A brewery needs to purchase glass bottles that canwithstand an internal pressure of at least 150 pounds per squareinch (psi). A prospective bottle vendor claims its productionprocess yields bottles with a mean strength of 157 psi and astandard deviation of 3 psi.

a. Assume that the strength of thevendor’s bottles is normally distributed. Calculate the probabilitythat a single bottle chosen randomly from this vendor’s factorywould fail to meet the brewer’s standard, assuming the vendor’sclaim is true.

b.         The brewer testsa sample of 40 bottles from this vendor and finds the mean strengthof the sample to be 155.7. Assuming the vendor’s strength claim tobe true, what is the probability of obtaining such a sample with amean this far or farther below the claimed mean? What does thisanswer suggest about the veracity of the vendor’s claim?

c.          Which ofthe following changes would make observing a sample as described inpart b. more likely:

           i)          Reducingthe claimed mean to 156, holding the standard deviation at 3;or

           ii)         Reducing thestandard deviation to 2, holding the mean at 157?

           [Note: You do not have to calculate new probabilities toanswer this. But be sure you explain the reasoning behind youranswers fully using the appropriate formulas]