Consider a machine that fills bottles with beer. Output is known to be normally distributed with a standard deviation σ of 0.2 ounces. The bottle labels state that the bottles contain 10 ounces of beer. (Thus, consider 10 to be the lower specification limit, or LSL.) Management can calibrate the machine to fill each bottle with a specified mean μ amount of beer. Management wants to keep μ relatively close to the LSL so that the company does not spend too much money filling bottles with extra beer. At the same time, management hopes to exceed the LSL most of the time so that customers are not shortchanged and so that the company does not find itself in legal trouble. Using the standard normal distribution, a Z-value can be computed to indicate the number of standard deviations that the LSL is below the mean: Z = (LSL − μ) /σ. This will be a negative value. Using Excel, the probability of output being less than the LSL is then =NORMSDIST(Z).

(a) Create an Excel model to calculate the probability of output being below the lower specification limit for any given value of μ.

(b) What is the probability of output being below the LSL for μ = 10.1 ounces?

(c) Use Goal Seek to find the value of μ that will cause output to fall below the LSL only 5% of the time.