document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1188882>Question 1188882</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A doctor at a local hospital is interested in estimating the birth weight of infants. How large a sample must she select if she desires to be 90% confident that her estimate is within 4 ounces of the true mean? Assume that s = 7 ounces based on earlier studies. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820230 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=820230\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i get the following:\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( "when the standard deviation is 7, then the standard error is 7/sqrt(n).\r<BR>\n" );
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document.write( "n is the sample size.\r<BR>\n" );
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document.write( "critical z-score at 90% confidence interval is plus or minus 1.645.\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 error.\r<BR>\n" );
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document.write( "you are looking for (x - m) to be equal to plus or minus 4.\r<BR>\n" );
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document.write( "on the high side, your formula becomes:\r<BR>\n" );
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document.write( "1.645 = 4 / (7 / sqrt(n)).\r<BR>\n" );
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document.write( "multiply both sides of this equation by 7 / sqrt(n) to get:\r<BR>\n" );
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document.write( "1.645 * 7 / sqrt(n) = 4.\r<BR>\n" );
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document.write( "multiply both sides of this equation by sqrt(n) and divide both sides of this equation by 4 to get:\r<BR>\n" );
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document.write( "1.645 * 7 / 4 = sqrt(n)\r<BR>\n" );
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document.write( "solve for sqrt(n) t9 get:\r<BR>\n" );
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document.write( "sqrt(n) = 2.87875.\r<BR>\n" );
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document.write( "solve for standard error to get:\r<BR>\n" );
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document.write( "standard error = 7 / sqrt(n) = 7 / 2.87875 = 2.431610942.\r<BR>\n" );
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document.write( "that's the standard error that should get you a margin of error of plus or minus 4.\r<BR>\n" );
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document.write( "on the high side, (x - m) would be equal to 4.<BR>\n" );
document.write( "on the low side, (x - m) would be equal to -4.\r<BR>\n" );
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document.write( "1.645 * 2.431618942 = 4<BR>\n" );
document.write( "-1.645 * 2.431618942 = -4\r<BR>\n" );
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document.write( "this result was achieved with a sample size of 2.87875 ^ 2 = 8.287201563.\r<BR>\n" );
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document.write( "since a sample size can't be a fraction of a whole number, then you need to go up to the next highest integer to get a minimum sample size of at least 9.\r<BR>\n" );
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document.write( "with a sample size of at least 9, your margin of error will be within plus or minus 4.<BR>\n" );
document.write( "in fact, it will be within less than plus or minus 4.\r<BR>\n" );
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document.write( "with a sample size of 9, standard error = 7 / sqrt(sample size) = 7/sqrt(9) = 7/3 = 2.333333333.\r<BR>\n" );
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document.write( "1.645 = (x - m) / 2.333333333.....<BR>\n" );
document.write( "(x - m) = 1.645 * 2.3333333..... = 3.838333333.....<BR>\n" );
document.write( "-1.645 = (x - m) / 2.3333333.....<BR>\n" );
document.write( "(x - m) = -1.645 * 2.333333..... = -3.83833333.....\r<BR>\n" );
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document.write( "the mean can be anything and you will always get a margin of error within plus or minus 3.83833333..... as long as the sample size is greater than or equal to 9.\r<BR>\n" );
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document.write( "as an example, assume the mean is 100.\r<BR>\n" );
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document.write( "on the high side, 1.645 = (x - 100) / 2.33333..... <BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "X = 1.645 * 2.3333..... + 100 = 103.8383333.....<BR>\n" );
document.write( "on the low side, x = -1.645 * 2.333..... + 100 = 96.16166666.....\r<BR>\n" );
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document.write( "103 - 100 = 3.83833333......<BR>\n" );
document.write( "100 - 96.16166666..... = 3.83833333.....\r<BR>\n" );
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document.write( "note that, every time you increase the sample size, the standard error gets smaller as long as the standard deviation remains the same.\r<BR>\n" );
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document.write( "it has to be recalculated every time the sample size is changed.\r<BR>\n" );
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document.write( "as another example, assume the sample size is 16.<BR>\n" );
document.write( "the standard error becomes 7 / sqrt(16) = 7/4 = 1.75.\r<BR>\n" );
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document.write( "1.645 = (x - m) / 1.75 gets you (x - m) = 1.75 * 1.645 = 2.87875.\r<BR>\n" );
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document.write( "the standard error gets smaller and the margin of error becomes smaller as well.\r<BR>\n" );
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document.write( "the solution to this problem is that the sample size needs to be greater than or equal to 9.<BR>\n" );
document.write( "this allows the margin of error to be less than 4.\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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