Question

A sample of Alzheimer's patients is tested to assess the amount of time in stage IV sleep these patients get in a 24-hour period. Number of minutes spent in Stage IV sleep is recorded for 61 patients. The mean stage IV sleep over a 24 hour period of time for these 61 patients was 48 minutes with a standard deviation of 14 minutes.

(a)     Compute 95% confidence interval for mean stage IV sleep. Interpret this confidence interval.

(b) It has been believed that individuals suffering from Alzheimer's Disease may spend less time per night in the deeper stages of sleep. Test the hypothesis at 5% significance level if the true mean stage IV sleep of Alzheimer patients is less than 50 minutes.

1. Provide the hypothesis statement
2. Calculate the test statistic value
3. Determine the probability value

(c) Could the confidence interval in part (a) be used to test the hypothesis in part (b)? Why or why not?

Answer 1

a)

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   60          
't value='   tα/2=   2.0003   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   14.0000   / √   61   =   1.792516
margin of error , E=t*SE =   2.0003   *   1.79252   =   3.585566
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    48.00   -   3.585566   =   44.414434
Interval Upper Limit = x̅ + E =    48.00   -   3.585566   =   51.585566
95%   confidence interval is (   44.41   < µ <   51.59   )

b)

Ho :   µ =   50                  
Ha :   µ <   50       (Left tail test)          
                          
Level of Significance ,    α =    0.05                  
sample std dev ,    s =    14.0000                  
Sample Size ,   n =    61                  
Sample Mean,    x̅ =   48.0000                  
                          
degree of freedom=   DF=n-1=   60                  
                          
Standard Error , SE = s/√n =   14.0000   / √    61   =   1.7925      
t-test statistic= (x̅ - µ )/SE = (   48.000   -   50   ) /    1.7925   =   -1.12   
                          
p-Value   =   0.1345   [Excel formula =t.dist(t-stat,df) ]              
Decision:   p-value>α, Do not reject null hypothesis                       

c)

Yes confidence interval can be used to test the hypthesis as 50 comes under the confidence interval.

Please revert back in case of any doubt.

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