Question

In: Statistics and Probability

Suppose that, in baseball, the distance of fly balls hit to the outfield is normally distributed...

Suppose that, in baseball, the distance of fly balls hit to the outfield is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. If 49 fly balls are randomly sampled:

a) What is the probability that a single fly ball travels more than 270 feet?

b) What is the probability that all 49 balls traveled an average of more than 270 feet? Sketch the graph of the distribution with 2 axes, one representing distance and the other representing z-score. Shade the region corresponding to the probability.

c) Find the 70th percentile of the distribution of the average of 49 fly balls to the nearest foot.

Solutions

Expert Solution

a)

µ =    250                                  
σ =    50                                  
n=   1                                  
                                      
X =   270                                  
                                      
Z =   (X - µ )/(σ/√n) = (   270   -   250   ) / (    50   / √   1   ) =   0.400
                                      
P(X ≥   270   ) = P(Z ≥   0.40   ) =   P ( Z <   -0.400   ) =    0.3446      

b)

µ =    250                                      
σ =    50                                      
n=   49                                      
                                          
X =   270                                      
                                          
Z =   (X - µ )/(σ/√n) = (   270   -   250   ) / (    50   / √   49   ) =   2.800  
                                          
P(X ≥   270   ) = P(Z ≥   2.80   ) =   P ( Z <   -2.800   ) =    0.0026           (answer)

c)

µ =    250                              
σ =    50                              
n=   1                              
proportion=   0.7000                              
                                  
Z value at    0.7   =   0.524   (excel formula =NORMSINV(   0.70   ) )          
z=(x-µ)/(σ/√n)                                  
so, X=z * σ/√n +µ=   0.524   *   50   / √    1   +   250   =   276.22 ≈ 276


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