TM 56 Inferences of mu1mu2 In matched pairs design The manag

TM 5.6. Inferences of mu1-mu2 In matched pairs design. The manager of a fleet of automobiles is testing two brands of radial tires and assigns one lire of each brand at random to the two rear wheels of eight cars and runs the cars until the tires wear out. The data (in kilometers) follow: Find a 95% confidence interval on the difference in mean lift. Which brand would you prefer? Test against using a = 0.05. What is your conclusion?

Solution

A)

Let ud = u2 - u1.              
Formulating the null and alternative hypotheses,              
              
Ho:   ud   =   0  
Ha:   ud   =/   0  
At level of significance =    0.05          
As we can see, this is a    two   tailed test.      
              
Calculating the standard deviation of the differences (third column):              
              
s =    1114.375688          
              
Thus, the standard error of the difference is sD = s/sqrt(n):              
              
sD =    393.9913029          
              
Calculating the mean of the differences (third column):              
              
XD =    868.375          

For the   0.95   confidence level,      
              
alpha/2 = (1 - confidence level)/2 =    0.025      
df = n - 1 = 8 - 1 = 7
  
t(alpha/2) =    2.364624252          
              
lower bound = [X1 - X2] - t(alpha/2) * sD =    -63.26638981          
upper bound = [X1 - X2] + t(alpha/2) * sD =    1800.01639          
              
Thus, the confidence interval is              
              
(   -63.26638981   ,   1800.01639   ) [ANSWER]

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b)
              
As t = [XD - uD]/sD, where uD = the hypothesized difference =    0   , then      
              
t =    2.204046114          
              
As df = n - 1 =    7          
              
Then the critical value of t is              
              
tcrit =    +/-   2.364624252      
              
As |t| < 2.364, then   WE FAIL TO REJECT THE NULL HYPOTHESIS.          

Thus, there is no significant evidence that the mean difference is not 0. Thus, we cannot prefer a brand over the other. [CONCLUSION]