document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1026798>Question 1026798</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A muffler company advertises that you will receive a rebate if it takes longer than 30 minutes to replace a muffler. Experience has shown that the time taken to replace a muffler is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2.5 minutes.<BR>\n" );
document.write( "a) What proportion of mufflers take between 22 and 26 minutes to replace?<BR>\n" );
document.write( "b) What should be the rebate time of 30 minutes be changed to if the company wishes to provide only 1% of customers with a rebate? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642170 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=642170\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "A muffler company advertises that you will receive a rebate if it takes longer than 30 minutes to replace a muffler. Experience has shown that the time taken to replace a muffler is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2.5 minutes.<BR>\n" );
document.write( "a) What proportion of mufflers take between 22 and 26 minutes to replace?<BR>\n" );
document.write( "b) What should be the rebate time of 30 minutes be changed to if the company wishes to provide only 1% of customers with a rebate?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( " <BR>\n" );
document.write( "You will need a normal distribution table, or a calculator or software that provides the same information.  Also learn how to use the table, most of which gives the left-tail.  (check: P(Z=1)=0.8413447)<BR>\n" );
document.write( " <BR>\n" );
document.write( "Assuming (as given) normal distribution and no distinction between mechanics, then we have<BR>\n" );
document.write( "&mu;=30 minutes, and <BR>\n" );
document.write( "&sigma;=2.5 minutes.<BR>\n" );
document.write( "recall that<BR>\n" );
document.write( "Z=(X-&mu;)/&sigma;\r<BR>\n" );
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document.write( "(a) proportion of mufflers that take 22 and 26 minutes to replace: <BR>\n" );
document.write( "P(22&le;x&le;26)<BR>\n" );
document.write( "=P((22-25)/2.5&le;Z&le;(26-25)/2.5)<BR>\n" );
document.write( "=P(-1.2&le;Z&le;0.4)<BR>\n" );
document.write( "=P(0.4)-P(-1.2)<BR>\n" );
document.write( "=0.6554-0.1151<BR>\n" );
document.write( "=0.5404<BR>\n" );
document.write( " <BR>\n" );
document.write( "(b) New rebate time:<BR>\n" );
document.write( "Here we need to find Z such that<BR>\n" );
document.write( "P(z&le;Z)=0.99<BR>\n" );
document.write( "From tables, P(z<2.326348)=0.99, therefore <BR>\n" );
document.write( "Z=2.325348=(x-25)/2.5<BR>\n" );
document.write( "x=30.81587, or 30 minutes and 49 seconds.<BR>\n" );
document.write( "Answer, the rebate time limit should be changed to 30 minutes and 49 seconds.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );