Thirty sixth graders were randomly selected from a school di

Thirty sixth graders were randomly selected from a school district. Then, they were divided into 15 matched pairs, each pair having equal IQ\'s. One member of each pair was randomly selected to receive special training. Then, all of the students were given an IQ test. Test results are summarized below. Do these results provide evidence that the special training helped or hurt student performance? Use a 0.05 level of significance. Assume that the mean differences are approximately normally distributed.

Solution

4.

Here are the entries for the Difference column:

-2
-4
-2
-3
-3
-2
5
4
3
2
1
0
-1
-5
-4

Let ud = u2 - u1.              

u1 = with training
u2 = without training

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 (4th column):              
              
s =    3.104528183          
              
Thus, the standard error of the difference is sD = s/sqrt(n):              
              
sD =    0.80158573          
              
Calculating the mean of the differences (third column):              
              
XD =    -0.733333333          
              
As t = [XD - uD]/sD, where uD = the hypothesized difference =    0   , then      
              
t =    -0.914853279          
              
As df = n - 1 =    14          
              
Then the critical value of t is              
              
tcrit =    +/-   2.144786688      
              
As |t| < 2.1448,   WE FAIL TO REJECT THE NULL HYPOTHESIS.          
              
Thus, there is no significant difference the means of the matched pairs. [CONCLUSION]