document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198875>Question 1198875</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The following data are a random sample of turtles and their weights (ounces).<BR>\n" );
document.write( "{18,16,20,12,11,10,16,17}<BR>\n" );
document.write( "Assume the underlying population distribution is approximately normal and has no outliers. Find the following, rounding your intermediate steps and answers to 3 decimal places.\r<BR>\n" );
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document.write( "a. Find the sample standard deviation.<BR>\n" );
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document.write( "b. Find the 95% confidence interval for the population standard deviation.<BR>\n" );
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document.write( "c. Find the 99% confidence interval for the population standard deviation.<BR>\n" );
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document.write( "d. As expected, the answer for part c. is ____ than the answer for part b.\r<BR>\n" );
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document.write( "e. The lower bound is ____ the upper bound is from s because of the shape of the χ2 distribution.<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 #848224 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848224\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a. Sample Standard Deviation**\r<BR>\n" );
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document.write( "1. **Calculate the sample mean (x̄):**<BR>\n" );
document.write( "   x̄ = (18 + 16 + 20 + 12 + 11 + 10 + 16 + 17) / 8 = 15\r<BR>\n" );
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document.write( "2. **Calculate the squared differences from the mean:**<BR>\n" );
document.write( "   (18-15)² = 9<BR>\n" );
document.write( "   (16-15)² = 1<BR>\n" );
document.write( "   (20-15)² = 25<BR>\n" );
document.write( "   (12-15)² = 9<BR>\n" );
document.write( "   (11-15)² = 16<BR>\n" );
document.write( "   (10-15)² = 25<BR>\n" );
document.write( "   (16-15)² = 1<BR>\n" );
document.write( "   (17-15)² = 4\r<BR>\n" );
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document.write( "3. **Sum the squared differences:**<BR>\n" );
document.write( "   9 + 1 + 25 + 9 + 16 + 25 + 1 + 4 = 89\r<BR>\n" );
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document.write( "4. **Calculate the sample variance (s²):**<BR>\n" );
document.write( "   s² = Σ(x - x̄)² / (n - 1) = 89 / (8 - 1) = 12.714\r<BR>\n" );
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document.write( "5. **Calculate the sample standard deviation (s):**<BR>\n" );
document.write( "   s = √s² = √12.714 = 3.566\r<BR>\n" );
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document.write( "**Therefore, the sample standard deviation (s) is 3.566**\r<BR>\n" );
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document.write( "**b. 95% Confidence Interval for the Population Standard Deviation**\r<BR>\n" );
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document.write( "* **Find the chi-square values:**<BR>\n" );
document.write( "    * Degrees of freedom (df) = n - 1 = 8 - 1 = 7<BR>\n" );
document.write( "    * Using a chi-square table or calculator:<BR>\n" );
document.write( "        * χ²_lower (for 0.025 area in the right tail) = 2.167<BR>\n" );
document.write( "        * χ²_upper (for 0.025 area in the left tail) = 18.475\r<BR>\n" );
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document.write( "* **Calculate the confidence interval:**<BR>\n" );
document.write( "    * Lower bound: √[(n - 1) * s² / χ²_upper] = √[(7 * 12.714) / 18.475] = 2.172<BR>\n" );
document.write( "    * Upper bound: √[(n - 1) * s² / χ²_lower] = √[(7 * 12.714) / 2.167] = 6.390\r<BR>\n" );
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document.write( "* **95% Confidence Interval: (2.172, 6.390)**\r<BR>\n" );
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document.write( "**c. 99% Confidence Interval for the Population Standard Deviation**\r<BR>\n" );
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document.write( "* **Find the chi-square values:**<BR>\n" );
document.write( "    * Degrees of freedom (df) = 7<BR>\n" );
document.write( "    * Using a chi-square table or calculator:<BR>\n" );
document.write( "        * χ²_lower (for 0.005 area in the right tail) = 0.989<BR>\n" );
document.write( "        * χ²_upper (for 0.005 area in the left tail) = 20.278\r<BR>\n" );
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document.write( "* **Calculate the confidence interval:**<BR>\n" );
document.write( "    * Lower bound: √[(7 * 12.714) / 20.278] = 1.974<BR>\n" );
document.write( "    * Upper bound: √[(7 * 12.714) / 0.989] = 9.446\r<BR>\n" );
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document.write( "* **99% Confidence Interval: (1.974, 9.446)**\r<BR>\n" );
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document.write( "**d. As expected, the answer for part c. is wider** than the answer for part b.\r<BR>\n" );
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document.write( "**e. The lower bound is lower** and **the upper bound is higher** from s because of the shape of the chi-square distribution.\r<BR>\n" );
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document.write( "* The chi-square distribution is skewed to the right. <BR>\n" );
document.write( "* For a higher confidence level (like 99%), we need a wider interval to capture more of the possible values of the population standard deviation. This results in a lower lower bound and a higher upper bound compared to the 95% confidence interval.<BR>\n" );
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