Question

Assume that the machines' productivity of a factory has a normal distribution. The mean is 1600 units and the standard deviation is 150 units. The factory is testing four machines' performance. What is the probability that at least one of them reach the productivity of 1300? And what is the probability that none of them reach 1450? Please provide clear explanation, thank you!

Answer 1

    Let X be the productivity of a machine                      
   X follows normal distribution mean μ and standard deviation σ                      
   Given    μ = 1600 units       σ = 150 units          
                          
   Let A, B, C, D be the productivities of the 4 machines                      
a)    To find P(atleast 1 of the 4 machines reaches the productivity of 1300)                      
   P(atleast 1 of the 4 machines reaches the productivity of 1300)                      
       = 1 - P(none of the 4 machines reaches the productivity of 1300)                  
       = 1 - [P(A < 1300) * P(B < 1300) * P(C < 1300) * P(D < 1300)                  
       = 1 - P(X < 1300)4                  
   We use Excel function NORM.DIST to find the probabilities                      
       = 1 - [NORM.DIST(1300, 1600, 150, TRUE)]4                  
       = 1 - (0.02275)4                  
       = 0.9999                  
   P(atleast 1 of the 4 machines reaches the productivity of 1300) =                    
                          
b)   To find P(none of the 4 machines reaches the productivity of 1450)                      
   P(none of the 4 machines reaches the productivity of 1450)                      
       = P(A < 1450) * P(B < 1450) * P(C < 1450) * P(D < 1450)                  
       = P(X < 1450)4                  
   We use Excel function NORM.DIST to find the probabilities                      
       = [NORM.DIST(1450, 1600, 150, TRUE)]4                  
       = (0.15866)4                  
       = 0.0006                  
   P(none of the 4 machines reaches the productivity of 1450) =                    

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