document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1026699>Question 1026699</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 #642141 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=642141\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> mean is 25.<BR>\n" );
document.write( "standard deviation is 2.5.\r<BR>\n" );
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document.write( "use z-score to normalize.\r<BR>\n" );
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document.write( "z = (x-m)/s\r<BR>\n" );
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document.write( "x is the raw score.<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation.\r<BR>\n" );
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document.write( "x = 22<BR>\n" );
document.write( "m = 25<BR>\n" );
document.write( "s = 2.5<BR>\n" );
document.write( "z = (22 - 25) / 2.5 = -1.2\r<BR>\n" );
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document.write( "x = 26<BR>\n" );
document.write( "m = 25<BR>\n" );
document.write( "s = 2.5<BR>\n" );
document.write( "z = (26-25)/2.5 = 1/2.5 = .4\r<BR>\n" );
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document.write( "look in the z-score table for area under the distribution curve to the left of those z-scores.\r<BR>\n" );
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document.write( "area to the left for z-score of -1.2 = .0548<BR>\n" );
document.write( "area to the left for z-score of .4 = .6554\r<BR>\n" );
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document.write( "area in between = .6554 - .0548 = .6006\r<BR>\n" );
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document.write( ".6006 of mufflers take between 22 and 26 minutes to replace.\r<BR>\n" );
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document.write( "in order to have only 1% of the customers get a rebate, look in the z-score table for the z-score that has 99% of the area under the normal distribution curve to the left of it.\r<BR>\n" );
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document.write( "that area will have a z-score of 2.33 which has .9901 area to the left of it.\r<BR>\n" );
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document.write( "that's just a little bit less than .01 to the right of it, because 1 - .9901 = .0099 which is a touch under .01.\r<BR>\n" );
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document.write( "to find the raw score associated with that, use the z-score formula again.\r<BR>\n" );
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document.write( "z = 2.33<BR>\n" );
document.write( "x = what you want to find.<BR>\n" );
document.write( "m = 25<BR>\n" );
document.write( "s = 2.5\r<BR>\n" );
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document.write( "z = (x-m)/s becomes 2.33 = (x-25)/2.5\r<BR>\n" );
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document.write( "solve for x to get x = 2.33 * 2.5 + 25 which is equal to 30.825.\r<BR>\n" );
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document.write( "customers would get a rebate if it takes more than 30.825 minutes.\r<BR>\n" );
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document.write( "that would be approximately 30 minutes and 50 seconds.\r<BR>\n" );
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document.write( "that's pretty close to 31, so they would probably round that off to 31 minutes.\r<BR>\n" );
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document.write( "at the current existing rate, you would calculate the proportion using the z-score formula again.\r<BR>\n" );
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document.write( "m = 25<BR>\n" );
document.write( "s = 2.5<BR>\n" );
document.write( "x = 30\r<BR>\n" );
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document.write( "z = (30 - 25) / 2.5 = 2.\r<BR>\n" );
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document.write( "with a z-score of 2, the area to the left of the z-score is .9772, and so the area to the right side of the z-score is 1 - .9772 = .0228.\r<BR>\n" );
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document.write( "that's roughly 2.3% that get the rebate.\r<BR>\n" );
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document.write( "that's the percent that's currently getting the rebate.\r<BR>\n" );
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document.write( "increase to 31 minutes and z=(x-m)/2.5 becomes z = 2.4 which gets an area to the left of that z-score equal to .9918 which gets an area of 1 - .9918 = .0082 to the right of it.\r<BR>\n" );
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document.write( "that's .8% who would then get the rebate.\r<BR>\n" );
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document.write( "the z-score table i used can be found here:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf\" target = \"_blank\">http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf</a>\r<BR>\n" );
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