document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204120>Question 1204120</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The College of Podiatrists states that  48% of women wear shoes that are too small for their feet. A researcher wants to be  98% confident that this proportion is within  0.05 of the true proportion. How large a sample is necessary?\r<BR>\n" );
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document.write( "Which answer is correct?\r<BR>\n" );
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document.write( "718, 413, 540, or 286? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840167 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=840167\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> population mean proportion is .48.<BR>\n" );
document.write( "98% confidence interval with two tails requires critical z-score of plus or minus 2.326347.<BR>\n" );
document.write( "margin of error is plus or minus .05.<BR>\n" );
document.write( "z-score formula is z = (x-m)/s<BR>\n" );
document.write( "high side z-score = 2.326347.<BR>\n" );
document.write( "(x-m) is equal to .05 which is the high side margin of error.<BR>\n" );
document.write( "s = standard error = sqrt(.48 * .52 / n) = sqrt(.2496/n)<BR>\n" );
document.write( "n is the sample size.<BR>\n" );
document.write( "z-score formula becomes 2.326347 = .05 / sqrt(.2496/n)<BR>\n" );
document.write( "mutiply both sides of the equation by sqrt(.2496/n) and divide both sides of the formula by 2.326347 to get:<BR>\n" );
document.write( "sqrt(.2496/n) = .05 / 2.326347<BR>\n" );
document.write( "square both sides of the equation to get:<BR>\n" );
document.write( ".2496/n = (.05/2.326347)^2<BR>\n" );
document.write( "solve for n to get:<BR>\n" );
document.write( "n = .2496 / (.05/2.326347)^2 = 540.323134.<BR>\n" );
document.write( "standard error becomes equal to sqrt(.2496/540.323134) = .021493.<BR>\n" );
document.write( "z-score formula becomees:<BR>\n" );
document.write( "2.326347 = (x-m) / .021493.<BR>\n" );
document.write( "solve for (x-m) to get:<BR>\n" );
document.write( "(x-m) = 2.326347 * .021493 = .050000.<BR>\n" );
document.write( "that's your margin of error.\r<BR>\n" );
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document.write( "use the calculator at <a  rel=nofollow HREF= \"https://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">https://davidmlane.com/hyperstat/z_table.html</a> to test it out.\r<BR>\n" );
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document.write( "calculator says that 98% confidence interval is between .43 and .53.<BR>\n" );
document.write( ".48 minus .43 = .05<BR>\n" );
document.write( ".53 minus .48 = .05<BR>\n" );
document.write( "margin of error is .05 as desired.\r<BR>\n" );
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document.write( "here are the results from that calculator.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2023/100831.jpg\">\r<BR>\n" );
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document.write( "your sample size needs to be an integer, so round 540.323134 to 541 and that's your solution.<BR>\n" );
document.write( "you will get a margin of error slightly less than .05 when you do that.<BR>\n" );
document.write( "your revised standared error will be equal to sqrt(.2496/541) = .021479.<BR>\n" );
document.write( "your margin of error becomes .021479 * 2.326347 = .049970 which is slightly less than .05 = within .05, as required.<BR>\n" );
document.write( " </SPAN>\n" );
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