document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=360241>Question 360241</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A local medical research association proposes to sponsor a footrace. The average time it takes to run the course is 45.8 minutes with a standard deviation of 3.6 minutes. If the association decides to include only the top 25% of the racers, what should be the cut off time in the tryout run? Assume the variable is normally distributed. Would a person who runs the course in 40 minutes qualify? <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #257120 by </B><A HREF=/tutors/aboutme.mpl?userid=neatmath><B>neatmath(302)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=neatmath><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=neatmath><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=257120\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In this case, we only need to find the running time that corresponds with the LOWER 25% z-score (since the lower times are the better times)!  Tricky, tricky! <br>\r<BR>\n" );
document.write( "\n" );
document.write( "Remember, the z-score is simply (x-xbar)/s where xbar is the mean, x is our value of interest, and s is the standard deviation.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "So first we need to find the z-score that corresponds with the lower 25% value.  We can do this in several ways, by consulting z-score tables or z-score calculators.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "My z-score calculator tells me we are looking for the z-score of about -0.67 to capture 25% of values to the left and 75% of values to the right.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "Then we just need to convert this z-score back into a running time, like this:<br>\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-45.8%29%2F3.6=-0.67 ALIGN=MIDDLE  ALT=\"%28x-45.8%29%2F3.6=-0.67\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-45.8%29=-0.67%2A3.6 ALIGN=MIDDLE  ALT=\"%28x-45.8%29=-0.67%2A3.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-45.8%29=-2.412 ALIGN=MIDDLE  ALT=\"%28x-45.8%29=-2.412\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=45.8-2.412 ALIGN=MIDDLE  ALT=\"x=45.8-2.412\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=43.388 ALIGN=MIDDLE  ALT=\"x=43.388\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore our cutoff time should be about 43.388 minutes.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "Notice that even though 43.388 is less than the mean, it still reflects the point where the upper 25% of the values end.  This is kind of tricky, because usually the higher numbers are associated with the \"upper\" values.<br>\r<BR>\n" );
document.write( "\n" );
document.write( "It should be immediately obvious that a runner who runs the course in 40 minutes (which is LESS than the 43.388 cutoff time) on the tryout run should qualify for the race!<br>\r<BR>\n" );
document.write( "\n" );
document.write( "I hope this helps!<br> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );