document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206930>Question 1206930</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A political candidate has asked his/her assistant to conduct a poll to determine the percentage of people in the community that supports him/her. If the candidate wants a 5% margin of error at a 90% confidence level, what size of sample is needed? Be sure to round accordingly.\r<BR>\n" );
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document.write( "The candidate would need to survey  people in the community in order to be within a 5% margin of error at a 90% confidence level.<BR>\n" );
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document.write( "Answer: <font color=red size=4>271</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "The margin of error for a proportion is<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)\r<BR>\n" );
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document.write( "Let's isolate n.<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "sqrt(phat*(1-phat)/n) = E/z<BR>\n" );
document.write( "phat*(1-phat)/n = (E/z)^2<BR>\n" );
document.write( "n/( phat*(1-phat) ) = (z/E)^2<BR>\n" );
document.write( "n = phat*(1-phat)*(z/E)^2<BR>\n" );
document.write( "This formula comes up a lot so it's best to either memorize it, or write it down in your notes. The same can be said about the original formula mentioned.\r<BR>\n" );
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document.write( "At 90% confidence, the z critical value is roughly z = 1.645<BR>\n" );
document.write( "This is another thing to memorize or have on a reference sheet somewhere.\r<BR>\n" );
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document.write( "The error we want to have is E = 0.05 to represent being at most 5 percentage points off of the true proportion, ie a \"5% margin of error\".\r<BR>\n" );
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document.write( "We aren't told the value of phat, so the default is to go with phat = 0.5 which is the most conservative estimate. <BR>\n" );
document.write( "It's right at the midpoint of phat = 0 and phat = 1.\r<BR>\n" );
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document.write( "Here are the input values<BR>\n" );
document.write( "z = 1.645 (approximate)<BR>\n" );
document.write( "E = 0.05<BR>\n" );
document.write( "phat = 0.5\r<BR>\n" );
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document.write( "Then we'll have the following<BR>\n" );
document.write( "n = phat*(1-phat)*(z/E)^2<BR>\n" );
document.write( "n = 0.5*(1-0.5)*(1.645/0.05)^2<BR>\n" );
document.write( "n = 270.6025<BR>\n" );
document.write( "n = <font color=red>271</font>\r<BR>\n" );
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document.write( "Always make sure to round UP to the nearest whole number when it comes to minimum sample size problems.<BR>\n" );
document.write( "Refer to part (c) of <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1206824.html\">this question</a> for more info.<BR>\n" );
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