document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086260>Question 1086260</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The prices (in dollars) for a graphing calculator are shown below for 8 online vendors. Estimate the population standard deviation in price with 90% confidence.<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 #700438 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=700438\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Confidence Level = C = 90% = 0.90\r<BR>\n" );
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document.write( "alpha = 1-C = 1-0.90 = 0.10\r<BR>\n" );
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document.write( "With confidence intervals, we have a shaded region in the middle and two unshaded tails (see the image labeled \"chi-square distribution\" below)\r<BR>\n" );
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document.write( "The shaded region has area of 0.90 and the two unshaded regions combine to 0.10. The total area under the entire curve is 1.\r<BR>\n" );
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document.write( "Each tail has area of alpha/2 = 0.10/2 = 0.05\r<BR>\n" );
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document.write( "The sample size is n = 8, so the degrees of freedom are df = n-1 = 8-1 = 7\r<BR>\n" );
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document.write( "Let's define\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \chi_L^2\"> = left Chi-Square critical value\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \chi_R^2\"> = right Chi-Square critical value\r<BR>\n" );
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document.write( "The goal is to find the critical values such that \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P\left(\chi_L^2 \le \chi^2 \le \chi_R^2\right) = 0.90\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \chi^2\"> is the Chi-Square symbol\r<BR>\n" );
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document.write( "Again, see the image labeled \"chi-square distribution\" below.\r<BR>\n" );
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document.write( "Use a calculator or table to find the Chi-Square critical values.\r<BR>\n" );
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document.write( "I'm going to use a stats table.\r<BR>\n" );
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document.write( "According to <a  rel=nofollow HREF=\"https://people.richland.edu/james/lecture/m170/tbl-chi.html\">this table</a>, it states that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \chi_L^2=2.167\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \chi_R^2=14.067\">\r<BR>\n" );
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document.write( "You're probably wondering how I got these values. Well first you would highlight everything in the row that starts with 7 (to indicate df = 7). Then mark the columns 0.95 and 0.05 as I've done so in red\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/qZLYS8m.png\">\r<BR>\n" );
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document.write( "The two values at the row/column intersections are 2.167 and 14.067 in that order (from left to right)\r<BR>\n" );
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document.write( "These two values 2.167 and 14.067 are set up in such a way that \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P\left(2.167 \le \chi^2 \le 14.067\right) \approx 0.90\">\r<BR>\n" );
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document.write( "as shown by the diagram below\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/bcq3LCg.png\">\r<BR>\n" );
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document.write( "Point A = location of the left chi-square critical value<BR>\n" );
document.write( "Point B = location of the right chi-square critical value<BR>\n" );
document.write( "area under the curve between A and B = 0.90 = 90% = confidence level\r<BR>\n" );
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document.write( "Now that we have the Chi-Square critical values, let's find the standard deviation. To make things simple, it's best to use a calculator to compute s. We can't find sigma (because we're trying to approximate it based on the sample) but we can find the sample estimate s. \r<BR>\n" );
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document.write( "Using a <a  rel=nofollow HREF=\"http://www.calculator.net/standard-deviation-calculator.html\">calculator</a>, the approximate value of s is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+7.59699 ALIGN=MIDDLE  ALT=\"s+=+7.59699\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "It's possible to use other calculators such as the TI83. If you wish to use that, type the eight data values into a blank list (say L1) and then you can do a one-var stats on the data list. See <a  rel=nofollow HREF=\"http://bas.utk.edu/stat-201-help/calculators/TI83sum.pdf\">this page</a> for a similar example. \r<BR>\n" );
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document.write( "Finally we can compute the confidence interval estimate for sigma\r<BR>\n" );
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document.write( "I'm going to use the formula drawn from <a  rel=nofollow HREF=\"http://www.milefoot.com/math/stat/ci-variances.htm\">this page</a> (though I'm going to use slightly different notation for the chi-square symbols)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{\frac{(n-1)*s^2}{\chi_R^2}} \le \sigma \le \sqrt{\frac{(n-1)*s^2}{\chi_L^2}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{\frac{(8-1)*(7.59699)^2}{14.067}} \le \sigma \le \sqrt{\frac{(8-1)*(7.59699)^2}{2.167}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{\frac{7*(7.59699)^2}{14.067}} \le \sigma \le \sqrt{\frac{7*(7.59699)^2}{2.167}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{\frac{7*57.7142570601}{14.067}} \le \sigma \le \sqrt{\frac{7*57.7142570601}{2.167}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{\frac{403.9997994207}{14.067}} \le \sigma \le \sqrt{\frac{403.9997994207}{2.167}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{28.7196843264875} \le \sigma \le \sqrt{186.432763922796}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 5.35907495063164 \le \sigma \le 13.6540383741513\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 5.359 \le \sigma \le 13.654\">\r<BR>\n" );
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document.write( "We are 90% confident that the population standard deviation (sigma = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is between 5.359 and 13.654.</font> </SPAN>\n" );
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