The fracture strength of a certain type of manufactured glas

The fracture strength of a certain type of manufactured glass is normally distributed with a mean of 512 MPa with a standard deviation of 14 MPa.

What is the probability that a randomly chosen sample of glass will break at less than 512 MPa?

What is the probability that a randomly chosen sample of glass will break at more than 536 Mpa?

What is the probability that a randomly chosen sample of glass will break at less than 541 MPa?

Solution

a)

As 512 is the mean, by symmetry, it is 0.5. [ANSWER, 0.5]

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b)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    536      
u = mean =    512      
          
s = standard deviation =    14      
          
Thus,          
          
z = (x - u) / s =    1.714285714      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.714285714   ) =    0.043238133 [ANSWER]

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c)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    541      
u = mean =    512      
          
s = standard deviation =    14      
          
Thus,          
          
z = (x - u) / s =    2.071428571      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   2.071428571   ) =    0.980840618 [ANSWER]