Question

1) state your hypotheses; 2) draw a graph showing the critical z-scores and the rejection region; 3) calculate z-obtained; 4) make a conclusion.

Does regular exercise improve cognitive skills? A sample of 67 3rd grade students was randomly assigned into 2 groups. One group (N=32) participated in an afterschool fitness program 3 days a week. Another group (N = 35) attended a regular afterschool care program. After 3 months, the mean cognitive test score in the first group was 138 with a standard deviation of 24, and the mean cognitive test score in the second group was 135 with a standard deviation of 27. Use α=0.05.

Answer 1

For sample 1 :  
x̅1 =   138, s1=   24, n1 =   32
For sample 2 :  
x̅2 =   135, s2 =   27, n2 =    35

α =    0.05

Null and Alternative hypothesis:              
Ho : µ1 = µ2        
H1 : µ1 > µ2        
              
df  =n1+n2-2 = 65          
              
Critical value :              
At α = 0.05  and df = 65, right tailed critical value, t_c= ABS(T.INV(0.05   ,    65   ) )=   1.669

Rejection region: Reject Ho, if t > 1.669

c) Pooled variance :              
S²p = ((n1-1)*s1² + (n2-1)*s2² )/(n1+n2-2)              
=   656.0307692          
Test statistic:              
t = (x̅1 - x̅2)/[√(s²p(1/n1 + 1/n2 )]              
=    0.4789          
              
d) Decision: As t = 0.4789 < 1.669, we fail to reject the null hypothesis.

Conclusion:   
There is not enough evidence to claim that regular exercise improve cognitive skills at 0.05 significance level.