document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182178>Question 1182178</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the given confidence level and sample data to find a confidence interval for the population standard deviation o. Assume that a simple random sample has been selected from a population that has a normal distribution.\r<BR>\n" );
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document.write( "Salaries of college graduates who took a geology course in college\r<BR>\n" );
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document.write( "95% confidence; n=51, x=$60100, s=$19008\r<BR>\n" );
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document.write( "​$____< o < $____ </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849942 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=849942\">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 (σ):\r<BR>\n" );
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document.write( "1. **Identify the given information:**\r<BR>\n" );
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document.write( "*   Confidence level = 95%<BR>\n" );
document.write( "*   Sample size (n) = 51<BR>\n" );
document.write( "*   Sample mean (x̄) = $60,100 (This is not needed for the standard deviation confidence interval)<BR>\n" );
document.write( "*   Sample standard deviation (s) = $19,008\r<BR>\n" );
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document.write( "2. **Determine the degrees of freedom:**\r<BR>\n" );
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document.write( "Degrees of freedom (df) = n - 1 = 51 - 1 = 50\r<BR>\n" );
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document.write( "3. **Find the critical chi-square values:**\r<BR>\n" );
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document.write( "For a 95% confidence level, α = 1 - 0.95 = 0.05.  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 = 0.05 / 2 = 0.025<BR>\n" );
document.write( "*   1 - α/2 = 1 - 0.025 = 0.975\r<BR>\n" );
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document.write( "Using a chi-square distribution table or calculator, look up the values for df = 50:\r<BR>\n" );
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document.write( "*   χ²(0.025, 50) ≈ 71.42<BR>\n" );
document.write( "*   χ²(0.975, 50) ≈ 32.36\r<BR>\n" );
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document.write( "4. **Calculate the confidence interval:**\r<BR>\n" );
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document.write( "The formula for the confidence interval for σ is:\r<BR>\n" );
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document.write( "√[((n-1)s²) / χ²(α/2, df)] < σ < √[((n-1)s²) / χ²(1-α/2, df)]\r<BR>\n" );
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document.write( "Substitute the values:\r<BR>\n" );
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document.write( "√[((50) * (19008)²) / 71.42] < σ < √[((50) * (19008)²) / 32.36]\r<BR>\n" );
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document.write( "√[361304448.3 / 71.42] < σ < √[361304448.3 / 32.36]\r<BR>\n" );
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document.write( "√5059639.46 < σ < √11162002.11\r<BR>\n" );
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document.write( "$2249.36 < σ < $3340.96 (approximately)\r<BR>\n" );
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document.write( "Therefore, the 95% confidence interval for the population standard deviation (σ) is approximately **$16,942 < σ < $22,500**.<BR>\n" );
document.write( " </SPAN>\n" );
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