Question

A small study was conducted to test the effect of bright light and melatonin on the mood, sleep patterns and overall well-being of female dementia patients. The group of patients was given a daily dose of 2.5 mg of melatonin (a hormone that controls circadian rhythm) and exposed to natural sun light and bright fluorescent lights during normal day light hours.

Melatonin	No Melatonin
16	15
18	11
8	12
10	10
16	13
16	14
14	7
8	13
21	12

Listed above are the subjective scores for well-being of the patients during the course of the study and 3 months before the treatments. Assume symmetry.

Analyze appropriately at alpha of 0.05 to determine if there is any significant difference between the effects of bright light and melatonin.

Suppose that you mistakenly believed that the data sets were independent samples taken from two different populations: different female dementia patients were tested for the effects of bright light and melatonin.

Carry out the t test and determine whether you would have found that female dementia patients who were given bright light and melatonin would have a significantly higher subjective scores for well-being.

Explain the cause of any differences in your results when compared to the previous problem.

Answer 1

1)

Ho :   µd=   0                  
Ha :   µd ╪   0                  
                          
Level of Significance ,    α =    0.05       claim:µd=0          
                          
sample size ,    n =    9                  
                          
mean of sample 1,    x̅1=   14.111                  
                          
mean of sample 2,    x̅2=   11.889                  
                          
mean of difference ,    D̅ =ΣDi / n =   2.222                  
                          
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    4.8677                  
                          
std error , SE = Sd / √n =    4.8677   / √   9   =   1.6226      
                          
t-statistic = (D̅ - µd)/SE = (   2.222222222   -   0   ) /    1.6226   =   1.370
                          
Degree of freedom, DF=   n - 1 =    8                  

                          
p-value =        0.208028   [excel function: =t.dist.2t(t-stat,df) ]               
Conclusion:     p-value>α , Do not reject null hypothesis                      

there is not any significant difference between the effects of bright light and melatonin.

2)

Ho :   µ1 - µ2 =   0                  
Ha :   µ1-µ2 >   0                  
                          
Level of Significance ,    α =    0.05                  
                          
Sample #1   ---->   Melatonin
mean of sample 1,    x̅1=   14.11                  
standard deviation of sample 1,   s1 =    4.54                  
size of sample 1,    n1=   9                  
                          
Sample #2   ---->   Without Melatonin
mean of sample 2,    x̅2=   11.89                  
standard deviation of sample 2,   s2 =    2.37                  
size of sample 2,    n2=   9                  
                          
difference in sample means =    x̅1-x̅2 =    14.1111   -   11.9   =   2.22  
                          
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    3.6209                  
std error , SE =    Sp*√(1/n1+1/n2) =    1.7069                  
                          
t-statistic = ((x̅1-x̅2)-µd)/SE = (   2.2222   -   0   ) /    1.71   =   1.302
                          
Degree of freedom, DF=   n1+n2-2 =    16                  

p-value =        0.105693   [excel function: =T.DIST.RT(t stat,df) ]              
Conclusion:     p-value>α , Do not reject null hypothesis                      
               

there is not any significant difference between the effects of bright light and melatonin.

Thanks in advance!

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