document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190707>Question 1190707</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A medical researcher conducted a study to understand the efficacy of a new medicine. for the study he took a sample of 55 patients and found the margin of error for the estimate to be 12.As he aware that the margin of error is related to the size of sample being considered, if the researcher wants to have more precision in his measurement of efficacy by reducing the margin of error of estimate to 4,what is the sample size he needs to consider? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #822425 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=822425\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If the doctor was trying to estimate the population mean mu, then the margin of error would be<BR>\n" );
document.write( "E = z*s/sqrt(n)<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "E = margin of error<BR>\n" );
document.write( "z = critical z value based on the confidence level<BR>\n" );
document.write( "s = standard deviation<BR>\n" );
document.write( "n = sample size<BR>\n" );
document.write( "You probably have come across this formula when setting up confidence intervals.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the values of z and s aren't mentioned, we'll just hold them to fixed unknown constants.<BR>\n" );
document.write( "It'll turn out that they don't matter.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's solve the mentioned formula for n<BR>\n" );
document.write( "E = z*s/sqrt(n)<BR>\n" );
document.write( "E*sqrt(n) = z*s<BR>\n" );
document.write( "sqrt(n) = z*s/E<BR>\n" );
document.write( "n = (z*s/E)^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's now plug in the first error mentioned 12<BR>\n" );
document.write( "n = (z*s/E)^2<BR>\n" );
document.write( "n = (z*s/12)^2<BR>\n" );
document.write( "n = A/(12^2) ... let A = (z*s)^2<BR>\n" );
document.write( "n = A/144<BR>\n" );
document.write( "This margin of error E = 12 corresponds directly to the sample size n = 55, so,<BR>\n" );
document.write( "n = A/144<BR>\n" );
document.write( "A = 144n<BR>\n" );
document.write( "A = 144*55<BR>\n" );
document.write( "A = 7920\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can then determine what n must be if we want E = 4<BR>\n" );
document.write( "n = (z*s/E)^2<BR>\n" );
document.write( "n = A/(E^2) ... again let A = (z*s)^2<BR>\n" );
document.write( "n = 7920/(4^2)<BR>\n" );
document.write( "n = 495<BR>\n" );
document.write( "The value of A doesn't change because the confidence level doesn't change (which in turn means the z value stays the same). Also, the standard deviation is fixed as well.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: 495<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );