document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=540988>Question 540988</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this problem:<BR>\n" );
document.write( "Write an equation for the following points<BR>\n" );
document.write( "x      y<BR>\n" );
document.write( "80    160<BR>\n" );
document.write( "84    181<BR>\n" );
document.write( "68    121<BR>\n" );
document.write( "81    166<BR>\n" );
document.write( "91    200 <BR>\n" );
document.write( "94    227<BR>\n" );
document.write( "75    140<BR>\n" );
document.write( "88    189 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #354061 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=354061\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The answer depends on what you need the equation for.<BR>\n" );
document.write( "Early in high school, my son was asked to plot a line that looked good for the points (based on just what his eyes were telling him). Late in high school he was asked o pick the best fit for the data, using linear regression.<BR>\n" );
document.write( "The data looks linear, with some scatter. I do not see a reason to suspect that the variation of y with x is anything but linear. However, the data does not fit the line well enough for my usual purposes (and I like to see a wider range of x values). It could be that the measurement procedure leads to some scatter. It could be analyst error or sloppy technique. It could be something we are not controlling, like degradation of the compound. It could be any of a bunch of unexpected phenomena that I have observed before, or any of a bunch of unexpected phenomena that I have not observed before yet.<BR>\n" );
document.write( "I work as an analytical chemist. If you came out of the lab and showed me that data, I would tabulate it and plot it using Excel spreadsheet software. In fact I did, and got <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2=0.975 ALIGN=MIDDLE  ALT=\"r%5E2=0.975\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the linear fit. I would want to know how the data was generated and what it will be used for. I would be asking a lot of questions to figure out if there were errors or artifacts. If it was some non-critical measurement, like verifying that the concentration of some flavoring in a solution is within 10% of the listed value, for information only, I'd let it pass. The expression I got for the linear fit was y=3.9874x-149.03.<BR>\n" );
document.write( "That said, some statistician may say that the data fits a different type of curve better, and fill pages with calculations and conclusions. There is a danger in trying to draw too many conclusions from comparatively little data. With a polynomial of degree 6, you are guaranteed to be able to fit any 7 points with some curve, just like you are guaranteed a perfect linear fit with just two points. Unfortunately, the curve may wave and weave between the points, making no sense. That's what happened when I asked Excel to fit your 8 points into a degree 6 polynomial. The degree 5 one looked more reasonable, but I still think it's abusing statistics. Both polynomials gave <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2=1 ALIGN=MIDDLE  ALT=\"r%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. As statistically expected, when the degree and complexity of the model equation decreased, so did the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2 ALIGN=MIDDLE  ALT=\"r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to 0.996, 0.985, and 0.984 for degree 4, 3, and 2 polynomials, and 0.987 and 0.986 for exponential and power functions.<BR>\n" );
document.write( "I do not remember the formulas for linear regression anymore. Spreadsheet software has been doing regression for me for the last 30 years, and even shows me the formulas in the help menu. </SPAN>\n" );
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