document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1187716>Question 1187716</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A study is conducted to estimate the mean completion time of a task to within 3 minutes 6 seconds of the population mean. Suppose completion time is normally distributed with a standard deviation of 9 minutes 3 seconds. What minimum sample size is required at the 90.9% confidence level?\r<BR>\n" );
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document.write( "You must use a critical z value rounded to two (2) decimal places. <BR>\n" );
document.write( "Do not round other intermediate results. Report final answer as an integer.\r<BR>\n" );
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document.write( "Minimum sample size required = </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #819177 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=819177\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the standard deviation is 9 minutes 3 seconds.<BR>\n" );
document.write( "you want the margin of error to be plus or minus 3 minutes 6 seconds.\r<BR>\n" );
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document.write( "it probably makes sense to convert everything to seconds.\r<BR>\n" );
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document.write( "standard deviation of 9 minutes 3 seconds is equal to 543 seconds.<BR>\n" );
document.write( "margin of error of 3 minutes 6 seconds is equal to 186 seconds.\r<BR>\n" );
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document.write( "the formula for the standard error is:\r<BR>\n" );
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document.write( "s = standard deviation / square root of sample size.\r<BR>\n" );
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document.write( "90.9% confidence level gives you a two tail alpha of 100 - 90.9 = 9.1% / 2 = 4.55% on each end of the confidence interval.\r<BR>\n" );
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document.write( "that's equivalent to an area of .0455 on each end of the confidence interval.\r<BR>\n" );
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document.write( "that give you a critical z-score of plus or minus 1.690146137.\r<BR>\n" );
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document.write( "you could probably round that off to 1.69 but we'll keep it at full value from the calculator for now.\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 raw mean<BR>\n" );
document.write( "s is the standard error.\r<BR>\n" );
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document.write( "the standard error is equal to the standard deviation divided by the square root of the sample size.\r<BR>\n" );
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document.write( "since the standard deviation is equal to 543 seconds, then the standard error is equal to 543 / sqrt(sample size).\r<BR>\n" );
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document.write( "if we want the margin of error to be equal to 186 seconds, then (x - m) must be equal to 186.\r<BR>\n" );
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document.write( "the z-score formula becomes:\r<BR>\n" );
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document.write( "1.690146137 = 186 / (543 / sqrt(sample size).\r<BR>\n" );
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document.write( "this is equivalent to:\r<BR>\n" );
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document.write( "1.690146137 = 186 * sqrt(sample size) / 543,\r<BR>\n" );
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document.write( "solve for sqrt(sample size) to get:\r<BR>\n" );
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document.write( "sqrt(sample size) = 1.690146137 * 543 / 186.\r<BR>\n" );
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document.write( "that becomes:\r<BR>\n" );
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document.write( "sqrt(sample size) = 4.934136303.\r<BR>\n" );
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document.write( "square both sides of that equation to get:\r<BR>\n" );
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document.write( "sample size = 24.34570106.\r<BR>\n" );
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document.write( "calculate the standard error using the square root of the sample size to get:\r<BR>\n" );
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document.write( "standard error = 543 / 4.934136303 = 110.0496554.\r<BR>\n" );
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document.write( "with that standard error, you should be able to calculate the margin of error to get what you wanted it to be.\r<BR>\n" );
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document.write( "on the high side, z = (x - m) / s becomes:\r<BR>\n" );
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document.write( "1.690146137 = (x - m) / 110.0496554.\r<BR>\n" );
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document.write( "solve for (x - m) to get:\r<BR>\n" );
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document.write( "(x - m) = 186.\r<BR>\n" );
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document.write( "on the low side, z = (x - m) / s becomes:\r<BR>\n" );
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document.write( "1.690146137 = (x - m) / 110.0496554.\r<BR>\n" );
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document.write( "solve for (x - m) to get:\r<BR>\n" );
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document.write( "(x - m) = -186.\r<BR>\n" );
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document.write( "looks like the margin of error is exactly what you wanted.\r<BR>\n" );
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document.write( "this margin or error should be the same regardless of what the mean is, as ong as the standard error is equal to 110.0496554.\r<BR>\n" );
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document.write( "to see if that's true, pick any mean that looks reasonable and solve for the confidence interval.\r<BR>\n" );
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document.write( "assume the mean is 5000 seconds.\r<BR>\n" );
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document.write( "the critical z-score formula becomes:\r<BR>\n" );
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document.write( "1.690146137 = (x - 5000) / 110.0496554.\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = 1.690146137 * 110.0496554 + 5000 = 5186.\r<BR>\n" );
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document.write( "that's 186 greater than 5000.\r<BR>\n" );
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document.write( "on the low side, the formula becomes:\r<BR>\n" );
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document.write( "-1.690146137 = (x - 5000) / 110.0496554.\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = -1.690146137 * 110.0496554 + 5000 = 4814\r<BR>\n" );
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document.write( "5000 - 4814 = 186, which is what you wanted.\r<BR>\n" );
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document.write( "your solution is that the minimum sample size required is 24.34570106.\r<BR>\n" );
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document.write( "round that the next highest integer to get:\r<BR>\n" );
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document.write( "the minimum sample size required is 25.\r<BR>\n" );
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document.write( "that's what i get.\r<BR>\n" );
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document.write( "let me know if you have any questions.\r<BR>\n" );
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document.write( "theo\r<BR>\n" );
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document.write( " </SPAN>\n" );
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