document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196665>Question 1196665</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( "The residual associated with the bear whose length is 149.0 cm and weight is 85 kg is _______kg. (round your answer to three digits after the decimal)<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 #829606 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=829606\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<font color=red>Answer: -24.960</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "x = total length (cm)<BR>\n" );
document.write( "y = weight (kg)\r<BR>\n" );
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document.write( "Use technology to find the equation of the regression line<BR>\n" );
document.write( "I used GeoGebra to get y = 1.69417x - 142.47092 approximately.<BR>\n" );
document.write( "You could use a spreadsheet program or any linear regression calculator to get the same thing. <BR>\n" );
document.write( "I'll go into further detail where this equation comes from in the next section below.\r<BR>\n" );
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document.write( "Plug in x = 149.0 to find that,<BR>\n" );
document.write( "y = 1.69417x - 142.47092<BR>\n" );
document.write( "y = 1.69417*149.0 - 142.47092<BR>\n" );
document.write( "y = 109.96041\r<BR>\n" );
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document.write( "The length of x = 149.0 cm leads to a predicted weight of about y = 109.96041 kg<BR>\n" );
document.write( "The true weight associated with this x value should be y = 85 kg instead. \r<BR>\n" );
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document.write( "The residual is the error between the observed y value and predicted y value<BR>\n" );
document.write( "residual = (observed y value) - (predicted y value)<BR>\n" );
document.write( "residual = 85 - 109.96041<BR>\n" );
document.write( "residual = -24.96041<BR>\n" );
document.write( "residual = <font color=red>-24.960</font><BR>\n" );
document.write( "This is approximate and rounded to three decimal places (nearest thousandth)\r<BR>\n" );
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document.write( "---------------------------------------------\r<BR>\n" );
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document.write( "This section will go into further detail where the regression line came from. <BR>\n" );
document.write( "Depending on your teacher, this section is optional. \r<BR>\n" );
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document.write( "In real world settings, you won't need to know the formulas. It's much more efficient to use calculators, software, or spreadsheets. <BR>\n" );
document.write( "However, it's still good to know what's going on under the hood. I'll leave out the proofs and derivations of each formula. <BR>\n" );
document.write( "Those are better suited for calculus and linear algebra settings. \r<BR>\n" );
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document.write( "Here's the original data set of x and y values paired up together<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >y</TD></tr><tr><TD >139</TD><TD >110</TD></tr><tr><TD >138</TD><TD >60</TD></tr><tr><TD >139</TD><TD >90</TD></tr><tr><TD >120.5</TD><TD >60</TD></tr><tr><TD >149</TD><TD >85</TD></tr><tr><TD >141</TD><TD >100</TD></tr><tr><TD >141</TD><TD >95</TD></tr><tr><TD >150</TD><TD >85</TD></tr><tr><TD >166</TD><TD >155</TD></tr><tr><TD >151.5</TD><TD >140</TD></tr><tr><TD >129.5</TD><TD >105</TD></tr><tr><TD >150</TD><TD >110</TD></tr>\n" );
document.write( "</TABLE>We'll form the following columns:<BR>\n" );
document.write( "x^2<BR>\n" );
document.write( "xy\r<BR>\n" );
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document.write( "The x^2 column is where we square each x value<BR>\n" );
document.write( "eg: 139 squares to 139^2 = 139*139 = 19321\r<BR>\n" );
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document.write( "The xy column has us multiply each x and y value together (separately per row).<BR>\n" );
document.write( "Eg: 139.0*110 = 15290 in the first row of this column.\r<BR>\n" );
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document.write( "I strongly recommend using spreadsheet software rather than doing it all by hand.\r<BR>\n" );
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document.write( "Here's what all that looks like<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >y</TD><TD >x^2</TD><TD >xy</TD></tr><tr><TD >139</TD><TD >110</TD><TD >19321</TD><TD >15290</TD></tr><tr><TD >138</TD><TD >60</TD><TD >19044</TD><TD >8280</TD></tr><tr><TD >139</TD><TD >90</TD><TD >19321</TD><TD >12510</TD></tr><tr><TD >120.5</TD><TD >60</TD><TD >14520.25</TD><TD >7230</TD></tr><tr><TD >149</TD><TD >85</TD><TD >22201</TD><TD >12665</TD></tr><tr><TD >141</TD><TD >100</TD><TD >19881</TD><TD >14100</TD></tr><tr><TD >141</TD><TD >95</TD><TD >19881</TD><TD >13395</TD></tr><tr><TD >150</TD><TD >85</TD><TD >22500</TD><TD >12750</TD></tr><tr><TD >166</TD><TD >155</TD><TD >27556</TD><TD >25730</TD></tr><tr><TD >151.5</TD><TD >140</TD><TD >22952.25</TD><TD >21210</TD></tr><tr><TD >129.5</TD><TD >105</TD><TD >16770.25</TD><TD >13597.5</TD></tr><tr><TD >150</TD><TD >110</TD><TD >22500</TD><TD >16500</TD></tr>\n" );
document.write( "</TABLE>Next we add up the values of each column<BR>\n" );
document.write( "P = sum of the x values = 1714.5<BR>\n" );
document.write( "Q = sum of the y values = 1195<BR>\n" );
document.write( "R = sum of the x^2 values = 246447.75<BR>\n" );
document.write( "S = sum of the xy values = 173257.5\r<BR>\n" );
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document.write( "The linear regression equation 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( "To calculate m and b, we use these two formulas<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%28Q%2AR-P%2AS%29%2F%28n%2AR-P%5E2%29 ALIGN=MIDDLE  ALT=\"b+=+%28Q%2AR-P%2AS%29%2F%28n%2AR-P%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where P,Q,R,S were mentioned in the previous paragraph above.\r<BR>\n" );
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document.write( "The numerators are different, but the denominators are identical.<BR>\n" );
document.write( "The n refers to the sample size. It's the number of x,y pairs of values. In this case we have n = 12 such items.\r<BR>\n" );
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document.write( "So,<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%2812%2A173257.5-1714.5%2A1195%29%2F%2812%2A246447.75-%281714.5%29%5E2%29 ALIGN=MIDDLE  ALT=\"m+=+%2812%2A173257.5-1714.5%2A1195%29%2F%2812%2A246447.75-%281714.5%29%5E2%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=m+=+1.6941680312382 ALIGN=MIDDLE  ALT=\"m+=+1.6941680312382\"   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=m+=+1.69417 ALIGN=MIDDLE  ALT=\"m+=+1.69417\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is the approximate slope\r<BR>\n" );
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document.write( "And,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%28Q%2AR-P%2AS%29%2F%28n%2AR-P%5E2%29 ALIGN=MIDDLE  ALT=\"b+=+%28Q%2AR-P%2AS%29%2F%28n%2AR-P%5E2%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=b+=+%281195%2A246447.75+-+1714.5%2A173257.5%29%2F%2812%2A246447.75-%281714.5%29%5E2%29 ALIGN=MIDDLE  ALT=\"b+=+%281195%2A246447.75+-+1714.5%2A173257.5%29%2F%2812%2A246447.75-%281714.5%29%5E2%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=b+=+-142.470924129823 ALIGN=MIDDLE  ALT=\"b+=+-142.470924129823\"   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=b+=+-142.47092 ALIGN=MIDDLE  ALT=\"b+=+-142.47092\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is the approximate y intercept.\r<BR>\n" );
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document.write( "Therefore, the template y = mx+b updates to the approximation of y = 1.69417x - 142.47092<BR>\n" );
document.write( "This was the linear regression equation (aka line of best fit) mentioned in the previous section. <BR>\n" );
document.write( "Follow the steps mentioned in the previous section to get an <font color=red>answer of -24.960</font><BR>\n" );
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