document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196666>Question 1196666</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Because it is not practical to weigh bears in the field, researchers sought to develop a model to predict a bear's weight based on its length.  Here are the results for a sample:\r<BR>\n" );
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document.write( "Total Length (cm)	Weight (kg)<BR>\n" );
document.write( "139.0	110<BR>\n" );
document.write( "138.0	60<BR>\n" );
document.write( "139.0	90<BR>\n" );
document.write( "120.5	60<BR>\n" );
document.write( "149.0	85<BR>\n" );
document.write( "141.0	100<BR>\n" );
document.write( "141.0	95<BR>\n" );
document.write( "150.0	85<BR>\n" );
document.write( "166.0	155<BR>\n" );
document.write( "151.5	140<BR>\n" );
document.write( "129.5	105<BR>\n" );
document.write( "150.0	110<BR>\n" );
document.write( "If the bear length was instead measured in inches (there are 0.3937 inches in 1 cm) and the weight was measured in pounds (there are 2.2 pounds in one kg), then the slope of the regression line would _______.\r<BR>\n" );
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document.write( "Group of answer choices\r<BR>\n" );
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document.write( "decrease by a factor of 0.179 (0.179 = 0.3937/2.2)\r<BR>\n" );
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document.write( "not change\r<BR>\n" );
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document.write( "increase by a factor of 0.179 (0.179 = 0.3737/2.2)\r<BR>\n" );
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document.write( "decrease by a factor of 5.588 (5.588 = 2.2/0.3937)\r<BR>\n" );
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document.write( "increase by a factor of 5.588 (5.588 = 2.2/0.3937) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829616 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=829616\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The regression line is of the form y = mx+b<BR>\n" );
document.write( "m = slope<BR>\n" );
document.write( "b = y intercept\r<BR>\n" );
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document.write( "The slope formula for a linear regression is this particularly complex equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large m = \frac{n*\left(\sum \text{x}y\right)-\left(\sum \text{x}\right)*\left(\sum y\right)}{n*\left(\sum \text{x}^2\right)-\left(\sum \text{x}\right)^2}\">\r<BR>\n" );
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document.write( "We don't need to calculate the value of m itself. We just need to see how m changes.<BR>\n" );
document.write( "The x values represent the total length in cm<BR>\n" );
document.write( "To go from cm to inches, we multiply by 0.3937 since 1 cm = 0.3937 inches approximately\r<BR>\n" );
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document.write( "This means each x updates to  0.3937x\r<BR>\n" );
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document.write( "Let's replace each x with 0.3937x and do a bit of algebra like so:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large m = \frac{n*\left(\sum \text{x}y\right)-\left(\sum \text{x}\right)*\left(\sum 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 m = \frac{n*\left(\sum 0.3937\text{x}y\right)-\left(\sum 0.3937\text{x}\right)*\left(\sum y\right)}{n*\left(\sum (0.3937\text{x})^2\right)-\left(\sum 0.3937\text{x}\right)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large m = \frac{0.3937n*\left(\sum \text{x}y\right)-0.3937\left(\sum \text{x}\right)*\left(\sum y\right)}{0.3937^2n*\left(\sum \text{x}^2\right)-0.3937^2\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 m = \frac{0.3937\left(n*\left(\sum \text{x}y\right)-\left(\sum \text{x}\right)*\left(\sum y\right)\right)}{0.3937^2\left(n*\left(\sum \text{x}^2\right)-\left(\sum \text{x}\right)^2\right)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large m = \frac{n*\left(\sum \text{x}y\right)-\left(\sum \text{x}\right)*\left(\sum y\right)}{0.3937\left(n*\left(\sum \text{x}^2\right)-\left(\sum \text{x}\right)^2\right)}\">\r<BR>\n" );
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document.write( "This shows that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=newSlope+=+%281%2F%280.3937%29%29%2A%28oldSlope%29 ALIGN=MIDDLE  ALT=\"newSlope+=+%281%2F%280.3937%29%29%2A%28oldSlope%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=newSlope+=+%28oldSlope%29%2F%280.3937%29 ALIGN=MIDDLE  ALT=\"newSlope+=+%28oldSlope%29%2F%280.3937%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and this occurs after we scaled the x values to get 0.3937x (i.e multiplied each x value by 0.3937 to convert from cm to inches)\r<BR>\n" );
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document.write( "Through similar steps we can replace each y with 2.2y and have this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large m = \frac{n*\left(\sum \text{x}y\right)-\left(\sum \text{x}\right)*\left(\sum 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 m = \frac{n*\left(\sum \text{x}*2.2y\right)-\left(\sum \text{x}\right)*\left(\sum 2.2y\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 m = \frac{2.2n*\left(\sum \text{x}y\right)-2.2\left(\sum \text{x}\right)*\left(\sum 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 m = \frac{2.2\left(n*\left(\sum \text{x}y\right)-\left(\sum \text{x}\right)*\left(\sum y\right)\right)}{n*\left(\sum \text{x}^2\right)-\left(\sum \text{x}\right)^2}\">\r<BR>\n" );
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document.write( "So we arrive at the fact <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=newSlope+=+2.2%2A%28oldSlope%29 ALIGN=MIDDLE  ALT=\"newSlope+=+2.2%2A%28oldSlope%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > when we scaled each y value by 2.2 (i.e. multiplied each weight by 2.2 to convert from kg to pounds)\r<BR>\n" );
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document.write( "Put the two ideas together and we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=newSlope+=+%282.2%2F0.3937%29%2A%28oldSlope%29 ALIGN=MIDDLE  ALT=\"newSlope+=+%282.2%2F0.3937%29%2A%28oldSlope%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=newSlope+=+5.588%2A%28oldSlope%29 ALIGN=MIDDLE  ALT=\"newSlope+=+5.588%2A%28oldSlope%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The value 5.588 is approximate\r<BR>\n" );
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document.write( "You can think of it like this:<BR>\n" );
document.write( "slope = rise/run<BR>\n" );
document.write( "slope = (change in y)/(change in x)<BR>\n" );
document.write( "new slope = (2.2*(change in y))/(0.3937*(change in x))<BR>\n" );
document.write( "new slope = (2.2/0.3937)*( (change in y)/(change in x) )<BR>\n" );
document.write( "new slope = (2.2/0.3937)*(old slope)<BR>\n" );
document.write( "<font color=red>new slope = 5.588*(old slope)</font>\r<BR>\n" );
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document.write( "-----------------------------------------------------------\r<BR>\n" );
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document.write( "Answer: <font color=red>Increase by a factor of 5.588<BR>\n" );
document.write( "Since 5.588 = 2.2/0.3937</font>\r<BR>\n" );
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document.write( "Note: you likely wont need to write down those complicated formulas for your homework. Those are for illustrative purposes mostly. <BR>\n" );
document.write( "You could use the much faster method that the tutor @ewatrrr talked about.<BR>\n" );
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