The following experiment was conducted to compare two coatin

The following experiment was conducted to compare two coatings designed to improve the durability of the soles of jogging shoes. A 1/8th inch thick layer of coating 1 was applied to one of a pair of shoes (randomly chosen from the pair), and a layer of equal thickness of coating 2 was applied to the other shoe of the pair. Ten joggers were given pairs of shoes treated in this manner and were instructed to record the number of miles covered in each shoe before the 1/8th inch coating was worn through in any one place. The results are given in the Data Table below:

                       Data Table

Jogger                         Coating 1                            Coating 2

1                                      892                                      958

2                                      904                                      953

3                                      775                                      765

4                                      435                                      510

5                                      946                                      895

6                                      853                                      884

7                                      780                                      895

8                                      695                                      725

9                                      825                                      858

10                                    750                                      812

* At the 0.05 level of significance, do the data provide sufficient evidence to indicate a difference between the mean number of miles of wear that a runner might expect from the two coatings?

****College graduate statistical methods problem. Please explain in detail as I am having difficulty working through, thanks!!

Solution

Consider the table, where the third column is the difference:

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 =    44.48561225          
              
Thus, the standard error of the difference is sD = s/sqrt(n):              
              
sD =    14.06758578          
              
Calculating the mean of the differences (third column):              
              
XD =    40          
              
As t = [XD - uD]/sD, where uD = the hypothesized difference =    0   , then      
              
t =    2.843416107          
              
As df = n - 1 =    9          
              
Then the critical value of t is              
              
tcrit =    +/-   2.262157163      
              
As |t| > 2.262, we REJECT THE NULL HYPOTHESIS.

Thus, there is significant evidence that indicate a difference between the mean number of miles of wear that a runner might expect from the two coatings. [CONCLUSION]