document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1188786>Question 1188786</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (1 point) The heights of Vulcans - an imaginary humanoid in Star Trek- are normally distributed. Suppose that a simple random sample of 11 Vulcans have a standard deviation of 25.8. Find the confidence Interval for the standard deviation of the entire population with 80% confidence.\r<BR>\n" );
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document.write( "1. Find the critical values   𝜒2𝐿=𝜒21−𝛼/2     and     𝜒2𝑅=𝜒2𝛼/2   that correspond to 80% degree of confidence and the sample size 𝑛=11.<BR>\n" );
document.write( "𝜒2𝐿= <BR>\n" );
document.write( " 𝜒2𝑅= \r<BR>\n" );
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document.write( "2. Find the upper and lower limits of 80% confidence Interval for the standard deviation of the entire population.\r<BR>\n" );
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document.write( "The lower limit of the 80% confidence interval = \r<BR>\n" );
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document.write( "The upper limit of the 80% confidence interval = <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 #849296 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=849296\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the confidence interval for the standard deviation of Vulcan heights:\r<BR>\n" );
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document.write( "**1. Find the critical chi-square values:**\r<BR>\n" );
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document.write( "* **Degrees of freedom (df):** df = n - 1 = 11 - 1 = 10<BR>\n" );
document.write( "* **Confidence level:** 80%, so α = 1 - 0.80 = 0.20<BR>\n" );
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( "Now, look up the chi-square values in a chi-square table or use a calculator for df = 10:\r<BR>\n" );
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document.write( "* χ²(0.90, 10) = χ²_L ≈ 4.865  (Lower critical value)<BR>\n" );
document.write( "* χ²(0.10, 10) = χ²_R ≈ 15.987 (Upper critical value)\r<BR>\n" );
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document.write( "**2. Calculate the confidence interval limits:**\r<BR>\n" );
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document.write( "* **Sample standard deviation (s):** 25.8<BR>\n" );
document.write( "* **Sample size (n):** 11\r<BR>\n" );
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document.write( "* **Lower Limit:**<BR>\n" );
document.write( "   sqrt[ (n-1) * s² / χ²_R ] = sqrt[ (10 * 25.8²) / 15.987 ] ≈ sqrt(419.92) ≈ 20.49\r<BR>\n" );
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document.write( "* **Upper Limit:**<BR>\n" );
document.write( "   sqrt[ (n-1) * s² / χ²_L ] = sqrt[ (10 * 25.8²) / 4.865 ] ≈ sqrt(1390.94) ≈ 37.29\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "1. χ²_L ≈ 4.865<BR>\n" );
document.write( "   χ²_R ≈ 15.987\r<BR>\n" );
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document.write( "2. Lower Limit ≈ 20.49<BR>\n" );
document.write( "   Upper Limit ≈ 37.29\r<BR>\n" );
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document.write( "Therefore, you are 80% confident that the population standard deviation of Vulcan heights is between approximately 20.49 and 37.29.<BR>\n" );
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