document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=332017>Question 332017</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Silver Gym’s aerobic class grew so large that the class had to be split in half and offered on two different nights. Silver also hired a new aerobic leader to handle the Tuesday night class, whereas an existing staff member continued to handle the Thursday night class. Concerned about customer satisfaction with the new leader, Silver Gym’s management decided to secure a leader-approval rating from each class after four weeks. Results are shown in the table below:<BR>\n" );
document.write( "·        <BR>\n" );
document.write( "·                             Tuesday                 Thursday<BR>\n" );
document.write( "       Class size              38                      41<BR>\n" );
document.write( "       Mean Approval Rating    87.3                    89.7<BR>\n" );
document.write( "       Standard Deviation      7.4                     6.3\r<BR>\n" );
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document.write( "Did the new leader of the Tuesday night class receive a lower approval rating than the leader of the Thursday night class? Set up and test your hypotheses at a = 0.05. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #237967 by </B><A HREF=/tutors/aboutme.mpl?userid=jrfrunner><B>jrfrunner(365)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jrfrunner><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=jrfrunner><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=237967\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let Mu(n)= Population mean rating of new trainer<BR>\n" );
document.write( "Let Mu(e)= Population mean rating of existing trainer<BR>\n" );
document.write( "Let Xbar(n) = sample average rating for new trainer<BR>\n" );
document.write( "Let Xbar(e) = sample average rating for existing trainer<BR>\n" );
document.write( "Let S(n)=sample std dev for new trainer<BR>\n" );
document.write( "Let S(e)=sample std dev for existing trainer<BR>\n" );
document.write( "Let n(n)= sample size for new trainer<BR>\n" );
document.write( "Let n(e)= sample size for exiting trainer<BR>\n" );
document.write( "--<BR>\n" );
document.write( "to answer<BR>\n" );
document.write( "Did the new leader of the Tuesday night class receive a lower approval rating than the leader of the Thursday night class? <BR>\n" );
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document.write( "Ho: Mu(n)=Mu(e)<BR>\n" );
document.write( "Ha: Mu(n) < Mu(e)\r<BR>\n" );
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document.write( "at alpha =0.05<BR>\n" );
document.write( "lower tail test<BR>\n" );
document.write( "---<BR>\n" );
document.write( "Find the test statistic<BR>\n" );
document.write( "This is a two independent samples t test.  There are two t tests, one if the population variances are equal and one if the population variances are not equal<BR>\n" );
document.write( "----<BR>\n" );
document.write( "To determine the t test to use, you should conduct a side hypothesis test on the population variances<BR>\n" );
document.write( "Ho: Variance (n) =Variance (e)<BR>\n" );
document.write( "Ha: Variance (n) not equla variance (e)\r<BR>\n" );
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document.write( "test statistic:  F=max(var(n),var(e))/min(var(n),var(e)) = (7.4/6.3)^2  =1.38<BR>\n" );
document.write( "critical value is F(alpha, dof numerator, dof denominator) = F(.05,37,40)=1.71\r<BR>\n" );
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document.write( "since test statistic=1.38 < critical value=1.71 cannot reject Ho: variances are equal.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "Since we now established that its likely the pop variances are equal, we may pool the asmple variances\r<BR>\n" );
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document.write( "pooled sample variance = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Sp%5E2=%28%2838-1%29%2A7.4%5E2%2B%2841-1%29%2A6.3%5E3%29%29 ALIGN=MIDDLE  ALT=\"Sp%5E2=%28%2838-1%29%2A7.4%5E2%2B%2841-1%29%2A6.3%5E3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >/(38+41-2)=46.93<BR>\n" );
document.write( "----<BR>\n" );
document.write( "test statistic: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t=%28Xbar%28n%29-Xbar%28e%29%29%2F%28Sp%2Asqrt%281%2Fn%28n%29+%2B+1%2Fn%28e%29%29%29 ALIGN=MIDDLE  ALT=\"+t=%28Xbar%28n%29-Xbar%28e%29%29%2F%28Sp%2Asqrt%281%2Fn%28n%29+%2B+1%2Fn%28e%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%2887.3-89.7%29%2F%286.85%2Asqrt%281%2F38%2B1%2F41%29%29 ALIGN=MIDDLE  ALT=\"t=%2887.3-89.7%29%2F%286.85%2Asqrt%281%2F38%2B1%2F41%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =-2.4/1.54=-1.558\r<BR>\n" );
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document.write( "critical value = t(1-0.05,38+41-2)=-1.66\r<BR>\n" );
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document.write( "Since our test statistic= -1.558 > critical value = -1.66, it falls in the fail to reject Ho.\r<BR>\n" );
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document.write( "pvalue = P(t<=-1.558) = 0.062 > alpha =0.05<BR>\n" );
document.write( "---<BR>\n" );
document.write( "Conclusion\r<BR>\n" );
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document.write( "Cannot reject the hypothesis that the mean ratings between trainers is the same.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );