document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178022>Question 1178022</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm working on this problem: <BR>\n" );
document.write( "The lifetime of a certain car battery is reported to be 154 weeks with a standard deviation of 8 weeks and is normally distributed.  If the company offers a 3-year guarantee, compute the percentage of batteries that will be returned for refund for this battery (52 weeks = 1 year).<BR>\n" );
document.write( "I don't think I did it correctly at all.... do I calculate the P(z<156-154/(8/100^.5)?\r<BR>\n" );
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document.write( "Please help. I think I just need a nudge in the right direction. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #807194 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=807194\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the lifetime of a battery is reported to be 154 weeks with a standard deviation of 8 weeks and is normally distributed.\r<BR>\n" );
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document.write( "that would be the average lifetime.\r<BR>\n" );
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document.write( "the company offers a 3 year guarantee.\r<BR>\n" );
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document.write( "3 years * 52 weeks a year is equal to 156 weeks.\r<BR>\n" );
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document.write( "if the battery dies in less than 156 weeks, then the battery can be returned for a refund.\r<BR>\n" );
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document.write( "you are looking for the probability that the life of the battery will be less than 156 weeks.\r<BR>\n" );
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document.write( "you want to find the z-score.\r<BR>\n" );
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document.write( "z-score formula is z = (x - m) / s\r<BR>\n" );
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document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation in this case.\r<BR>\n" );
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document.write( "with the numbers you have, the formula becomes:\r<BR>\n" );
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document.write( "z = (156 - 154) / 8.\r<BR>\n" );
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document.write( "solve for z to get z = 2/8 = .25.\r<BR>\n" );
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document.write( "find the area to the left of a z-score of .25 is equal to .5987062744.\r<BR>\n" );
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document.write( "based on the data, it is expected that approximately 60% of the batteries will be returned for a refund.\r<BR>\n" );
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document.write( "it looks like you were calculating a standard error rather than just using the standard deviation.\r<BR>\n" );
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document.write( "standard error is used when you have a sample of a certain size.\r<BR>\n" );
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document.write( "you then take the standard deviation and divide it by the square root of the sample size.\r<BR>\n" );
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document.write( "in this problem, you don't have a sample as far as i can see.<BR>\n" );
document.write( "it looks like you are dealing with the population.<BR>\n" );
document.write( "in that case, you just use the standard deviation rather than the standard error.\r<BR>\n" );
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document.write( "your calculation is p(z < (156-154)/(8/sqrt(100)).<BR>\n" );
document.write( "it looks like you think you are dealing with a sample of 100 batteries.<BR>\n" );
document.write( "i didn't see that in the problem statement.<BR>\n" );
document.write( "based on what i saw, your formula should have been p(z < (156-154)/8.\r<BR>\n" );
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document.write( "if in fact, you did have a sample of size 100, then i think your formula is correct.\r<BR>\n" );
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document.write( "the standard error would be 8 / sqrt(100) = 8/10 = .8\r<BR>\n" );
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document.write( "the formula would become z = (156-154)/.8 = 2.5\r<BR>\n" );
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document.write( "the probability of getting less than a z-score of 2.5 would be much higher.<BR>\n" );
document.write( "it would be .99379...<BR>\n" );
document.write( "pretty much all of the batteries would need to be returned for a refund.<BR>\n" );
document.write( "that doesn't sound right.\r<BR>\n" );
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document.write( "if they were looking for a small percentage of refunds to be given, i would think that the average life of the batteries would need to be much higher, or the guarantee would need to be much lower.\r<BR>\n" );
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document.write( "for example, if they were looking for a refund percentage of 5%, with a guarantee of 3 years, then the average life of the battery should have been much higher.\r<BR>\n" );
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document.write( "5% on the left side of the normal distribution curve would need a z-score of -1.645.<BR>\n" );
document.write( "z-score formula is z = (x-m)/s<BR>\n" );
document.write( "using the standard deviation of 8, and an x of 156, the formula would become:<BR>\n" );
document.write( "-1.645 = (156 - m) / 8<BR>\n" );
document.write( "solve for m to get m = 169.16.\r<BR>\n" );
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document.write( "the average life of the battery would have had to be 169.16 if you were dealing with the population.\r<BR>\n" );
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document.write( "in that case, z would be equal to (156 - 169.16) / 8 = -1.645 and the area to the left of that z-score would be equal to .-5, meaning that 5% of the batteries would need to be returned for a refund.\r<BR>\n" );
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document.write( "if you are talking about the average life of a sample of 100, then you would get a different result.\r<BR>\n" );
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document.write( "in that case, you would use the standard error rather than the standard deviation.\r<BR>\n" );
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document.write( "the standard error would be 8/sqrt(100) = 8/10 = .8\r<BR>\n" );
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document.write( "your desired z-score is still -1.645 if you only want to give a refund to 5% of the customers.\r<BR>\n" );
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document.write( "the z-score formula would become -1.645 = (156 - m) / .8<BR>\n" );
document.write( "solve for m to get m = 157.316.\r<BR>\n" );
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document.write( "the average life of the sample of 100 would need to be 157.316 weeks.\r<BR>\n" );
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document.write( "your z-score formula would becomes z = (156 - 157.216) / .8 = -1.645.\r<BR>\n" );
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document.write( "bottom line is you need to check your problem statement again to see if something is missing.\r<BR>\n" );
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document.write( "let me know what you find and what you think your calculations need to be if the problem statement is different than the one i'm seeing.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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