Question

In Project 1 this Spring, you studied recording times in minutes for a sample of Dr. Marion’s 33 recordings of Gustav Mahler’s 1st Symphony: 55 49 50 52 53 56 55 53 57 59 52 55 50 50 62 53 58 58 63 51 52 51 54 65 60 57 57 56 54 56 47 48 52

Using σ = 4, and your sample mean, estimate the probabilities that:

i. Mean recording time of 64 recordings exceeds 55 minutes.

ii. Total recording time of 36 recordings exceeds 30 hours.

Answer 1

a)

µ =    54.545                                  
σ = 4
n=   64                                  
                                      
X =   55                                  
                                      
Z =   (X - µ )/(σ/√n) = (   55   -   54.545   ) / ( 4 / √   64   ) =   0.91
                                      
P(X ≥   55   ) = P(Z ≥   0.9 ) =   P ( Z <   -0.91 ) = 0.1814
excel formula for probability from z score is =NORMSDIST(Z)      

b)

Total time = 30 *60 minutes = 1800minutes

Mean = 54,545 * 36 = 1963.62                  

µ =    1963.62                  
σ = 4 * sqrt(36) = 24   
                      
P ( X ≥   1800.00   ) = P( (X-µ)/σ ≥ (1800-1963.62) / 24)              
= P(Z ≥   -6.8175) = P( Z <   6.8175) =

0.9999857

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