document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192241>Question 1192241</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A researcher wishes to estimate the proportion of college students who own a car. He wants to be 95% confident and be accurate within 8% of the true proportion. find the minimum sample necessary. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824168 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824168\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "In your stats class, you should have come across the margin of error for a confidence interval involving proportions.\r<BR>\n" );
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document.write( "That particular margin of error formula is:<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)\r<BR>\n" );
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document.write( "Let's solve for 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\r<BR>\n" );
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document.write( "At 95% confidence, the z critical value is about z = 1.96 which is determined through a Z table or a calculator.\r<BR>\n" );
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document.write( "Unfortunately we don't know what phat is, so we'll have to go with the conservative estimate of phat = 0.5\r<BR>\n" );
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document.write( "The desired error we want is E = 0.08, since we want to be within 8% of the true proportion p.<BR>\n" );
document.write( "The phrasing \"within\" is the same as saying \"at most\". So we want the error to be at most 0.08, ie we want <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E+%3C=+0.08 ALIGN=MIDDLE  ALT=\"E+%3C=+0.08\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that E is positive.\r<BR>\n" );
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document.write( "Plug in those items mentioned<BR>\n" );
document.write( "n = phat*(1-phat)(z/E)^2<BR>\n" );
document.write( "n = 0.5*(1-0.5)(1.96/0.08)^2<BR>\n" );
document.write( "n = 150.0625\r<BR>\n" );
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document.write( "Despite this value of n being very close to 150, we <b><font color=blue>always round up</font></b> when it comes to minimum sample size problems like this. <BR>\n" );
document.write( "Why up? Because it ensures we clear the hurdle. <BR>\n" );
document.write( "The larger n gets, the smaller E gets.<BR>\n" );
document.write( "Intuitively this fits with the idea that a larger sample is more representative; hence it would more accurately capture the parameter we're after (leading to reduced error).\r<BR>\n" );
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document.write( "Let's say we go with n = 150<BR>\n" );
document.write( "Calculating the margin of error gets us<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 1.96*sqrt(0.5*(1-0.5)/150)<BR>\n" );
document.write( "E = 0.08001666493091<BR>\n" );
document.write( "This error is a bit too high, as it is larger than 0.08<BR>\n" );
document.write( "Though you could argue that rounding it to two decimal places gets us E = 0.08 just fine.\r<BR>\n" );
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document.write( "Standard practice is to round up to ensure E is under 0.08<BR>\n" );
document.write( "So if we go for n = 151 then we have<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 1.96*sqrt(0.5*(1-0.5)/151)<BR>\n" );
document.write( "E = 0.07975126895958<BR>\n" );
document.write( "We've cleared the hurdle and E is now smaller than 0.08\r<BR>\n" );
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document.write( "Answer: Min sample size = <font color=red>151</font><BR>\n" );
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