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Penicillin is produced by the Penicillium fungus,which is
Chapter 4, Problem 11E(choose chapter or problem)
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.
Questions & Answers
QUESTION:
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.
ANSWER:
Answer:
Step 1 of 2:
(a)
In this question, we are asked to find on what proportion of days will the process shut down.
Sugar concentration in batches of brothe is normally distributed.
The optimum sugar concentration (mean ) is and standard deviation is
If the concentration exceeds , the fungus dies and the process must be shut down for the day.
Let represent the proportion of days.
We need to find .
Hence we can write,
Now we will calculate the score because our distribution is approximately normal.
thus the score of 6.0 is
=
= 1.833
= 1.8334
From the z table, the area to the left of is 0.9664.
Therefore
Hence the process will be shut down on 3.36% of days.