document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197734>Question 1197734</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The financial aid officer at a South African university wishes to estimate the mean cost of textbooks per semester for students. For the estimate to be useful it should be within R{13} of the true population mean. How large a sample should be used in order to be 95% confident of achieving this level of accuracy if we know the population standard deviation is R{92}? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831432 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=831432\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> population standard deviation is 92.<BR>\n" );
document.write( "you want to get a margin of error of 13.<BR>\n" );
document.write( "standard error = standard deviation / square root of sample size.<BR>\n" );
document.write( "s = 92 / sqrt(n)<BR>\n" );
document.write( "n is the sample size<BR>\n" );
document.write( "you want margin of error to be equal to 13.<BR>\n" );
document.write( "z-score formula is z = (x - m) / s<BR>\n" );
document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard error.<BR>\n" );
document.write( "you want (x - m) to be equal to plus or minus 13.<BR>\n" );
document.write( "because the normal distribution is symmetric about the mean, you only need to find one side and will automatically get the other side with a change of sign.<BR>\n" );
document.write( "at 95% confidence interval, the critical z-score on the high side of the confidence interval is equal to z = 1.96.<BR>\n" );
document.write( "z-score formula becomes:<BR>\n" );
document.write( "1.96 = 13 / (92 / sqrt(n))<BR>\n" );
document.write( "multiply both sides of this equation by (92 / sqrt(n)) to get:<BR>\n" );
document.write( "1.96 * 92 / sqrt(n) = 13<BR>\n" );
document.write( "multiply both sides by sqrt(n) and divide both sides by 13 to get:<BR>\n" );
document.write( "1.96 * 92 / 13 = sqrt(n)<BR>\n" );
document.write( "solve for sqrt(n) to get:<BR>\n" );
document.write( "sqrt(n) = 13.87076923.<BR>\n" );
document.write( "when sqrt(n) = that, s (standard error) = 92 / 13.87076923 = 6.632653061.<BR>\n" );
document.write( "critical z-score formula becomes:<BR>\n" );
document.write( "z = 13 / 6.632653061 = 1.96, confirming you will get the 95% confidence interval of between z = -1.96 and 1.96 (you only derived the high side z-score; the low side z-score is the same with a change of sign)<BR>\n" );
document.write( "you can use the z-score formula to see if what you got is correct.<BR>\n" );
document.write( "i used an online z-score calculator at <a  rel=nofollow HREF= \"https://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">https://davidmlane.com/hyperstat/z_table.html</a><BR>\n" );
document.write( "the mean can be anything and the standard deviation is 92 and the standard error is 6.632653061 and the result will be a margin of error of 13.<BR>\n" );
document.write( "here are some examples.<BR>\n" );
document.write( "the first is using the z-score with a mean of 0 and a standard deviation of 1.<BR>\n" );
document.write( "the rest are using a mean of 100, 200, 5000, with a standard deviation of 6.632653061.<BR>\n" );
document.write( "note that the calculator says standard deviation, but when you are using the calculator with the mean of a sample, standard deviation is really standard error.<BR>\n" );
document.write( "in all of the examples except the first, the standard error was 6.632653061.<BR>\n" );
document.write( "this was truncated by the calculator to 6 decimal digits.<BR>\n" );
document.write( "i had no control over that.\r<BR>\n" );
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document.write( "regardless of the mean, the margin of error will be 13 as long as the standard error is equal to 6.632653061.\r<BR>\n" );
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document.write( "since sqrt(n) = 13.87076923, then n = that squared = 192.3982391.<BR>\n" );
document.write( "the margin of error will be less than or equal to 13 when the sample size is greater than or equal to 192.3982391.<BR>\n" );
document.write( "since sample size needs to be an integer, then minimum sample size will be 193.<BR>\n" );
document.write( "this will result in a margin of error slightly less than 13.\r<BR>\n" );
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document.write( "let me know if you have any questions.<BR>\n" );
document.write( "theo<BR>\n" );
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