document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202200>Question 1202200</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A political candidate has asked you to conduct a poll to determine what percentage of people support her. If the candidate only wants a 0.5% margin of error at a 97.5% confidence level, what size of sample is needed? When finding the z-value, round it to four decimal places. </SPAN>\n" );
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document.write( "Answer: <font color=red size=4>50239</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "If you are using a TI83 or TI84, then refer to this article<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.statology.org/invnorm-ti-84/\">https://www.statology.org/invnorm-ti-84/</a>\r<BR>\n" );
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document.write( "A 97.5% confidence level means the main body has an area of 0.975<BR>\n" );
document.write( "Each tail has an area of (1-0.975)/2 = 0.0125<BR>\n" );
document.write( "Then use the TI84 to input <font color=\"blue\">invNorm(0.0125,0,1)</font> to get the approximate result of -2.2414; we'll apply absolute value to make it positive.\r<BR>\n" );
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document.write( "The z critical value is roughly z = 2.2414\r<BR>\n" );
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document.write( "If you are using a spreadsheet to find the z value, then the command to use is called <font color=\"blue\">NORMSINV</font><BR>\n" );
document.write( "The more general inverse normal function is called NORMINV<BR>\n" );
document.write( "The S refers to the standard normal<BR>\n" );
document.write( "More info found here<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/topic/excel-statistical-functions-normsinv-3b33f03c-c50b-9d84-5269-0dc85692d349\">https://support.microsoft.com/en-us/topic/excel-statistical-functions-normsinv-3b33f03c-c50b-9d84-5269-0dc85692d349</a><BR>\n" );
document.write( "and here<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/norminv-function-87981ab8-2de0-4cb0-b1aa-e21d4cb879b8\">https://support.microsoft.com/en-us/office/norminv-function-87981ab8-2de0-4cb0-b1aa-e21d4cb879b8</a><BR>\n" );
document.write( "Each article is about excel, but those functions apply to other spreadsheet programs as well.\r<BR>\n" );
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document.write( "The input <font color=\"blue\">=NORMSINV((1-0.975)/2)</font> will give the z value we're after.<BR>\n" );
document.write( "So will <font color=\"blue\">=NORMSINV(0.0125)</font><BR>\n" );
document.write( "Don't forget about the equal sign up front when executing commands in a spreadsheet.\r<BR>\n" );
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document.write( "The margin of error is 0.5% meaning we have E = 0.005\r<BR>\n" );
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document.write( "The political candidate wants to know the value of n to make this confidence interval<BR>\n" );
document.write( "phat - E < p < phat + E<BR>\n" );
document.write( "phat - 0.005 < p < phat + 0.005\r<BR>\n" );
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document.write( "We're not told phat so we assume a conservative estimate of phat = 0.5 which is right in the middle of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=p%3C=1 ALIGN=MIDDLE  ALT=\"0%3C=p%3C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The formula for the margin of error of a proportion is<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)\r<BR>\n" );
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document.write( "Solving for n gets this formula<BR>\n" );
document.write( "n = phat*(1-phat)*(z/E)^2<BR>\n" );
document.write( "which will determine the min sample size needed\r<BR>\n" );
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document.write( "The inputs are<BR>\n" );
document.write( "phat = 0.5<BR>\n" );
document.write( "z = 2.2414<BR>\n" );
document.write( "E = 0.005\r<BR>\n" );
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document.write( "n = phat*(1-phat)*(z/E)^2<BR>\n" );
document.write( "n = 0.5*(1-0.5)*(2.2414/0.005)^2<BR>\n" );
document.write( "n = 50238.7396<BR>\n" );
document.write( "n = <font color=red>50239</font><BR>\n" );
document.write( "For problems like this, always round UP to the nearest whole number.<BR>\n" );
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