There is concern about the thickness of bearing plates being produced. If the thickness exceeds 150 mm ...

There is concern about the thickness of bearing plates being produced. If the thickness exceeds 150 mm then the production line must be shut down and the machine recalibrated. Suppose the thickness produced varies considerably throughout a shift, and has a normal distribution with mean µ = 160 mm and standard deviation σ = 10 mm

a. What is the probability that the thickness measurement of a single random sample will fail to detect that the machine is out of calibration?
 

b. If 5 samples are randomly drawn throughout a shift, what is the probability that the average of the 5 measurements will be less than 150 mm and hence fail to indicate that the machine is out of calibration?
 

c. How many measurements are required in a single shift so that there is a 1 percent probability of failing to detect that the machine is out of calibration?