Question

An experimenter examining decision-making ability asked 10 children to solve as many problems as they could in 10 minutes. One group was told that this was a test of their innate problem-solving ability; a second group was told that this was just a time-filling task.

a. Does the mean number of problems solved differ between the two groups? Perform the correct hypothesis test with α = .05 using the 5 steps.

Innate	Time-filling
4	11
5	6
8	9
3	7
7	9

b. How large is the treatment effect? Answer this using percentage of variance explained and provide a brief explanation.

c. Are the variances homogenous? Numerically justify your answer.

Answer 1

A)

Ho :   µ1 - µ2 =   0          
Ha :   µ1-µ2 ╪   0          
                  
Level of Significance ,    α =    0.05          
                  
Sample #1   ---->   1          
mean of sample 1,    x̅1=   5.40          
standard deviation of sample 1,   s1 =    2.073644135          
size of sample 1,    n1=   5          
                  
Sample #2   ---->   2          
mean of sample 2,    x̅2=   8.400          
standard deviation of sample 2,   s2 =    1.95          
size of sample 2,    n2=   5          
                  
difference in sample means = x̅1-x̅2 =    5.400   -   8.4000   =   -3.0000
                  
std error , SE =    √(s1²/n1+s2²/n2) =    1.2728          
t-statistic = ((x̅1-x̅2)-µd)/SE = (   -3.0000   /   1.2728   ) =   -2.3570
                  
Degree of freedom, = 7          
                  
                     
p-value =        0.0506 (excel function: =T.DIST.2T(t stat,df) )      
Conclusion:     p-value>α , Do not reject null hypothesis              

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B)

effect size=t²/(t²+df)  

r²=   0.410

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C)

Null and alternative hypothesis:  
Hₒ : σ₁² = σ₂²  
H₁ : σ₁² ≠ σ₂²  

Test statistic:  
F = s₁² / s₂² = 4301.476 / 3.798601 =    1132.38
Degree of freedom:  
df₁ = n₁-1 =    4
df₂ = n₂-1 =    4

P-value :  
P-value = 2*F.DIST.RT(1132.3843, 4, 4) =    0.0000
Conclusion:  
As p-value < α, we reject the null hypothesis.  

VARIANCES ARE NOT EQUAL

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