Question

Q: Calculate the mean, standard deviation, and 95% confidence limits for each set.

A	B	C	D	E	F
3.5	70.24	.812	2.7	70.65	.514
3.1	70.22	.792	3.0	70.63	.503
3.1	70.10	.794	2.6	70.64	.486
3.3		.900	2.8	70.21	.497
2.5			3.2		.472

Q2: A type of steel contains 1.12% nickel and the standard deviation is 0.03%. The following data are collected (in percent)

a) 1.10 b) 1.08 c) 1.09 d) 1.12 e) 1.09

Is there an indication of bias in the method at the 95% level?

Answer 1

1)

for A

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   0.3742
Sample Size ,   n =    5
Sample Mean,    x̅ = ΣX/n =    3.1000

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   4          
't value='   tα/2=   2.776   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   0.3742   / √   5   =   0.1673
margin of error , E=t*SE =   2.7764   *   0.1673   =   0.4646
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    3.10   -   0.464588   =   2.6354
Interval Upper Limit = x̅ + E =    3.10   -   0.464588   =   3.5646
95%   confidence interval is (   2.64   < µ <   3.56   )

------------

for B)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   0.0757
Sample Size ,   n =    3
Sample Mean,    x̅ = ΣX/n =    70.1867

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   2          
't value='   tα/2=   4.303   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   0.0757   / √   3   =   0.0437
margin of error , E=t*SE =   4.3027   *   0.0437   =   0.1881
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    70.19   -   0.188096   =   69.9986
Interval Upper Limit = x̅ + E =    70.19   -   0.188096   =   70.3748
95%   confidence interval is (   70.00   < µ <   70.37   )

-------------

for C)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   0.0511
Sample Size ,   n =    4
Sample Mean,    x̅ = ΣX/n =    0.8245

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   3          
't value='   tα/2=   3.182   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   0.0511   / √   4   =   0.0256
margin of error , E=t*SE =   3.1824   *   0.0256   =   0.0814
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    0.82   -   0.081360   =   0.7431
Interval Upper Limit = x̅ + E =    0.82   -   0.081360   =   0.9059
95%   confidence interval is (   0.74   < µ <   0.91   )

--------------

for D)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   0.2408
Sample Size ,   n =    5
Sample Mean,    x̅ = ΣX/n =    2.8600

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   4          
't value='   tα/2=   2.776   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   0.2408   / √   5   =   0.1077
margin of error , E=t*SE =   2.7764   *   0.1077   =   0.2990
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    2.86   -   0.299032   =   2.5610
Interval Upper Limit = x̅ + E =    2.86   -   0.299032   =   3.1590
95%   confidence interval is (   2.56   < µ <   3.16   )

---------

for E)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   0.2152
Sample Size ,   n =    4
Sample Mean,    x̅ = ΣX/n =    70.5325

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   3          
't value='   tα/2=   3.182   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   0.2152   / √   4   =   0.1076
margin of error , E=t*SE =   3.1824   *   0.1076   =   0.3424
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    70.53   -   0.342360   =   70.1901
Interval Upper Limit = x̅ + E =    70.53   -   0.342360   =   70.8749
95%   confidence interval is (   70.19   < µ <   70.87   )

----------

for F)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   0.0161
Sample Size ,   n =    5
Sample Mean,    x̅ = ΣX/n =    0.4944

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   4          
't value='   tα/2=   2.776   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   0.0161   / √   5   =   0.0072
margin of error , E=t*SE =   2.7764   *   0.0072   =   0.0200
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    0.49   -   0.019994   =   0.4744
Interval Upper Limit = x̅ + E =    0.49   -   0.019994   =   0.5144
95%   confidence interval is (   0.47   < µ <   0.51   )