Question

A fuel injector assembly plant that runs two shifts per day simulated to estimate throughput. To validate the model, the resulting simulated throughput will be compared against the observed system behavior. By searching the previous year’s database records, it was estimated that the average throughput of the fuel injector assembly plant was 35.6 on a given day. Eight independent replications of the model were run, each of 30 days’ duration, with the following results for average throughput of the assembly plant:
38.9 32.0 37.4 35.1 39.8 31.9 34.2 36.5
(a) Develop and conduct a statistical test to evaluate whether model output is consistent with system behavior. Use the level of significance α = 0.05.
(b) If it is important to detect a difference of 2 throughput per day, what sample size is needed to have a power of 0.90? Interpret your results in terms of model validity or invalidity. (Use α = 0.05)

Answer 1

a)

Ho :   µ =   35.6                  
Ha :   µ ╪   35.6       (Two tail test)          
                          
Level of Significance ,    α =    0.050                  
sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   2.9596                  
Sample Size ,   n =    8                  
Sample Mean,    x̅ = ΣX/n =    35.7250                  
                          
degree of freedom=   DF=n-1=   7                  
                          
Standard Error , SE = s/√n =   2.9596   / √    8   =   1.0464      
t-test statistic= (x̅ - µ )/SE = (   35.725   -   35.6   ) /    1.0464   =   0.119
                          
  
p-Value   =   0.9083   [Excel formula =t.dist(t-stat,df) ]              
Decision:   p-value>α, Do not reject null hypothesis                       
Conclusion: There is not enough evidence that     model output is consistent with system behavior

b)

hypothesis mean,   µo =    35.6                              
                                      
Level of Significance ,    α =    0.05                              
std dev =    σ =    2.9596
power =    1-ß =    0.9                              
ß=       0.1                              
δ=   µ - µo =    2                              
                                      
Z (α/2)=       1.9600   [excel function: =normsinv(α/2)                          
                                     
Z (ß) =        1.2816   [excel function: =normsinv(ß)                         
                                      
sample size needed =    n = ( ( Z(ß)+Z(α/2) )*σ / δ )² = ( (   1.2816   +   1.960   ) *   3.0   /   2   ) ² =   23.01
                                      
so, sample size =        24