Trying to improve the techniques I use in the classroom and

Trying to improve the techniques I use in the classroom and reduce the anxiety of the students, I decided to try an experiment. I decided that I would incorporate more relaxation techniques prior to exams. For a comparison, I subjected the students to twice the exams (evil I know). In essence, I did a pre-and post-test. The scores are above. at 0.05, can it be concluded that the relaxation techniques helped improve test scores?

Solution

85   87   2
72   70   -2
91   92   1
56   68   12
80   79   -1
94   93   -1
82   86   4
78   72   -6
68   70   2

Let

group 2 = relax
group 1 = no relax

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    right   tailed test.      
              
Calculating the standard deviation of the differences (third column):              
              
s =    4.273774855          
              
Thus, the standard error of the difference is sD = s/sqrt(n):              
              
sD =    1.424591618          
              
Calculating the mean of the differences (third column):              
              
XD =    1.222222222          
              
As t = [XD - uD]/sD, where uD = the hypothesized difference =    0   , then      
              
t =    0.857945678          
              
As df = n - 1 =    8          
              
Then the critical value of t is              
              
tcrit =    +   2.306004135      
              
Thus, comparing t < tcrit, we   WE FAIL TO REJECT THE NULL HYPOTHESIS.          
              

Thus, there is no significant evidence that the relaxation techniques helped improve test scores. [CONCLUSION]