Question

P1: A startup company uses a novel synthetic biology approach to produce opioids by using engineered yeast. The facility has k=8 reactors working in parallel. They produce opioids in batches. The whole process takes a day with the technician Jessie starting the bioreactor in the morning and Mr. White finishing the process by purifying the opioid in the afternoon. The yield of the process (for one reactor) is well described by a normal distribution with mean 800 mg and standard deviation 180 mg. The reactors are identical and operated independently from each other. The total (all reactors combined) yield for today’s batch was 6000 mg. Mr. White suspects that Jessie is stealing opioid and confronts him. The answer from Jessie was: “Dude, 6000 total mg equals an average of 750 mg per reactor. This is less than one sigma away from the expected yield. Random variation dude!” Mr. White smells baloney.

a) How many times a year you expect a reactor to produce 750 mg or less? (consider 1 year = 365 days) Please show your calculation.

b) How many times a year you expect to see an average yield of 750 mg or less? (consider 1 year = 365 days) Please show your calculation.

c) Does Jessie have a point? Discuss briefly.

Answer 1

a)

for normal distribution z score =(X-μ)/σ
here mean=       μ=	800
std deviation   =σ=	180.0000

P( a reactor to produce 750 mg or less )

probability =

P(X<750)

=

P(Z<-0.28)=

0.3897

expected number of times a year you expect a reactor to produce 750 mg or less =365*8*0.3897=1138

b)

sample size       =n=	8
std error=σx̅=σ/√n=	63.6396

P( an average yield of 750 mg or less )

probability =

P(X<750)

=

P(Z<-0.79)=

0.2148

  expected number of times a year you expect to see an average yield of 750 mg or less =365*0.2148=78

c)Yes as probability of above event is higher than 0.05 ; therefore it can not be termed as unusual