document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1188789>Question 1188789</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (1 point) The weights of unicorns are normally distributed. Suppose that a simple random sample of 12 unicorns have a standard deviation of 20.8. Estimate the standard deviation of the entire population with 80% confidence.\r<BR>\n" );
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document.write( "LCL = <BR>\n" );
document.write( "UCL = <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 #849294 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849294\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the confidence interval for the population standard deviation:\r<BR>\n" );
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document.write( "**1. Identify the given information:**\r<BR>\n" );
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document.write( "* Sample size (n) = 12<BR>\n" );
document.write( "* Sample standard deviation (s) = 20.8<BR>\n" );
document.write( "* Confidence level = 80%\r<BR>\n" );
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document.write( "**2. Find the degrees of freedom:**\r<BR>\n" );
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document.write( "Degrees of freedom (df) = n - 1 = 12 - 1 = 11\r<BR>\n" );
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document.write( "**3. Find the chi-square values:**\r<BR>\n" );
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document.write( "For an 80% confidence level, the alpha (α) is 1 - 0.80 = 0.20.  We need to find the chi-square values for α/2 and 1-α/2.\r<BR>\n" );
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document.write( "* α/2 = 0.20 / 2 = 0.10<BR>\n" );
document.write( "* 1 - α/2 = 1 - 0.10 = 0.90\r<BR>\n" );
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document.write( "Using a chi-square table or calculator, look up the values for df = 11:\r<BR>\n" );
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document.write( "* χ²(0.10, 11) ≈ 19.675<BR>\n" );
document.write( "* χ²(0.90, 11) ≈ 5.578\r<BR>\n" );
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document.write( "**4. Calculate the lower confidence limit (LCL):**\r<BR>\n" );
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document.write( "LCL = sqrt[ (n-1) * s² / χ²(α/2, df) ]<BR>\n" );
document.write( "LCL = sqrt[ (11 * 20.8²) / 19.675 ]<BR>\n" );
document.write( "LCL = sqrt[ 4758.72 / 19.675 ]<BR>\n" );
document.write( "LCL = sqrt[241.85]<BR>\n" );
document.write( "LCL ≈ 15.55\r<BR>\n" );
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document.write( "**5. Calculate the upper confidence limit (UCL):**\r<BR>\n" );
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document.write( "UCL = sqrt[ (n-1) * s² / χ²(1-α/2, df) ]<BR>\n" );
document.write( "UCL = sqrt[ (11 * 20.8²) / 5.578 ]<BR>\n" );
document.write( "UCL = sqrt[ 4758.72 / 5.578 ]<BR>\n" );
document.write( "UCL = sqrt[853.14]<BR>\n" );
document.write( "UCL ≈ 29.21\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "LCL ≈ 15.55<BR>\n" );
document.write( "UCL ≈ 29.21\r<BR>\n" );
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document.write( "Therefore, we are 80% confident that the population standard deviation of unicorn weights is between approximately 15.55 and 29.21.<BR>\n" );
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