document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200392>Question 1200392</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable preliminary estimation for the population proportion. You would like to be 98% confident that you estimate is within 1.5% of the true population proportion. How large of a sample size is required? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834573 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=834573\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> at 98% two tailed confidence interval, the critical z-score is plus or minus 2.326347877.<BR>\n" );
document.write( "the standard error is equal to the standard deviation divided by the square root of the sample size.<BR>\n" );
document.write( "you want the margin of error to be maximum of .015.<BR>\n" );
document.write( "your critical z-score formula becomes z = .015 / s which becomes 2.326347877 = .015 / s, where s is the standard error.<BR>\n" );
document.write( "solve for s to get s = .015 / 2.326347877 = .0064478749.<BR>\n" );
document.write( "the formula for s is s = sqrt(p * q / n)<BR>\n" );
document.write( "s is maximum when p = .5.<BR>\n" );
document.write( "you get s = sqrt(.5*.5/n)<BR>\n" );
document.write( "since s = .0064478749, then you get .0064478749 = sqrt(.25/n) = sqrt(.25) / sqrt(n).<BR>\n" );
document.write( "solve for sqrt(n) to get sqrt(n) = sqrt(.25) / .0064478749 = 77.54492924.<BR>\n" );
document.write( "solve for n to get n = that squared = 6013.21605.<BR>\n" );
document.write( "when n = that, maximum s will be sqrt(.25/6013.21605) = .0064478749.<BR>\n" );
document.write( "this indicates that the sample size shuld be greater than 6013.21605.<BR>\n" );
document.write( "nearest integer greater than 6013.21605 = 6014.<BR>\n" );
document.write( "regardless of what the mean proportin is, the margin of error should be less tha .015 when the sample size is 6014 or more.<BR>\n" );
document.write( "to test this out, we'll look at some possible mean proportiong to see if this is true.<BR>\n" );
document.write( "at p = .5, you get s = sqrt(.5*.5/6014) = .0064474546.<BR>\n" );
document.write( "you get 2.326347877 = (x-m) / .0064474546<BR>\n" );
document.write( "solve for (x-m) to get (x-m) = 2.326347877 * .0064474546 = .014999 which is less than .015.<BR>\n" );
document.write( "when p = .9, you get s = sqrt(.9*.1)/6014) = .0038604720.<BR>\n" );
document.write( "your z-score formule becomes 2.326347877 = (x-m) / .008604720.<BR>\n" );
document.write( "solve for (x-m) to get (x-m) = .0089994134 which is less than .015.<BR>\n" );
document.write( "you should be guaranteed that the magin of error will be less than .015 when the sample size is 6014 or greater.<BR>\n" );
document.write( "i had to do this many times to make sure i did it correctly.<BR>\n" );
document.write( "i'm reasonably certain that i finally got it right.<BR>\n" );
document.write( "my best guess is a sample size of 6014 or greater will satisfy the requirement that the margin of error is less than .015 regardless of the value of p.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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