document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1011308>Question 1011308</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Natalie performs a chemistry experiment where she records the temperature of an ongoing reaction.  The solution is 93.5 degree C after 3 min.; 90 degree C after 5 min.;84.8 degree C after 9 min.;70.2 degree C after 18 min.; 54.4 degree after 30 min.; 42.5 degree C after 37 min.; and 24.9 degree C after 48 min. Perform a linear regression on this data to complete the following\r<BR>\n" );
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document.write( "what does the value of the correlation coefficient tell you about the correlation of the data?\r<BR>\n" );
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document.write( "write the equation of the best fitting line.\r<BR>\n" );
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document.write( "on average how much does the temperature decrease every 5 min.\r<BR>\n" );
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document.write( "If Natalie's solution is expected to freeze at -7 degrees C, how many minutes into the experiment should the solution freeze?\r<BR>\n" );
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document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #627269 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=627269\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What is expected as \"Perform a linear regression on this data\" depends on your course/book/teacher.<BR>\n" );
document.write( "Before calculators (yes, I am that old),<BR>\n" );
document.write( "we would do a lot of pencil and paper calculations,<BR>\n" );
document.write( "using complicated formulas.<BR>\n" );
document.write( "Maybe you are expected to use the functions in your calculator.<BR>\n" );
document.write( "Those function uses formulas you do not need to remember,<BR>\n" );
document.write( "to calculate results such as<BR>\n" );
document.write( "correlation coefficient,<BR>\n" );
document.write( "slope, and<BR>\n" );
document.write( "y-intercept<BR>\n" );
document.write( "of the linear regression best fit line.<BR>\n" );
document.write( "Since I am sitting in front of my computer,<BR>\n" );
document.write( "I would use the spreadsheet program I have installed (Microsoft Excel)<BR>\n" );
document.write( "to tabulate, graph, and calculate the needed results.<BR>\n" );
document.write( " <BR>\n" );
document.write( "My tabulation, graph and results look like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C8%2C%0D%0Aminutes%2C3%2C5%2C9%2C18%2C30%2C37%2C48%2C%0D%0Adegrees%2C93.5%2C90%2C84.8%2C70.2%2C54.4%2C42.5%2C24.9%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C360%2C-20%2C80%2C-10%2C110%2Cgrid%280%29%2C%0D%0Acircle%283%2C93.5%2C1%29%2Ccircle%285%2C90%2C1%29%2C%0D%0Acircle%289%2C84.8%2C1%29%2Ccircle%2818%2C70.2%2C1%29%2C%0D%0Acircle%2830%2C54.4%2C1%29%2Ccircle%2837%2C42.5%2C1%29%2C%0D%0Acircle%2848%2C24.9%2C1%29%2Cgreen%28line%280%2C98.02%2C68%2C-4.36%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "correlation coefficient: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=-0.9995 ALIGN=MIDDLE  ALT=\"r=-0.9995\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded to the first decimal place that is not a 9),<BR>\n" );
document.write( "slope: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-1.50556 ALIGN=MIDDLE  ALT=\"m=-1.50556\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded to 5 decimal places),<BR>\n" );
document.write( "y-intercept: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=98.0192 ALIGN=MIDDLE  ALT=\"b=98.0192\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded to 4 decimal places),<BR>\n" );
document.write( "best fit line (y=mx+b): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=temperature=-1.50556%2Aminutes%2B98.0192 ALIGN=MIDDLE  ALT=\"temperature=-1.50556%2Aminutes%2B98.0192\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "The absolute value of the correlation coefficient is very close to exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that means that the line fits the data very closely (as we see in the graph),<BR>\n" );
document.write( "meaning that the linear model is right (that the temperature varies linearly with time).<BR>\n" );
document.write( "The fact that the correlation coefficient is negative<BR>\n" );
document.write( "shows that temperature decreases with time. <BR>\n" );
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document.write( "The slope tells us that the temperature decreases <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.50556 ALIGN=MIDDLE  ALT=\"1.50556\"   DISABLED_style= \"max-width:90%; height:auto;\" > degrees per minute,<BR>\n" );
document.write( "so on average, the temperature decreases <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A1.50556=about ALIGN=MIDDLE  ALT=\"5%2A1.50556=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%287.5%29 ALIGN=MIDDLE  ALT=\"highlight%287.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > degrees (rounded) every <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes.<BR>\n" );
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document.write( "The data allows us to predict temperature between 3 and 48 minutes,<BR>\n" );
document.write( "but we may feel somewhat confident that the relationship still holds for<BR>\n" );
document.write( "values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (seconds) close to the edges of the [3,48] interval.<BR>\n" );
document.write( "That is why I extended the best fit line in the graph to cover [0,68].<BR>\n" );
document.write( "The line seems to cross the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > x-axis at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=65 ALIGN=MIDDLE  ALT=\"x=65\"   DISABLED_style= \"max-width:90%; height:auto;\" > (65 minutes).<BR>\n" );
document.write( "We can solve the linear best fit equation for the time when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=temperature=-7degrees ALIGN=MIDDLE  ALT=\"temperature=-7degrees\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the solution freezes.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28temperature=-7%2Ctemperature=-1.50556%2Aminutes%2B98.0192%29 ALIGN=MIDDLE  ALT=\"system%28temperature=-7%2Ctemperature=-1.50556%2Aminutes%2B98.0192%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28temperature=-7%2C1.50556%2Aminutes=98.0192%2B7%29 ALIGN=MIDDLE  ALT=\"system%28temperature=-7%2C1.50556%2Aminutes=98.0192%2B7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28temperature=-7%2C1.50556%2Aminutes=105.0192%29 ALIGN=MIDDLE  ALT=\"system%28temperature=-7%2C1.50556%2Aminutes=105.0192%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28temperature=-7%2Ctemperature=105.0192%2F1.50556=65.1%29 ALIGN=MIDDLE  ALT=\"system%28temperature=-7%2Ctemperature=105.0192%2F1.50556=65.1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded).<BR>\n" );
document.write( "Since Natalie was measuring the time in whole minutes,<BR>\n" );
document.write( "we can report the time the solution freezes with the same precision as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2865minutes%29 ALIGN=MIDDLE  ALT=\"highlight%2865minutes%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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