Question

In: Statistics and Probability

1. The time required by a mechanic in a bicycle shop to assemble a certain type...

1. The time required by a mechanic in a bicycle shop to assemble a certain type of bicycle may be looked upon as a random variable with a mean distribution of 20.5 minutes and a standard deviation of 2.3 minutes. find the probabilities that the time required to assemble a bicycle is:

a. at least 20 minutes

b. at most 19.0 minutes

c. between 2.0 and 21.0 minutes

d. between 18.0 and 20 minutes

Solutions

Expert Solution

a)

µ =    20.5                  
σ =    2.3                  
                      
P ( X ≥   20   ) = P( (X-µ)/σ ≥ (20-20.5) / 2.3)              
= P(Z ≥   -0.22   ) = P( Z <   0.217   ) =    0.5860   (answer)

b)

µ =    20.5          
σ =    2.3          
              
P( X ≤    19   ) = P( (X-µ)/σ ≤ (19-20.5) /2.3)      
=P(Z ≤   -0.65   ) =   0.2571   (answer)

c)

µ =    20.5                                  
σ =    2.3                                  
we need to calculate probability for ,                                      
P (   2   < X <   21   )                      
=P( (2-20.5)/2.3 < (X-µ)/σ < (21-20.5)/2.3 )                                      
                                      
P (    -8.043   < Z <    0.217   )                       
= P ( Z <    0.217   ) - P ( Z <   -8.04   ) =    0.5860   -    0.0000   =    0.5860   (answer)

d)

we need to calculate probability for ,                                      
P (   18   < X <   20   )                      
=P( (18-20.5)/2.3 < (X-µ)/σ < (20-20.5)/2.3 )                                      
                                      
P (    -1.087   < Z <    -0.217   )                       
= P ( Z <    -0.217   ) - P ( Z <   -1.09   ) =    0.4140   -    0.1385   =    0.2754   (answer)


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