document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196551>Question 1196551</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random sample of 8 applicants for a certain foreign job is selected. The number of years that these applicants have studied French in high school or university, X, and the mark which they obtained in a proficiency test in French, Y, are recorded. The following information is given: ∑X =32, ∑Y =528, ∑XY=2136, ∑X2 =130 and ∑Y2 =34872 Fitting a regression line, which of the following statements is incorrect?\r<BR>\n" );
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document.write( "a.<BR>\n" );
document.write( "b₁ = 12\r<BR>\n" );
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document.write( "b.<BR>\n" );
document.write( "Ŷ = 12+18Xi\r<BR>\n" );
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document.write( "c.<BR>\n" );
document.write( "The predicted mark obtained if an applicant studied French for two years is 42.\r<BR>\n" );
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document.write( "d.<BR>\n" );
document.write( "b₀ = 18\r<BR>\n" );
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document.write( "e.<BR>\n" );
document.write( "Mean of X values = 4  Mean of Y values = 66 </SPAN>\n" );
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document.write( "X2 refers to X^2 or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2 ALIGN=MIDDLE  ALT=\"X%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Y2 refers to Y^2 or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y%5E2 ALIGN=MIDDLE  ALT=\"Y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Given info:<BR>\n" );
document.write( "n = 8 = sample size<BR>\n" );
document.write( "∑X =32, ∑Y =528, ∑XY=2136, ∑X2 =130 and ∑Y2 =34872<BR>\n" );
document.write( "Something like ∑X =32 means the X values add to 32.\r<BR>\n" );
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document.write( "The regression line is Ŷ = b₀ + b₁*Xi<BR>\n" );
document.write( "b₀ = y intercept<BR>\n" );
document.write( "b₁ = slope <BR>\n" );
document.write( "This is a different way to express the slope-intercept form y = mx+b.\r<BR>\n" );
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document.write( "The formulas to compute each are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{b}_{0} = \frac{\left(\sum \text{y}\right)\left(\sum \text{x}^2\right)-\left(\sum \text{x}\right)\left(\sum \text{x}\text{y}\right)}{n\left(\sum \text{x}^2\right) - \left(\sum \text{x}\right)^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{b}_{1} = \frac{n\left(\sum \text{x}\text{y}\right)-\left(\sum \text{x}\right)\left(\sum \text{y}\right)}{n\left(\sum \text{x}^2\right) - \left(\sum \text{x}\right)^2\">\r<BR>\n" );
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document.write( "Notes: <ul><li>The denominators are identical.</li><li>Since the formulas are ugly and complicated, it's best to use statistical or spreadsheet software when it comes to real world linear regression applications.</li></ul>Let's compute the y intercept\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{b}_{0} = \frac{\left(\sum \text{y}\right)\left(\sum \text{x}^2\right)-\left(\sum \text{x}\right)\left(\sum \text{x}\text{y}\right)}{n\left(\sum \text{x}^2\right) - \left(\sum \text{x}\right)^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{b}_{0} = \frac{(528)*(130)-(32)*(2136)}{8(130)-(32)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{b}_{0} = \frac{288}{16}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{b}_{0} = 18\">\r<BR>\n" );
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document.write( "Now onto the slope<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{b}_{1} = \frac{n\left(\sum \text{x}\text{y}\right)-\left(\sum \text{x}\right)\left(\sum \text{y}\right)}{n\left(\sum \text{x}^2\right) - \left(\sum \text{x}\right)^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{b}_{1} = \frac{8*(2136) - (32)*(528)}{8(130)-(32)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{b}_{1} = \frac{192}{16}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{b}_{1} = 12\">\r<BR>\n" );
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document.write( "The equation Ŷ = b₀ + b₁*Xi updates to Ŷ = 18+12Xi<BR>\n" );
document.write( "Therefore, choice B is the incorrect statement. The 12 and 18 needed to be swapped around.\r<BR>\n" );
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document.write( "Answer: <font color=red>Choice B)  Ŷ = 12+18Xi</font><BR>\n" );
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