document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1187210>Question 1187210</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Q No.6) <BR>\n" );
document.write( "Part a) <BR>\n" );
document.write( "The following table shows verbal reasoning test score, x, and an English test score, y, for each of a random sample of 8 children who took both tests.<BR>\n" );
document.write( "Child 	A 	B	 C	 D	 E	 F	 G	 H<BR>\n" );
document.write( "x 	112 	113 	110 	113 	112	 114 	109	 113<BR>\n" );
document.write( "y 	69 	65 	75 	70 	70 	75 	68 	76<BR>\n" );
document.write( "Calculate the Co-efficient of correlation “r” between the scores in verbal reasoning and English.\r<BR>\n" );
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document.write( " Part b) <BR>\n" );
document.write( "The age (X) and systolic blood pressure (Y) of 100 individuals gave the following sums:<BR>\n" );
document.write( "X i  4421, Yi  12130, X i2  208349, Yi2 1498976, X iYi  542735 <BR>\n" );
document.write( "Determine the equation of the least squares regression line of Y on X.\r<BR>\n" );
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document.write( "the missing letter is sumation (E)<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 #849313 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=849313\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Part a) Correlation Coefficient (r)\r<BR>\n" );
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document.write( "Here's how to calculate the correlation coefficient (r) for the verbal reasoning (x) and English (y) test scores:\r<BR>\n" );
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document.write( "Calculate the means of x and y:\r<BR>\n" );
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document.write( "Mean of x (x̄) = (112 + 113 + 110 + 113 + 112 + 114 + 109 + 113) / 8 = 111.75<BR>\n" );
document.write( "Mean of y (ȳ) = (69 + 65 + 75 + 70 + 70 + 75 + 68 + 76) / 8 = 71<BR>\n" );
document.write( "Calculate the deviations from the mean for x and y:\r<BR>\n" );
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document.write( "For each x value, calculate (x - x̄)<BR>\n" );
document.write( "For each y value, calculate (y - ȳ)<BR>\n" );
document.write( "Child	x	y	x - x̄	y - ȳ<BR>\n" );
document.write( "A	112	69	0.25	-2<BR>\n" );
document.write( "B	113	65	1.25	-6<BR>\n" );
document.write( "C	110	75	-1.75	4<BR>\n" );
document.write( "D	113	70	1.25	-1<BR>\n" );
document.write( "E	112	70	0.25	-1<BR>\n" );
document.write( "F	114	75	2.25	4<BR>\n" );
document.write( "G	109	68	-2.75	-3<BR>\n" );
document.write( "H	113	76	1.25	5\r<BR>\n" );
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document.write( "Export to Sheets<BR>\n" );
document.write( "Calculate the products of the deviations:\r<BR>\n" );
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document.write( "Multiply (x - x̄) and (y - ȳ) for each child.\r<BR>\n" );
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document.write( "Child	x - x̄	y - ȳ	(x - x̄)(y - ȳ)<BR>\n" );
document.write( "A	0.25	-2	-0.5<BR>\n" );
document.write( "B	1.25	-6	-7.5<BR>\n" );
document.write( "C	-1.75	4	-7<BR>\n" );
document.write( "D	1.25	-1	-1.25<BR>\n" );
document.write( "E	0.25	-1	-0.25<BR>\n" );
document.write( "F	2.25	4	9<BR>\n" );
document.write( "G	-2.75	-3	8.25<BR>\n" );
document.write( "H	1.25	5	6.25\r<BR>\n" );
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document.write( "Export to Sheets<BR>\n" );
document.write( "Calculate the sum of the products of deviations (Σ(x - x̄)(y - ȳ)):\r<BR>\n" );
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document.write( "Sum the values in the last column: -0.5 + (-7.5) + (-7) + (-1.25) + (-0.25) + 9 + 8.25 + 6.25 = 7\r<BR>\n" );
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document.write( "Calculate the sum of squared deviations for x and y:\r<BR>\n" );
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document.write( "Σ(x - x̄)² = (0.25)² + (1.25)² + (-1.75)² + (1.25)² + (0.25)² + (2.25)² + (-2.75)² + (1.25)² = 14.75<BR>\n" );
document.write( "Σ(y - ȳ)² = (-2)² + (-6)² + 4² + (-1)² + (-1)² + 4² + (-3)² + 5² = 96<BR>\n" );
document.write( "Calculate the correlation coefficient (r):\r<BR>\n" );
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document.write( "r = Σ(x - x̄)(y - ȳ) / sqrt[Σ(x - x̄)² * Σ(y - ȳ)²]<BR>\n" );
document.write( "r = 7 / sqrt(14.75 * 96)<BR>\n" );
document.write( "r = 7 / sqrt(1416)<BR>\n" );
document.write( "r ≈ 7 / 37.63<BR>\n" );
document.write( "r ≈ 0.186\r<BR>\n" );
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document.write( "Part b) Least Squares Regression Line\r<BR>\n" );
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document.write( "The equation of the least squares regression line of Y on X is given by:\r<BR>\n" );
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document.write( "y = a + bx\r<BR>\n" );
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document.write( "where:\r<BR>\n" );
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document.write( "b = (ΣXᵢYᵢ - (ΣXᵢ * ΣYᵢ) / n) / (ΣXᵢ² - (ΣXᵢ)² / n)<BR>\n" );
document.write( "a = ȳ - b * x̄<BR>\n" );
document.write( "Given:\r<BR>\n" );
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document.write( "ΣXᵢ = 4421<BR>\n" );
document.write( "ΣYᵢ = 12130<BR>\n" );
document.write( "ΣXᵢ² = 208349<BR>\n" );
document.write( "ΣYᵢ² = 1498976<BR>\n" );
document.write( "ΣXᵢYᵢ = 542735<BR>\n" );
document.write( "n = 100<BR>\n" );
document.write( "Calculate b:<BR>\n" );
document.write( "b = (542735 - (4421 * 12130) / 100) / (208349 - (4421)² / 100)<BR>\n" );
document.write( "b = (542735 - 536497.3) / (208349 - 195452.41)<BR>\n" );
document.write( "b = 6237.7 / 12896.59<BR>\n" );
document.write( "b ≈ 0.484\r<BR>\n" );
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document.write( "Calculate the means of X and Y:<BR>\n" );
document.write( "x̄ = ΣXᵢ / n = 4421 / 100 = 44.21<BR>\n" );
document.write( "ȳ = ΣYᵢ / n = 12130 / 100 = 121.3<BR>\n" );
document.write( "Calculate a:<BR>\n" );
document.write( "a = ȳ - b * x̄<BR>\n" );
document.write( "a = 121.3 - (0.484 * 44.21)<BR>\n" );
document.write( "a = 121.3 - 21.41<BR>\n" );
document.write( "a ≈ 99.89\r<BR>\n" );
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document.write( "Therefore, the equation of the least squares regression line of Y on X is:\r<BR>\n" );
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document.write( "y = 99.89 + 0.484x </SPAN>\n" );
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