document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1026627>Question 1026627</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A trucking firm suspects that the mean lifetime of a certain tire it uses is less than 36,000 miles. To check the claim, the firm randomly selects 54 of these tires and gets a mean lifetime of 35,630 miles with a population standards deviation of 1200 miles. At alpha=0.05, test the trucking firms claim.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #641935 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=641935\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> mean to test against is 36,000.<BR>\n" );
document.write( "population standard deviation is 1200.<BR>\n" );
document.write( "sample mean is 35,630.<BR>\n" );
document.write( "alpha = .05.<BR>\n" );
document.write( "sample size = 54.\r<BR>\n" );
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document.write( "z = (x-m)/se\r<BR>\n" );
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document.write( "z is the z-score.<BR>\n" );
document.write( "x is the sample mean.<BR>\n" );
document.write( "m is the mean to test against.<BR>\n" );
document.write( "se is the standard error.\r<BR>\n" );
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document.write( "standard error = standard deviation divided by square root of sample size.\r<BR>\n" );
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document.write( "se = 1200 / sqrt(54) = 163.3.\r<BR>\n" );
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document.write( "z = (35,630 - 36,000) / 163.3 = -370 / 163.3 = -2.27.\r<BR>\n" );
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document.write( "look up z-score in the z-score table to find that the area to the left of that z-score is equal to .0116.\r<BR>\n" );
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document.write( "your threshold alpha is .05.\r<BR>\n" );
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document.write( "the sample alpha is less than that.\r<BR>\n" );
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document.write( "this is an indication that the average life of the tire is actually less than 36,000 miles.\r<BR>\n" );
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document.write( "if the mean life of the tire was 36,000 miles, as claimed, then the likelihood of a sample having a mean life of 35,630 would only happen 1% of the time, on average.\r<BR>\n" );
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document.write( "to be sure, they would look at all the samples they took, and if more than 5% of those samples had a mean life less than the threshold, they would conclude that the tires do indeed have a mean life that is less than 36,000 miles.\r<BR>\n" );
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document.write( "the threshold of 5% can be found by looking at the z-score table and finding the z-score that has an area to the left of it that is .05 or less.\r<BR>\n" );
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document.write( "looking at the z-score table, i see that a z-score of -1.65 is just under the threshold of .05.\r<BR>\n" );
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document.write( "a z-score of -1.65 has an area to the left of it that is equal to .0495.\r<BR>\n" );
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document.write( "to find the raw score that relates to that, use the z-score formula again.\r<BR>\n" );
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document.write( "let x = the raw score.\r<BR>\n" );
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document.write( "z = (x-m) / se\r<BR>\n" );
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document.write( "z = -1.65<BR>\n" );
document.write( "m = 36,000<BR>\n" );
document.write( "se = 163.3\r<BR>\n" );
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document.write( "formula becomes:\r<BR>\n" );
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document.write( "-1.65 = (x - 36,000) / 163.3\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = -1.65 * 163.3 + 36,000.\r<BR>\n" );
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document.write( "you get x = 35,730.555\r<BR>\n" );
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document.write( "based on an alpha of .05, they would expect to get a sample with a mean less than 35,730 miles in less than 5% of the samples that they take.\r<BR>\n" );
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document.write( "since they take regular samples, they should be able to look at all the samples that they took and see how many of them had a mean of 35,730 or less.\r<BR>\n" );
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document.write( "if the percentage of those samples was greater than 5%, then they could reasonably conclude that the average life of the tires was actually less than 36,000 which is contrary to the manufacturer's claim.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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