document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137371>Question 1137371</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi! I'm struggling with one of my homework questions, perhaps because of how it's worded. Could someone help me out? \r<BR>\n" );
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document.write( "A manufacturer knows that their items have a normally distributed lifespan, with a mean of 3.1 years, and standard deviation of 0.6 years.\r<BR>\n" );
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document.write( "If 24 items are picked at random, 3% of the time their mean life will be less than how many years?\r<BR>\n" );
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document.write( "Give your answer to one decimal place.\r<BR>\n" );
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document.write( "Thanks in advance!<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #755258 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=755258\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the population mean is 3.1 years and the population standard deviation is .6 years.\r<BR>\n" );
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document.write( "your sample size is 24.\r<BR>\n" );
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document.write( "your standard error is therefore population standard deviation / square root of sample size = .6 / sqrt(24) = .1224744871.\r<BR>\n" );
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document.write( "you are looking to find the the z-score where the area to the left of it is 3% or .03.\r<BR>\n" );
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document.write( "you can use a z-score calculator to determine that the z-score associated with an area of .03 to the left of it is equal to -1.88079361.\r<BR>\n" );
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document.write( "you will want to find the raw score associated with that z-score.\r<BR>\n" );
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document.write( "the 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 error.\r<BR>\n" );
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document.write( "the formula becomes:\r<BR>\n" );
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document.write( "-1.88079361 = (x - 3.1) / .1224744871\r<BR>\n" );
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document.write( "solve for x to get x = .1224744871 * -1.88079361 + 3.1 = 2.869650767 years.\r<BR>\n" );
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document.write( "this can be seen visually using the following online normal distribution calculator.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">http://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
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document.write( "here's what it looks like.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/032601.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "what this is showing you is that, 3% of the time, the mean life of the sample of size 24 will be less than 2.869650767 years.\r<BR>\n" );
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document.write( "the calculator limits the number of digits that can be entered into it, therefore some discrepancies with my more detailed values will be noted.\r<BR>\n" );
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document.write( "i rounded my values to the maximum number of decimal digits allowed by the online calculator.\r<BR>\n" );
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document.write( "the answer wasn't affected by this rounding.\r<BR>\n" );
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document.write( "it said 3% as well.\r<BR>\n" );
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document.write( "the interpretation that you can take from this is that, if you took a very large number of samples of size 24, approximately 3% of the time the mean of the sample will be less than 2.869650767\r<BR>\n" );
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document.write( "the rest of the time,it will be more than 2.869650767 years.\r<BR>\n" );
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