Two men A and B who usually commute to work together decide

Two men, A and B, who usually commute to work together, decide to conduct an experiment to see whether one route is faster than the other. The men believe that their driving habits are approximately the same, and therefore they decide on the following procedure. Each morning for 2 weeks, A will drive to work on one route and B will use the other route. On the first morning, A will toss a coin. If heads appear, he will use route I; if tails appear, he will use route II. On the second morning, B will toss the coin: heads, route I; tails, route II. The times, recorded to the nearest minute, are shown in the following table. Assuming commute times are normal, find the 95% confidence interval estimate for the mean of the differences _d, where d = route I route II. (Round your answers to two decimal places.)

Solution

Getting the differences, x1 - x2,

1
7
-1
1
8
3
2
-3
4
2

Getting the mean and standard deviation,  
  
Xbar =    2.4
s =    3.339993347

Note that              
              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
              
X = sample mean =    2.4          
t(alpha/2) = critical t for the confidence interval =    2.262157163          
s = sample standard deviation =    3.339993347          
n = sample size =    10          
df = n - 1 =    9          
Thus,              
              
Lower bound =    0.010712693          
Upper bound =    4.789287307          
              
Thus, the confidence interval is              
              
(   0.010712693   ,   4.789287307   )

or

(   0.01 ,   4.79 )