document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177422>Question 1177422</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A survey reports it results by starting that standard error of the mean to be is 20.<BR>\n" );
document.write( "The population standard deviation is 500.<BR>\n" );
document.write( "1:How large is the sample used in this surveying?<BR>\n" );
document.write( "2: What is the probability that the sample mean will be within 25 of the population mean? <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #806377 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=806377\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> standard error is equal to population standard deviation divided by square root of sample size.\r<BR>\n" );
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document.write( "the formula is s = psd / sqrt(ss).\r<BR>\n" );
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document.write( "s is the standard error.<BR>\n" );
document.write( "psd is population standard deviation.<BR>\n" );
document.write( "ss is sample size.\r<BR>\n" );
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document.write( "when s = 20 and psd = 500, the formula becomes:\r<BR>\n" );
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document.write( "20 = 500 / sqrt(ss)\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) = 500 / 20 = 25\r<BR>\n" );
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document.write( "solve for ss to get:\r<BR>\n" );
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document.write( "ss = 25^2 = 625\r<BR>\n" );
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document.write( "that's the sample size.\r<BR>\n" );
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document.write( "the formula for z-score is:\r<BR>\n" );
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document.write( "z = (x - m) / s\r<BR>\n" );
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document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the raw mean<BR>\n" );
document.write( "s is the standard error.\r<BR>\n" );
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document.write( "when (x - m) = 25, and s = 20, the formula becomes:\r<BR>\n" );
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document.write( "z = 25 / 20 = 1.25\r<BR>\n" );
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document.write( "since the normal distribution curve is symmetric bout the mean, then the confidence interval is z-score of -1.25 to 1.25.\r<BR>\n" );
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document.write( "the probability is the area under the normal distribution curve between those 2 z-scores.\r<BR>\n" );
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document.write( "that probability is .7887003221.\r<BR>\n" );
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document.write( "the probability that the sample mean will be within 25 of the population mean is .7887003221.\r<BR>\n" );
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document.write( "this works regardless of the mean.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "assume the mean is 1500.\r<BR>\n" );
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document.write( "the z-score formula becomes plus or minus 1.25 = (x - 1500) / 20\r<BR>\n" );
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document.write( "solve for the raw score to get:\r<BR>\n" );
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document.write( "on the high side, x = 20 * 1.25 + 1500<BR>\n" );
document.write( "on the low side, x = 20 * -1.25 + 1500\r<BR>\n" );
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document.write( "since 20 * 1.25 is always equal to 25, then you get:\r<BR>\n" );
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document.write( "x = 1500 - 25 or x = 1500 + 25.\r<BR>\n" );
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document.write( "the sample mean will always be within 25 of the population mean if the z-score is plus or minus 1.25 and the standard error is 20, regardless of what the mean is.\r<BR>\n" );
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document.write( "standard error = psd / sqrt(ss)<BR>\n" );
document.write( "with a sample size of 625 and a psd of 500, this becomes:<BR>\n" );
document.write( "standard error = 500 / sqrt(625) = = 500 / 25 = 20 </SPAN>\n" );
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