document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170608>Question 1170608</A>: <I><SPAN STYLE=\"word-break: break-all;\"> With a sample size of 900, the standard error is 3. What should be the sample size so that we could be 95% confident that the population mean is within 4 of the sample mean?<BR>\n" );
document.write( "Answer=1945 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #795452 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=795452\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the standard error is equal to the standard deviation divided by the square root of the sample size.\r<BR>\n" );
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document.write( "with a sample size of 900, the standard error is 3.\r<BR>\n" );
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document.write( "this gets you the formula se = sd / sqrt(ss).<BR>\n" );
document.write( "se = standard error<BR>\n" );
document.write( "sd = standard deviation<BR>\n" );
document.write( "ss = sample size.\r<BR>\n" );
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document.write( "in your problem, the formula becomes 3 = sd / sqrt(900)<BR>\n" );
document.write( "solve for sd to get:<BR>\n" );
document.write( "sd = 3 * sqrt(900) = 3 * 30 = 90.\r<BR>\n" );
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document.write( "at 95% confidence level, the two tailed alpha will be .05/2 = .025.\r<BR>\n" );
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document.write( "the critical z-score becomes plus or minus 1.96\r<BR>\n" );
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document.write( "the z-score formula becomes plur or minus 1.96 = (x - m) / se\r<BR>\n" );
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document.write( "since se = sd / sqrt(ss), and since sd = 90, the formula becomes se = 90 / sqrt(ss).\r<BR>\n" );
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document.write( "the formula becomes plus or minus 1.96 = (x - m) / (90 / sqrt(ss)).\r<BR>\n" );
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document.write( "since you want the margin of error to be plus minus 4, then (x - m) must be equal to plus or minus 4, and the formula becomes:\r<BR>\n" );
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document.write( "plus or minus 1.96 = (plus or minus 4) / (90 / sqrt(ss))\r<BR>\n" );
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document.write( "this is the same as:\r<BR>\n" );
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document.write( "plus or minus 1.96 = (plus or minus 4) * sqrt(ss) / 90\r<BR>\n" );
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document.write( "solve for sqrt(ss) to get:\r<BR>\n" );
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document.write( "sqrt(ss) = 90 * (plus or minus 1.96) / (plus or minus 4)\r<BR>\n" );
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document.write( "just working on the plus sides of this equation, you get:\r<BR>\n" );
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document.write( "sqrt(ss) = 90 * 1.96 / 4.\r<BR>\n" );
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document.write( "that gets you sqrt(ss) = 44.1\r<BR>\n" );
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document.write( "solve for ss to get:\r<BR>\n" );
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document.write( "ss = 1944.81.\r<BR>\n" );
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document.write( "that's the answer your are looking for.\r<BR>\n" );
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document.write( "the other options will lead to the following results.\r<BR>\n" );
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document.write( "sqrt(ss) = 90 * -1.96 / -4 = 44.1, giving you the same answer.\r<BR>\n" );
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document.write( "sqrt(ss) = 90 * 1.96 / -4 = -44.1 which is not possible because it winds up being the square root of a negative number which is not real.\r<BR>\n" );
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document.write( "sqrt(ss) = 90 * -1.96 / 4 = -44.1 which is not real for the same reason.\r<BR>\n" );
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document.write( "the only real answer is 1944.81.\r<BR>\n" );
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document.write( "your solution is that the sample size is 1944.81.\r<BR>\n" );
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document.write( "to confirm, replace ss with 1944.81 in the problem to get:\r<BR>\n" );
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document.write( "plus or minus 1.96 = (x - m) / (90/sqrt(1944.81).\r<BR>\n" );
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document.write( "solve for (x - m) to get:\r<BR>\n" );
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document.write( "(x - m) = 1.96 * 90 / sqrt(1944.81) = 4 when 1.96 is positive.\r<BR>\n" );
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document.write( "(x - m) = -1.96 * 90 / sqrt(1944.81) = -4 when 1.96 is negative.\r<BR>\n" );
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document.write( "your margin of error is plus or minus 4.\r<BR>\n" );
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document.write( "this works with any mean as long as the standard error is equal to 90 / sqrt(1944.81) = 2.040800586.\r<BR>\n" );
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document.write( "for example, if the mean is 7500, then you get:\r<BR>\n" );
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document.write( "z = (7504 - 7500) / 2.040800586 = 1.96\r<BR>\n" );
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document.write( "z = (7496 - 7500) / 2.040800586 = -1.96\r<BR>\n" );
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document.write( "the confidence interval is from 7496 to 7504.\r<BR>\n" );
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document.write( "the margin of error is plus or minus 4.\r<BR>\n" );
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document.write( "normally, you would round the answer to the next highest integer which would make it equal to a sample size of 1945.\r<BR>\n" );
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document.write( "that makes your margin of error slightly less than 4.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "1.96 = (x - m) / (90/sqrt(1945)\r<BR>\n" );
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document.write( "solve for (x - m) to get:\r<BR>\n" );
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document.write( "(x - m) = 3.999804622.\r<BR>\n" );
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document.write( "that's slightly less than 4, but if you round it to the 3 decimal places, it will be equal to 4.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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