the equation of the form ŷ = a + bx, where:\r<BR>\n" );
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document.write( "* x = hours playing video games<BR>\n" );
document.write( "* y = GPA<BR>\n" );
document.write( "* b = slope<BR>\n" );
document.write( "* a = y-intercept\r<BR>\n" );
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document.write( "First, calculate the necessary sums:\r<BR>\n" );
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document.write( "* Σx = 10 + 3 + 0 + 2 + 5 + 4 + 7 = 31<BR>\n" );
document.write( "* Σy = 1.5 + 2.4 + 3.2 + 3.5 + 2.7 + 3 + 2.1 = 18.4<BR>\n" );
document.write( "* Σx² = 100 + 9 + 0 + 4 + 25 + 16 + 49 = 203<BR>\n" );
document.write( "* Σy² = 2.25 + 5.76 + 10.24 + 12.25 + 7.29 + 9 + 4.41 = 51.2<BR>\n" );
document.write( "* Σxy = 15 + 7.2 + 0 + 7 + 13.5 + 12 + 14.7 = 69.4<BR>\n" );
document.write( "* n = 7 (number of data points)\r<BR>\n" );
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document.write( "Now, calculate b (slope):\r<BR>\n" );
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document.write( "* b = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²)<BR>\n" );
document.write( "* b = (7 * 69.4 - 31 * 18.4) / (7 * 203 - 31²)<BR>\n" );
document.write( "* b = (485.8 - 570.4) / (1421 - 961)<BR>\n" );
document.write( "* b = -84.6 / 460<BR>\n" );
document.write( "* b ≈ -0.1839\r<BR>\n" );
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document.write( "Next, calculate a (y-intercept):\r<BR>\n" );
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document.write( "* a = (Σy - bΣx) / n<BR>\n" );
document.write( "* a = (18.4 - (-0.1839) * 31) / 7<BR>\n" );
document.write( "* a = (18.4 + 5.6999) / 7<BR>\n" );
document.write( "* a = 24.0999 / 7<BR>\n" );
document.write( "* a ≈ 3.4428\r<BR>\n" );
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document.write( "Therefore, the regression equation is:\r<BR>\n" );
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document.write( "* ŷ = 3.4428 - 0.1839x\r<BR>\n" );
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document.write( "**2. Predict GPA for 15 Hours of Video Games**\r<BR>\n" );
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document.write( "* x = 15<BR>\n" );
document.write( "* ŷ = 3.4428 - 0.1839 * 15<BR>\n" );
document.write( "* ŷ = 3.4428 - 2.7585<BR>\n" );
document.write( "* ŷ ≈ 0.6843\r<BR>\n" );
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document.write( "Rounded to two decimal places, the predicted GPA is 0.68.\r<BR>\n" );
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document.write( "**3. Calculate the Coefficient of Determination (R²)**\r<BR>\n" );
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document.write( "R² represents the percentage of variation in GPAs explained by the number of hours playing video games.\r<BR>\n" );
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document.write( "First, calculate the correlation coefficient (r):\r<BR>\n" );
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document.write( "* r = (nΣxy - ΣxΣy) / √((nΣx² - (Σx)²)(nΣy² - (Σy)²))<BR>\n" );
document.write( "* r = -84.6 / √((460)(7 * 51.2 - 18.4²))<BR>\n" );
document.write( "* r = -84.6 / √((460)(358.4 - 338.56))<BR>\n" );
document.write( "* r = -84.6 / √((460)(19.84))<BR>\n" );
document.write( "* r = -84.6 / √9126.4<BR>\n" );
document.write( "* r = -84.6 / 95.5322<BR>\n" );
document.write( "* r ≈ -0.8856\r<BR>\n" );
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document.write( "Now, calculate R²:\r<BR>\n" );
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document.write( "* R² = r²<BR>\n" );
document.write( "* R² = (-0.8856)²<BR>\n" );
document.write( "* R² ≈ 0.7843\r<BR>\n" );
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document.write( "Expressed as a percentage and rounded to two decimal places:\r<BR>\n" );
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document.write( "* R² ≈ 78.43%\r<BR>\n" );
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document.write( "**Answers**\r<BR>\n" );
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document.write( "* **Regression equation:** ŷ = 3.4428 - 0.1839x<BR>\n" );
document.write( "* **Predicted GPA (15 hours):** 0.68<BR>\n" );
document.write( "* **Percentage of variation explained:** 78.43%<BR>\n" );
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