document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1188020>Question 1188020</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A federal report indicated that 27% of children ages 2 to 5 years had a good diet—an increase<BR>\n" );
document.write( "over previous years. How large a sample is needed to estimate the true proportion of children with good diets within 2% with 95% confidence?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #819085 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=819085\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> p = .27<BR>\n" );
document.write( "q = .73<BR>\n" );
document.write( "s = sqrt(.27 * .73 / n)\r<BR>\n" );
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document.write( "p is the mean proportion.<BR>\n" );
document.write( "q is  1 minus the mean proportion.<BR>\n" );
document.write( "i think that s is the standard error of the mean proportion.<BR>\n" );
document.write( "that's because it is used as the standard error in the calculation, even though they call it the standard deviation.<BR>\n" );
document.write( "very confusing, but it works.\r<BR>\n" );
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document.write( "at 95% two tailed confidence interval, the critical alpha is .025 on each end.<BR>\n" );
document.write( "the critical z-score becomes plus or minus 1.96.\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 - m is the sample mean proportion minus the population mean proportion.<BR>\n" );
document.write( "s is the standard error.\r<BR>\n" );
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document.write( "since we want the margin of error to be plus or minus 2% of the mean, then (x - m) must be equal to .02 * the mean proportion.\r<BR>\n" );
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document.write( ".02 * .27 = .0054.\r<BR>\n" );
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document.write( "the z-score formula becomes:\r<BR>\n" );
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document.write( "1.96 = .0054 / sqrt(.27 * .73 / n)\r<BR>\n" );
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document.write( "multiply both sides of this equation by sqrt(.27 * .73 / n) and divide both sides of this equation by 1.96 to get:\r<BR>\n" );
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document.write( "sqrt(.27 * .73 / n) = .0054 / 1.96\r<BR>\n" );
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document.write( "square both sides of this equation to get:\r<BR>\n" );
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document.write( ".27 * .73 / n = (.0054 / 1.96) ^ 2\r<BR>\n" );
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document.write( "multiply both sides of this equation by n and divide both sides of this equation by .(.0054 / 1.96) ^ 2 to get:\r<BR>\n" );
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document.write( ".27 * .73 / (.0054 / 1.96) ^ 2 = n\r<BR>\n" );
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document.write( "solve for n to get:\r<BR>\n" );
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document.write( "n = 25966.37037.\r<BR>\n" );
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document.write( "that is the sample size required for the margin of error to be plus or minus 2% of the mean.\r<BR>\n" );
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document.write( "with that sample size, the standard error becomes sqrt(.27 * .73 / that) = .002755102.\r<BR>\n" );
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document.write( "with a mean of .27 and that standard error and a critical z-score of 1.96, the z-score formula becomes:\r<BR>\n" );
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document.write( "1.96 = (x - .27) / .002755102.\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = 1.96 * .002755102 + .27 = .2754.\r<BR>\n" );
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document.write( "the difference between .2754 and .27 is .0054.\r<BR>\n" );
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document.write( ".0054 / .27 = .02, as predicted with that sample size.\r<BR>\n" );
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document.write( "that's on the high side of the confidence interval.\r<BR>\n" );
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document.write( "on the low side of the confidence interval, the z-score formula becomes:\r<BR>\n" );
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document.write( "-1.96 = (x - .27) / .002755102.\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = -1.96 * .002755102 + .27 = .2646.\r<BR>\n" );
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document.write( "the difference between .27 and .2646 is .0054 again, which,  when divided by .0025 = .02.\r<BR>\n" );
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document.write( "if you multiply the population mean proportion by the sample size and you multiply the standard error by the sample size, your formula of:\r<BR>\n" );
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document.write( "p = .27<BR>\n" );
document.write( "q = 1 - .27 = .73<BR>\n" );
document.write( "s = sqrt(p * q / n) becomes:\r<BR>\n" );
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document.write( "m = .27 * n<BR>\n" );
document.write( "s = sqrt(p * q * n)\r<BR>\n" );
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document.write( "n = 25966.37037, same as we calculated above.\r<BR>\n" );
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document.write( "m = .27 * n = 7010.92<BR>\n" );
document.write( "s = sqrt(.27 * .73 * 25966.37037) = 71.54.\r<BR>\n" );
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document.write( "on the high side of the confidence interval, the critical z-score formula is:\r<BR>\n" );
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document.write( "1.96 = (x - 7010.92) / 71.54\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = 1.96 * 71.54 + 7010.92 = 7151.1384\r<BR>\n" );
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document.write( "7151.1384 minus 7010.92 = 140.2184 / 7010.92 = .02 which is off the mean by 2% of the mean.\r<BR>\n" );
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document.write( "on the low side of the confidence interval, the critical z-score formula is:\r<BR>\n" );
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document.write( "-1.96 = (x - 7010.92) / 71.54.\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = -1.96 * 71.54 + 7010.92 = 6870.7016\r<BR>\n" );
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document.write( "7010.92 minus 6870.7016 = 140.2184 / 7010.92 = .02 which is off the mean by 2% of the mean.\r<BR>\n" );
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document.write( "the formulas works.\r<BR>\n" );
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document.write( "your solution is that the sample size needs to be at least equal to 25966.37037 so that the margin of error will be at most 2% of the mean.\r<BR>\n" );
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document.write( "a sample size larger than that will give you a margin of error less than 2% of the mean.\r<BR>\n" );
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document.write( "for example, if the sample size is 30,000, then:\r<BR>\n" );
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document.write( "s = sqrt(.27 * .73 * 30000) = 76.896 rounded to 3 decimal places.<BR>\n" );
document.write( "m = .27 * 30,000 = 8100.<BR>\n" );
document.write( "the high side value of x will be 76.896 * 1.96 + 8100 = 8250.716 rounded to 3 decimal places.<BR>\n" );
document.write( "8250.716 minus 8100 = 150.716 rounded to 3 decimal places<BR>\n" );
document.write( "150.716 / 8100 = .0186 which rounds to .019 which is less than .02.<BR>\n" );
document.write( "the margin of error is less than 2% of the mean in this case, because the sample size was larger than 25,966.37037.\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 </SPAN>\n" );
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