document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201399>Question 1201399</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Lifetimes of AAA batteries are approximately normally distributed. A manufacturer wants to estimate the standard deviation of the lifetime of the AAA batteries it produces. A random sample of 15 AAA batteries produced by this manufacturer lasted a mean of 9.5 hours with a standard deviation of 1.7 hours. Find a 95% confidence interval for the population standard deviation of the lifetimes of AAA batteries produced by the manufacturer. Then give its lower limit and upper limit. Carry your intermediate computations to at least three decimal places. Round your answers to at least two decimal places. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835793 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=835793\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "sigma = population standard deviation\r<BR>\n" );
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document.write( "<font color=red>Goal: estimate sigma</font>\r<BR>\n" );
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document.write( "This estimate is written as a confidence interval in the format<BR>\n" );
document.write( "<font color=red>L < sigma < U</font><BR>\n" );
document.write( "where,<BR>\n" );
document.write( "L = lower boundary<BR>\n" );
document.write( "U = upper boundary\r<BR>\n" );
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document.write( "s = sample standard deviation, which helps estimate sigma<BR>\n" );
document.write( "s = 1.7 is given in the instructions\r<BR>\n" );
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document.write( "n = sample size = 15<BR>\n" );
document.write( "df = degrees of freedom<BR>\n" );
document.write( "df = n-1<BR>\n" );
document.write( "df = 15-1<BR>\n" );
document.write( "df = 14\r<BR>\n" );
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document.write( "Here is an article to check out<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://faculty.elgin.edu/dkernler/statistics/ch09/9-3.html\">https://faculty.elgin.edu/dkernler/statistics/ch09/9-3.html</a>\r<BR>\n" );
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document.write( "Check out example 1 to see how to determine the critical values based on the degrees of freedom (df) and on the confidence level.<BR>\n" );
document.write( "Example 1 uses df = 12, but the confidence level used is 95%. Meaning that you'll look at those same columns to get the XL and XR values (except just look at the row df = 14 instead)\r<BR>\n" );
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document.write( "If you were to look at the df = 14 row, and those columns mentioned, then you should find<BR>\n" );
document.write( "XL = 5.629 = left critical chi-square value<BR>\n" );
document.write( "XR = 26.119 = right critical chi-square value<BR>\n" );
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document.write( "This would indicate<BR>\n" );
document.write( "P(XL < X2 < XR) = 0.95<BR>\n" );
document.write( "P(5.629 < X2 < 26.119) = 0.95<BR>\n" );
document.write( "The area under the chi-square curve, between 5.629 and 26.119, is roughly 0.95<BR>\n" );
document.write( "About 95% of the area under the curve is between 5.629 and 26.119\r<BR>\n" );
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document.write( "Side note: A chi-square calculator can compute the left and right critical values needed.\r<BR>\n" );
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document.write( "Now we can compute the lower boundary of the confidence interval for sigma.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+matrix%281%2C2%2Clower%2Cboundary%29 ALIGN=MIDDLE  ALT=\"L+=+matrix%281%2C2%2Clower%2Cboundary%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+sqrt%28%28%28n-1%29%2As%5E2%29%2F%28XR%29%29 ALIGN=MIDDLE  ALT=\"L+=+sqrt%28%28%28n-1%29%2As%5E2%29%2F%28XR%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+sqrt%28%28%2815-1%29%2A%281.7%29%5E2%29%2F%2826.119%29%29 ALIGN=MIDDLE  ALT=\"L+=+sqrt%28%28%2815-1%29%2A%281.7%29%5E2%29%2F%2826.119%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+1.24461395614987 ALIGN=MIDDLE  ALT=\"L+=+1.24461395614987\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+1.245 ALIGN=MIDDLE  ALT=\"L+=+1.245\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now compute the upper boundary.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U+=+matrix%281%2C2%2Cupper%2Cboundary%29 ALIGN=MIDDLE  ALT=\"U+=+matrix%281%2C2%2Cupper%2Cboundary%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U+=+sqrt%28%28%28n-1%29%2As%5E2%29%2F%28XL%29%29 ALIGN=MIDDLE  ALT=\"U+=+sqrt%28%28%28n-1%29%2As%5E2%29%2F%28XL%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U+=+sqrt%28%28%2815-1%29%2A%281.7%29%5E2%29%2F%285.629%29%29 ALIGN=MIDDLE  ALT=\"U+=+sqrt%28%28%2815-1%29%2A%281.7%29%5E2%29%2F%285.629%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U+=+2.68100309220024 ALIGN=MIDDLE  ALT=\"U+=+2.68100309220024\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U+=+2.681 ALIGN=MIDDLE  ALT=\"U+=+2.681\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "It could be a bit confusing that XR goes for the left or lower boundary, while XL goes for the right or upper boundary.\r<BR>\n" );
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document.write( "The confidence interval format<BR>\n" );
document.write( "L < sigma < U<BR>\n" );
document.write( "is then updated to<BR>\n" );
document.write( "<font color=red size=4>1.245 < sigma < 2.681</font><BR>\n" );
document.write( "which is approximate.<BR>\n" );
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