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 problems working this problem, thanks!!

Solution

Set Up Hypothesis
Null Hypothesis , There Is No-Significance between them Ho: u1 = u2
Alternate Hypothesis, There Is Significance between them - H1: u1 != u2
Test Statistic
X(Mean)=785.5
Standard Deviation(s.d1)=145.19 ; Number(n1)=10
Y(Mean)=825.5
Standard Deviation(s.d2)=133.92; Number(n2)=10
we use Test Statistic (t) = (X-Y)/Sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =785.5-825.5/Sqrt((21080.1361/10)+(17934.5664/10))
to =-0.64
| to | =0.64
Critical Value
The Value of |t | with Min (n1-1, n2-1) i.e 9 d.f is 2.262
We got |to| = 0.64039 & | t | = 2.262
Make Decision
Hence Value of |to | < | t | and Here we Do not Reject Ho
P-Value: Two Tailed ( double the one tail ) - Ha : ( P != -0.6404 ) = 0.538
Hence Value of P0.05 < 0.538,Here We Do not Reject Ho

No evidence to indicate a difference between the mean number of miles