document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191258>Question 1191258</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately p = 37%. You would like to be 99% confident that your estimate is within 4% of the true population proportion.\r<BR>\n" );
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document.write( "How large of a sample size is required?<BR>\n" );
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document.write( "The margin of error for the confidence interval of a population proportion p is<BR>\n" );
document.write( "E = z*sqrt(phat(1-phat)/n)<BR>\n" );
document.write( "where z is the critical value, phat is the sample proportion, and n is the sample size.<BR>\n" );
document.write( "The term \"phat\" is often written as \"p hat\" or \"p-hat\", but I'm deciding to use a shorter version for the sake of compactness.<BR>\n" );
document.write( "phat estimates the population proportion p.\r<BR>\n" );
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document.write( "Solving for n gets us<BR>\n" );
document.write( "E = z*sqrt(phat(1-phat)/n)<BR>\n" );
document.write( "E/z = sqrt(phat(1-phat)/n)<BR>\n" );
document.write( "(E/z)^2 = phat(1-phat)/n<BR>\n" );
document.write( "n*(E/z)^2 = phat(1-phat)<BR>\n" );
document.write( "n = phat(1-phat)*(z/E)^2\r<BR>\n" );
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document.write( "Since we estimate p = 0.37, this will be plugged into each phat.<BR>\n" );
document.write( "At 99% confidence, the z critical value is roughly z = 2.576 (use a calculator or table to find this value).<BR>\n" );
document.write( "The desired error we want is E = 0.04\r<BR>\n" );
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document.write( "With all that in mind, we can then compute the minimum sample size n<BR>\n" );
document.write( "n = phat(1-phat)*(z/E)^2<BR>\n" );
document.write( "n = 0.37(1-0.37)*(2.576/0.04)^2<BR>\n" );
document.write( "n = 966.749616<BR>\n" );
document.write( "We round up to n = 967<BR>\n" );
document.write( "We always round up regardless how close or far the decimal value is to the nearest largest integer.<BR>\n" );
document.write( "For example, if we got the result n = 966.00001, we would still round up to n = 967<BR>\n" );
document.write( "This rounding up is to ensure we clear the hurdle needed to form the min sample size.\r<BR>\n" );
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document.write( "Answer: <font color=red>967</font>\r<BR>\n" );
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