Question

Consider the data below:

302.0207, 306.9998, 300.8687, 270.4524, 259.5252, 288.0257, 283.3269, 294.9006, 292.8622, 322.8245, 329.5085, 313.5145, 307.6594, 325.7587, 354.1881, 324.6396, 276.2345, 283.0315, 278.8270, 298.1269

a. What is your BOOTSTRAP estimate of the mean and standard error of the mean?

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b. What is your two-sided 95% BOOTSTRAP CI for the mean?

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c. What is your one-sided lower 99% BOOTSTRAP CI for the mean?

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:

d. What is your one-sided upper 62% BOOTSTRAP CI for the mean?

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Answer 1

a)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   23.1688          
Sample Size ,   n =    20          
Sample Mean,    x̅ = ΣX/n =    300.6648          
                  
degree of freedom=   DF=n-1=   19          
                  
Standard Error , SE = s/√n =   23.1688   / √    20   =   5.1807

b)

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   19          
't value='   tα/2=   2.0930   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   23.1688   / √   20   =   5.180712
margin of error , E=t*SE =   2.0930   *   5.18071   =   10.843356
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    300.66   -   10.843356   =   289.821414
Interval Upper Limit = x̅ + E =    300.66   -   10.843356   =   311.508126
95%   confidence interval is (   289.82   < µ <   311.51   )

c)

Level of Significance ,    α =    0.01          
degree of freedom=   DF=n-1=   19          
't value='   tα/2=   2.8609   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   23.1688   / √   20   =   5.180712
margin of error , E=t*SE =   2.8609   *   5.18071   =   14.821679
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    300.66   -   14.821679   =   285.843091

d)

Level of Significance ,    α =    0.38          
degree of freedom=   DF=n-1=   19          
't value='   tα/2=   0.8988   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   23.1688   / √   20   =   5.180712
margin of error , E=t*SE =   0.8988   *   5.18071   =   4.656514
                  
confidence interval is                   

Interval Upper Limit = x̅ + E =    300.66   -   4.656514   =   305.321284

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