Penicillin is produced by the Penicillium fungus which is gr

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 batch 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 oers 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 alternative broth result in fewer days of production lost? Explain with evidence.

Solution

Normal Distribution
Mean ( u ) =4.9
Standard Deviation ( sd )=0.6
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X > 6) = (6-4.9)/0.6
= 1.1/0.6 = 1.8333
= P ( Z >1.833) From Standard Normal Table
= 0.0334 ~ 3.34%
b)
P(X > 6) = (6-5.2)/0.4
= 0.8/0.4 = 2
= P ( Z >2) From Standard Normal Table
= 0.0228 ~ 2.28% days

The reason behind the lost in days, the change in mean rate
is lower, standard deviation lies in the lower interval .