Question

A government entity sets a Food Defect Action Level (FDAL) for the various foreign substances that inevitably end up in the foods we eat. The FDAL level for insect filth in peanut butter is 0.70 insect fragment (larvae, eggs, body parts, and so on) per gram. Suppose that a supply of peanut butter contains 0.70 insect fragment per gram. Compute the probability that the number of insect fragments in a 66-gram sample of peanut butter is (a) exactly two. Interpret the results. (b) fewer than two. Interpret the results. (c) at least two. Interpret the results. (d) at least one. Interpret the results. (e) Would it be unusual for a 6-gram sample of this supply of peanut butter to contain four or more insect fragments?

Answer 1

here expected number of insect fragments in a 6-gram sample =np=6*0.7 =4.2

a) from Poisson approximation:

probability that the number of insect  is  exactly two =P(X=2)=e^-4.2*4.2²/2! =0.1323

this means that if we take 100, 6 gram sample of  peanut butter then approximately 13 will gave exactly 2 number of insect

b)

P(X<2) =P(X=0)+P(X=1)=e^-4.2*4.2⁰/0!+e^-4.2*4.2¹/1! =0.0780

this means that if we take 100 , 6 gram sample of  peanut butter then approximately 8 will gave fewer than 2 number of insect

c)P(at least 2) =PX>=2)=1-P(X<2)=1-0.0780 =0.9220

this means that if we take 100 , 6 gram sample of  peanut butter then approximately 92 will gave at least 2 number of insect

d)

P(at least one)=P(X>=1)=1-P(X=0)=1-e^-4.2*4.2⁰/0! =1-0.0150 =0.9850

this means that if we take 100 , 6 gram sample of  peanut butter then approximately 99 will gave at least 1 number of insect

e)

No as probability of that is 0.6046 which is higher than 0.05 level