document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1122419>Question 1122419</A>: <I><SPAN STYLE=\"word-break: break-all;\"> n order to estimate the mean amount of time computer users spend on the internet each&#8203; month, how many computer users must be surveyed in order to be 95&#8203;% confident that your sample mean is within 13 minutes of the population&#8203; mean? Assume that the standard deviation of the population of monthly time spent on the internet is 221 min. What is a major obstacle to getting a good estimate of the population&#8203; mean? Use technology to find the estimated minimum required sample size. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #739644 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=739644\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the critical z-score for 95% confidence interval is z = plus or minus 1.959963986 according to my TI-84 Plus calculator.\r<BR>\n" );
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document.write( "you could use this z-score or you could round it off to plus or minus 1.96 which is probably what you'd do if you used the z-score table.\r<BR>\n" );
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document.write( "i'll use plus or minus 1.96.\r<BR>\n" );
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document.write( "the differences should be very small.\r<BR>\n" );
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document.write( "at 95% confidence interval, the z-score needs to be plus or minus 1.96.\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( "x is the raw score.<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard error of the distribution of sample means.\r<BR>\n" );
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document.write( "s = standard deviation / sqrt (sample size)\r<BR>\n" );
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document.write( "you want your estimate of population mean to be plus or minus 13 at 95% confidence interval.\r<BR>\n" );
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document.write( "this means the critical z-score should result in a critical raw score that is plus or minus 13 from the sample mean.\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( "you want (x - m) to be plus or minus 13.\r<BR>\n" );
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document.write( "working on the high side, you get 1.96 = 13 / s\r<BR>\n" );
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document.write( "s = standard deviation / sqrt(sample size) = 221 / sqrt(sample size)\r<BR>\n" );
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document.write( "formula becomes 1.96 = 13 / (221 / sqrt(sample size)\r<BR>\n" );
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document.write( "multiply both sides of this equation by 221 / sqrt(sample size) gets you:\r<BR>\n" );
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document.write( "1.96 * 221 / sqrt(sample size) = 13\r<BR>\n" );
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document.write( "divide both sides of this equation by 13 and multiply both sides of this equation by sqrt(sample size) gets you:\r<BR>\n" );
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document.write( "1.96 * 221 / 13 = sqrt(sample size)\r<BR>\n" );
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document.write( "solve for sqrt(sample size) gets you sqrt(sample size) = 33.32\r<BR>\n" );
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document.write( "solve for sample size gets you sample size = 33.32^2 = 1110.2224\r<BR>\n" );
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document.write( "round to the next highest integer gets you sample size = 1111.\r<BR>\n" );
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document.write( "that's the sample size required to get your margin of error to be within plus or minus 13.\r<BR>\n" );
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document.write( "to confirm, use the z-score formula again.\r<BR>\n" );
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document.write( "on the high side..... \r<BR>\n" );
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document.write( "1.96 = (x-m) / (221 / sqrt(1111)\r<BR>\n" );
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document.write( "multiply both sides of this equation by 221 / sqrt(1111) to get:\r<BR>\n" );
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document.write( "1.96 * 221 / sqrt(1111) = (x-m)\r<BR>\n" );
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document.write( "solve for (x-m) to get (x-m) = 12.995..... which is less than or equal to 13.\r<BR>\n" );
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document.write( "on the low side, -1.96 = (x-m) / 221 / sqrt(1111)\r<BR>\n" );
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document.write( "multiply both sides of this equation by 221 / sqrt(1111) to get:\r<BR>\n" );
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document.write( "-1.96 * 221 / sqrt(1111) = (x-m)\r<BR>\n" );
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document.write( "solve for (x-m) to get (x-m) = -12.995..... which is less than 13 from the mean.\r<BR>\n" );
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document.write( "looks like you got the sample size right.\r<BR>\n" );
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document.write( "the margin of error is less than or equal to plus or minus 13 at 95% confidence interval.\r<BR>\n" );
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document.write( "i believe this is accurate to the best of my knowledge.\r<BR>\n" );
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document.write( "the general formula for margin of error is z * standard deviation / sqrt (sample size).\r<BR>\n" );
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document.write( "in your problem, this becomes 13 = 1.96 * 221 / sqrt(sample size)\r<BR>\n" );
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document.write( "solve for sqrt(sample size) to get sqrt(sample size) = 1.96 * 221 / 13 = 33.32\r<BR>\n" );
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document.write( "sample size is 33.32^2 = 1110.something = 1111 rounded to next highest integer.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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