document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=664090>Question 664090</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You are given the sample mean and the standard deviation.  Use this information to construct the 90% and 95% confidence intervals for the population mean.  A random sample of 32 eight ounce drinks has a mean of 86.7 calories and a standard deviation of 43.9 calories.\r<BR>\n" );
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document.write( "The 90% CI is<BR>\n" );
document.write( "The 95% CI is </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #413125 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=413125\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> sample size is equal to 32<BR>\n" );
document.write( "sample mean is equal to 86.7 calories<BR>\n" );
document.write( "sample standard deviation is equal to 43.9 calories.\r<BR>\n" );
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document.write( "if this is a distribution of means, you have to calculate the standard error.<BR>\n" );
document.write( "standard error = standard deviation / square root of sample size.<BR>\n" );
document.write( "se = sd / sqrt(n)<BR>\n" );
document.write( "se = 43.9 / sqrt(32) = 7.76050\r<BR>\n" );
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document.write( "standard error is equal to 7.76050 calories.\r<BR>\n" );
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document.write( "since you are using the sample standard deviation rather than the population standard deviation, you have to use a T score rather than a Z score.\r<BR>\n" );
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document.write( "the degrees of freedom is equal to the sample size minus 1 which = 31.\r<BR>\n" );
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document.write( "The critical T value at 90% confidence interval with 31 degrees of freedom is equal to +/- 1.69552\r<BR>\n" );
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document.write( "The critical T value at 95% confidence interval with 31 degrees of freedom is equal to +/- 2.03951\r<BR>\n" );
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document.write( "you need to translate your T score to X score.\r<BR>\n" );
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document.write( "X score = T score * standard error of sample + mean of sample\r<BR>\n" );
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document.write( "at 90% confidence interval this becomes:\r<BR>\n" );
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document.write( "X score lower limit = -1.69552 * 7.76050 + 86.7 = 73.54192<BR>\n" );
document.write( "X score upper limit = 1.69552 * 7.76050 + 86.7 = 99.85808\r<BR>\n" );
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document.write( "at 95% confidence interval this becomes:\r<BR>\n" );
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document.write( "X score lower limit = -2.03951 * 7.76050 + 86.7 = 70.87246<BR>\n" );
document.write( "X score upper limit = 2.03951 * 7.76050 + 86.7 = 102.52762\r<BR>\n" );
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document.write( "you are 90% confident that the population score will be between 73.54192 and 99.85808.\r<BR>\n" );
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document.write( "you are 95% confident that the population score will be between 70.87246 and 102.52762\r<BR>\n" );
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document.write( "i'm reasonably confident this is correct.<BR>\n" );
document.write( "if you were using the population standard deviation then you would have used a Z score instead of a T score.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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