document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118451>Question 1118451</A>: <I><SPAN STYLE=\"word-break: break-all;\"> We want to estimate the population mean within 8, with a 80% level of confidence. The population standard deviation is estimated to be 21. How large a sample is required? (Round the intermediate calculation to 2 decimal places. Round the final answer to the nearest whole number.)\r<BR>\n" );
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document.write( "  Sample size	:<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 #733845 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=733845\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> at 80% confidence level, the critical alpha will be plus or minus .2/2 = .1.\r<BR>\n" );
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document.write( "at a critical alpha of .1, the critical z-score will be plus or minus 1.281551567 which you are instructed to round to two decimal places, making it plus or minus 1.28.\r<BR>\n" );
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document.write( "the standard error is equal to the standard deviation divided by sample size.\r<BR>\n" );
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document.write( "that would make the standard error equal to 21 / sqrt(n), where n is the sample size.\r<BR>\n" );
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document.write( "the formula for z-score is z = (x-m) / s\r<BR>\n" );
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document.write( "in this case:\r<BR>\n" );
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document.write( "z = the critical z-score<BR>\n" );
document.write( "x = the raw score<BR>\n" );
document.write( "m equal the mean<BR>\n" );
document.write( "s equal the standard error.\r<BR>\n" );
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document.write( "on the high side, the formula becomes 1.28 = (x-m) / (21/sqrt(n)).\r<BR>\n" );
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document.write( "since (x-m) on the high side equals 8,the formula becomes 1.28 = 8 / (21/sqrt(n)).\r<BR>\n" );
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document.write( "multiply both sides of this equation by (21/sqrt(n)) and you get 1.28 * (21/sqrt(n)) = 8.\r<BR>\n" );
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document.write( "multiply both sides of this equation by sqrt(n) and divide both sides of this equation by 8 to get 1.28 * 21 / 8 = sqrt(n).\r<BR>\n" );
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document.write( "solve for n to get n = 11.2896.\r<BR>\n" );
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document.write( "tha's the sample size required.\r<BR>\n" );
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document.write( "round it to the nearest whole integer to get 11.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( "to test this solution, it is recommended to use the unrounded values rather than the rounded values.\r<BR>\n" );
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document.write( "this way you'll get a range of plus or minus 8 exactly, or extremely close, if not.\r<BR>\n" );
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document.write( "the critical z-score is actually 1.281551567.\r<BR>\n" );
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document.write( "the sample size is actually 11.31698622.\r<BR>\n" );
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document.write( "the standard error is 21 / sqrt(11.31698622) = 6.242433164.\r<BR>\n" );
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document.write( "the formula is z = (x-m) / s.\r<BR>\n" );
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document.write( "z = plus or minus 1.281551567<BR>\n" );
document.write( "s = 6.242433164<BR>\n" );
document.write( "m can be any randomly generated number, such as 3524.\r<BR>\n" );
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document.write( "when z = plus, the formula becomes 1.281551567 = (x-3524) / 6.242433164.\r<BR>\n" );
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document.write( "solve for x to get x = 1.281551567 * 6.242433164 + 3524 = 3532\r<BR>\n" );
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document.write( "when z = minus, the formula becomes -1.281551567 = (x-3524) / 6.242433164.\r<BR>\n" );
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document.write( "solve for x to get x = -1.281551567 * 6.242433164 + 3524 = 3516.\r<BR>\n" );
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document.write( "3532 - 3524 = 8<BR>\n" );
document.write( "3516 - 3524 = -8\r<BR>\n" );
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document.write( "the difference is plus or minus 8.\r<BR>\n" );
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document.write( "it will always be plus or minus 8 regardless of the mean because 1.281551567 * 6.242433164 = 8.\r<BR>\n" );
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