Question

Construct a 98​% confidence interval to estimate the population mean with x =59 and σ=13 for the following sample sizes.

​a)	n	e=	33
​b)	n	=	42
​c)	n	=	60

​a) With 98​% ​confidence, when n=33 the population mean is between the lower limit of ? and the upper limit of ?

​(Round to two decimal places as​ needed.)

​b) With 98%confidence, when n=42 the population mean is between the lower limit of ? and the upper limit of ?

​(Round to two decimal places as​ needed.)

Answer 1

a)

Level of Significance ,    α =    0.02          
degree of freedom=   DF=n-1=   32          
't value='   tα/2=   2.4487   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   13.0000   / √   33   =   2.263010
margin of error , E=t*SE =   2.4487   *   2.26301   =   5.541381
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    59.00   -   5.541381   =   53.458619
Interval Upper Limit = x̅ + E =    59.00   -   5.541381   =   64.541381
98%   confidence interval is (   53.46   < µ <   64.54   )

b )

Level of Significance ,    α =    0.02          
degree of freedom=   DF=n-1=   41          
't value='   tα/2=   2.4208   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   13.0000   / √   42   =   2.005944
margin of error , E=t*SE =   2.4208   *   2.00594   =   4.855994
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    59.00   -   4.855994   =   54.144006
Interval Upper Limit = x̅ + E =    59.00   -   4.855994   =   63.855994
98%   confidence interval is (   54.14   < µ <   63.86   )

c)

Level of Significance ,    α =    0.02          
degree of freedom=   DF=n-1=   59          
't value='   tα/2=   2.3912   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   13.0000   / √   60   =   1.678293
margin of error , E=t*SE =   2.3912   *   1.67829   =   4.013182
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    59.00   -   4.013182   =   54.986818
Interval Upper Limit = x̅ + E =    59.00   -   4.013182   =   63.013182
98%   confidence interval is (   54.99   < µ <   63.01   )

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