Question

A trucking company would like to compare two different routes for efficiency. Truckers are randomly assigned to two different routes. Twenty truckers following Route A report an average of 49 ​minutes, with a standard deviation of 5 minutes. Twenty truckers following Route B report an average of 54 ​minutes, with a standard deviation of 3 minutes. Histograms of travel times for the routes are roughly symmetric and show no outliers.
​a) Find a​ 95% confidence interval for the difference in the commuting time for the two routes.
​b) Does the result in part​ (a) provide sufficient evidence to conclude that the company will save time by always driving one of the​ routes? Explain.
​a) The​ 95% confidence interval for the difference in the commuting time for the two routes muBminusmuA is ​(
nothing ​minutes,
nothing ​minutes).

Answer 1

SE = sqrt[ (s12/n1) + (s22/n2) ]              
(s12/n1)   0.4500          
(s22/n2)    1.2500          
SE   1.3038          
              
DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }              
[ (s12 / n1)2 / (n1 - 1) ]            0.011  
[ (s22 / n2)2 / (n2 - 1) ]           0.08  
(s12/n1 + s22/n2)2           2.89  
DF =    31          
              

t value at 95% = 2.040

x1 = 54 , s1 = 3 , n1 = 20
x2 = 49 , s2 = 5 , n2 = 20

CI = (x1 -x2) +/- t *sqrt(s1^2/n1+s2^2/n2)
= (54 - 49) +/- 2.040 *sqrt(3^2/20 + 5^2/20)
= (2.3408 , 7.6592 )

The 95% CI for the two routes muBminusmuA is ​(2.3408 , 7.6592 )

b)

Yes, because the confidence interval does not contain 0