document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=875125>Question 875125</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Test the claim that the mean age of the prison population in cone city is less than 26 years. Sample data are summarized as n=25, mean= 24.4, s=9.2 and a=.05\r<BR>\n" );
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document.write( "This is not for home just for study...thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #527983 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=527983\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hypothesis:\r<BR>\n" );
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document.write( "H0: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu=26 ALIGN=MIDDLE  ALT=\"mu=26\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "H1: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu%3C26 ALIGN=MIDDLE  ALT=\"mu%3C26\"   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=mu ALIGN=MIDDLE  ALT=\"mu\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the greek lowercase letter mu (population mean).\r<BR>\n" );
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document.write( "This is a one-tailed test to the left.\r<BR>\n" );
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document.write( "Decision Criteria:\r<BR>\n" );
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document.write( "We reject the null hypothesis H0 if the p value is less than 0.05 (the given significance level alpha)\r<BR>\n" );
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document.write( "Test Statistic:\r<BR>\n" );
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document.write( "We are using the T distribution because <BR>\n" );
document.write( "a) we do not know the population standard deviation (very common in most cases)<BR>\n" );
document.write( "b) n < 30\r<BR>\n" );
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document.write( "t = (xbar - mu)/(s/sqrt(n))<BR>\n" );
document.write( "t = (24.4 - 26)/(9.2/sqrt(25))<BR>\n" );
document.write( "t = -0.8695652173913<BR>\n" );
document.write( "t = -0.87\r<BR>\n" );
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document.write( "This is a one tailed test to the left, so we want to find the area to the left of t = -0.87. <BR>\n" );
document.write( "Due to symmetry, this is the same as finding the area to the right of t = 0.87\r<BR>\n" );
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document.write( "Use a <a  rel=nofollow HREF=\"http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">table</a> like this one to find the area to the right of 0.87 to be between 0.15 and 0.20<BR>\n" );
document.write( "How am I getting this? The sample size is n = 25, so the there are 25 - 1 = 24 degrees of freedom, df = 24. <BR>\n" );
document.write( "We look in the df = 24 row and we are looking for 0.87, but unfortunately it's not in the row; however, we know it's between  0.857 and 1.059. <BR>\n" );
document.write( "The probabilities for the one tailed test are 0.20 and 0.15 respectively, so that's how I got the area to be between 0.15 and 0.20\r<BR>\n" );
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document.write( "Anyways, the area to the right of t = 0.87 is between 0.15 and 0.20<BR>\n" );
document.write( "So the area to the left of t = -0.87 is between 0.15 and 0.20\r<BR>\n" );
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document.write( "This is the p-value. \r<BR>\n" );
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document.write( "The p-value is between 0.15 and 0.20\r<BR>\n" );
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document.write( "Decision:\r<BR>\n" );
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document.write( "The p-value is NOT less than 0.05 (the given significance level alpha). Even if the p-value was the smallest it can get (0.15), it is still not even close to being smaller than 0.05.\r<BR>\n" );
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document.write( "So we can't reject the null hypothesis H0\r<BR>\n" );
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document.write( "We fail to reject the null hypothesis H0\r<BR>\n" );
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document.write( "Interpretation:\r<BR>\n" );
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document.write( "We fail to reject the null hypothesis H0.\r<BR>\n" );
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document.write( "So we don't have enough significant statistical evidence to prove that the mean age of the prison population in cone city is less than 26 years.\r<BR>\n" );
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document.write( "That means we must assume that the mean age of the prison population in cone city is 26 years (until we can statistically prove otherwise). </SPAN>\n" );
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