Penicillin is produced by the Penicillium fungus, which is grown in a broth whose sugar content must be carefully controlled. The optimum sugar concentration is 4.9 mg/mL. If the concentration exceeds 6.0 mg/mL, the fungus dies and the process must be shut down for the day.
a. If sugar concentration in batches of broth is normally distributed with mean 4.9 mg/mL and standard deviation 0.6 mg/mL, on what proportion of days will the process shut down?
b. The supplier offers to sell broth with a sugar content that is normally distributed with mean 5.2 mg/mL and standard deviation 0.4 mg/mL. Will this broth result in fewer days of production lost? Explain.
(a) z = (6-4.9)/0.6 = 1.83. The area to the right of z= 1.83 is 1 ? 0.9664 = 0.0336.
The process will be shut down on 3.36% of days.
(b) z = (6-5.2)/0.4= 2.00 . The area to the right of z= 2.00 is 1 ? 0.9772 = 0.0228.
Since a process with this broth will be shut down on 2.28% of days, this broth will result in fewer days of production lost.
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