document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206917>Question 1206917</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the​ P-value method or the traditional method of testing hypotheses.<BR>\n" );
document.write( "Company A uses a new production method to manufacture aircraft altimeters. A simple random sample of new altimeters resulted in errors listed below. Use a 0.05 level of significance to test the claim that the new production method has errors with a standard deviation greater than 32.2​ ft, which was the standard deviation for the old production method. If it appears that the standard deviation is​ greater, does the new production method appear to be better or worse than the old​ method? Should the company take any​ action?<BR>\n" );
document.write( "-45​, 76​, -21​, -75​, -43​, 11​, 15​, 52​, -5​, -54​, -108​, -108  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844646 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=844646\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "sigma = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" > = population standard deviation<BR>\n" );
document.write( "s = sample standard deviation\r<BR>\n" );
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document.write( "It is easy to compute variable \"s\" because we have the sample of values given to us, and a calculator makes quick work of this computation.<BR>\n" );
document.write( "When using a calculator you should get s = 58.218176 approximately.\r<BR>\n" );
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document.write( "On the other hand, determining sigma is much more costly. <BR>\n" );
document.write( "This is because we need to take a census of a large population. <BR>\n" );
document.write( "This is true of many population parameters which is why statistics is needed.<BR>\n" );
document.write( "The idea is that variable \"s\" will estimate sigma.\r<BR>\n" );
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document.write( "The goal is to see if sigma = 32.2 or sigma > 32.2\r<BR>\n" );
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document.write( "Null Hypothesis: sigma = 32.2<BR>\n" );
document.write( "Alternate Hypothesis: sigma > 32.2\r<BR>\n" );
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document.write( "Some textbooks will replace the \"equals\" with \"less than or equal to\", but I want to be very narrow on what the null focuses on. The null should focus on one value only. \r<BR>\n" );
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document.write( "The \"greater than\" in the alternative hypothesis indicates we have a right-tailed test. \r<BR>\n" );
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document.write( "Since we're doing a hypothesis test on sigma, we'll need to do a Chi-Square Test.\r<BR>\n" );
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document.write( "Chi-Square Test Statistic = (n-1)*(s^2)/(sigma^2) <BR>\n" );
document.write( "= (12-1)*(58.218176^2)/(32.2^2) <BR>\n" );
document.write( "= 35.958 approximately\r<BR>\n" );
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document.write( "Now use a <a  rel=nofollow HREF=\"https://www.mathsisfun.com/data/chi-square-table.html\">Chi-Square Table</a> to determine the p-value is less than 0.001<BR>\n" );
document.write( "The sample size n = 12 leads to df = n-1 = 12-1 = 11.<BR>\n" );
document.write( "df = degrees of freedom<BR>\n" );
document.write( "The idea is to look through the df = 11 row and see if you can find the test statistic 35.958<BR>\n" );
document.write( "Unfortunately this value is not in this row but notice it's larger than 31.264 which corresponds to a p-value of 0.001; as the test statistic gets larger, the p-value will get smaller. \r<BR>\n" );
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document.write( "Since the p-value is smaller than alpha = 0.05, it means we have strong evidence to reject the null. <BR>\n" );
document.write( "Remember: \"If the p-value is low, then the null must go\".\r<BR>\n" );
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document.write( "Rejecting the null means we conclude that sigma > 32.2 appears to be the case.<BR>\n" );
document.write( "The new method is worse compared to the old method.<BR>\n" );
document.write( "This new production line has more errors in the altimeters and the company should take action to fix this. \r<BR>\n" );
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document.write( "If you prefer the traditional method over the p-value method, then use a <a  rel=nofollow HREF=\"https://www.mathsisfun.com/data/chi-square-table.html\">Chi-Square Table</a> to see that the critical value is roughly 19.675 when alpha = 0.05 and df = 11\r<BR>\n" );
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document.write( "P(chiSquare > 19.675) = 0.05 approximately when df = 11<BR>\n" );
document.write( "The value 19.675 is found in the table unlike earlier when 35.958 wasn't in the table.\r<BR>\n" );
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document.write( "Chi-Square critical value = 19.675<BR>\n" );
document.write( "Chi-Square test statistic = 35.958\r<BR>\n" );
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document.write( "We see the test statistic is larger than the critical value, so it's in the rejection region. <BR>\n" );
document.write( "Recall this is a right-tailed test.<BR>\n" );
document.write( "Therefore we reject the null and conclude sigma > 32.2<BR>\n" );
document.write( "The company should take action to fix the issue. <BR>\n" );
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