A random sample of five pairs of observations were selected

A random sample of five pairs of observations were selected from two

populations. The sample data are in the following table:

Pair

Value from Population 1

Value from Population 2

1

28

22

2

31

27

3

24

20

4

30

27

5

22

20

Test H_{o :} µ_d = 0 vs. H_A : µ_d 0 with 5% significance.

Solution

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 =    2.1514618          
              
Thus, the standard error of the difference is sD = s/sqrt(n):              
              
sD =    0.962162967          
              
Calculating the mean of the differences (third column):              
              
XD =    -3.8          
              
As t = [XD - uD]/sD, where uD = the hypothesized difference =    0   , then      
              
t =    -3.949434897          
              
As df = n - 1 =    4          
              
Then the critical value of t is              
              
tcrit =    +/-   2.776445105      
              
Thus, As t < -2.776, we   WE REJECT THE NULL HYPOTHESIS.          
              
Also, using p values,              
              
p =        0.016825982      
              
Also, comparing this p < 0.05,   WE REJECT THE NULL HYPOTHESIS.