Question

5) A concrete plant takes 20 independent samples of concrete each month. The samples are cured and strength tested to evaluate quality control at the plant. Let N20 be the number of samples which fall below a strength of 20 MPa. In general the fraction of such low strength samples is around 10%, and this is considered acceptable. However, if the samples taken in a particularmonth yield a value of N20 which exceeds its mean by more than two standard deviations, the plant operations are critically reviewed.

a) What is the probability of having to critically review the plant operations even though the actual fraction of low strength samples for that month is still 10%? (Answer: 0.04317)

b) If the actual fraction of low strength samples remains at 10%, what is the probability that the next critical review will take place in exactly 5 months? (Answer: 0.03619)

I am looking for the way to solve these problems as I already have the numerical, final answer. I believe I must use Poisson process but I am not sure how to set up the question. Any solution/explanation is greatly appreciated.

Answer 1

here this is binomial distribution with paramter p=0.1 and n=20

a)mean =np=20*0.1=2

and std deviaiton=sqrt(np(1-p))=1.34

hence 2 std deviation above values =mean+2*std deviation=2+2*1.34=4.68

P( probability of having to critically review the plant operations) =P(getting 5 or more failures)

=P(X>=5)=1-P(X<=4)

=1-(P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4))

=

-

=1-(0.12158+0.27017+0.28518+0.19012+0.08978) =1-0.95683=0.04317

b)

probability that the next critical review will take place in exactly 5 months

=P(in first 4 month defect are below critical value and in 5th month above critical value)

=(1-0.04317)⁴*0.04317=0.03619