Question

Based on a problem in Diez, Barr, and Cetinkaya-Rundel (2015). The fuel economy of randomly selected cars with automatic and manual transmissions, manufactured in 2012, are compared in the table below in miles per gallon (MPG) for city driving. The data are from the U.S. EPA. Test whether these data provide strong evidence of a difference between the average fuel efficiency of cars with manual and automatic transmissionsin terms of their average city 0.02. mileage. Assume that conditions for inference are satisfied. Use a significant level a Statistic Automatic Manual х 19.85 16.12 3.58 4.51 S 26 26 п Approximate answer: p-value = 0.002

Answer 1

Ho :   µ1 - µ2 =   0                  
Ha :   µ1-µ2 ╪   0                  
                          
Level of Significance ,    α =    0.02                  
                          
Sample #1   ---->   1                  
mean of sample 1,    x̅1=   16.120                  
standard deviation of sample 1,   s1 =    3.580                  
size of sample 1,    n1=   26                  
                          
Sample #2   ---->   2                  
mean of sample 2,    x̅2=   19.850                  
standard deviation of sample 2,   s2 =    4.510                  
size of sample 2,    n2=   26                  
                          
difference in sample means =    x̅1-x̅2 =    16.1200   -   19.9   =   -3.730  
                          
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    4.0716                  
std error , SE =    Sp*√(1/n1+1/n2) =    1.1293                  
                          
t-statistic = ((x̅1-x̅2)-µd)/SE = (   -3.7300   -   0   ) /    1.13   =   -3.3030
                          
Degree of freedom, DF=   n1+n2-2 =    50                  
p-value =        0.0018 (excel function: =T.DIST.2T(t stat,df) )              
Conclusion:     p-value <α , Reject null hypothesis