Question

Q.A manufacturer produces large quantities of colored mugs. It is known from previous records that 6% of the production will be green. A random sample of 10 mugs was taken from the production line.

1.Define a suitable distribution to model the number of green mugs in this sample.

2.Find the probability that there were exactly 3 green mugs in the sample.

3.A random sample of 125 mugs was taken. Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using a Poisson approximation and a Normal approximation

Answer 1

Ans:

1)Use normal distribution with n=10 and p=0.06

2)

P(x=k)=10C_k*0.06^k*(1-0.06)^10-k

P(x=3)=10C₃*0.06³*(1-0.06)⁷=0.0168

3)

Poisson approximation:

mean rate=np=125*0.06=7.5

P(10<=X<=13)=P(X<=13)-P(X<=9)

=Poisson(13,7.5,true)-Poisson(9,7.5,true)

=0.9784-0.7764

=0.2020

normal approximation:

standard deviation=sqrt(125*0.06*(1-0.06))=2.655

z(13.5)=(13.5-7.5)/2.655=2.26

z(9.5)=(9.5-7.5)/2.655=0.75

P(10<=X<=13)=P(z<=2.26)-P(z<=0.75)=0.9880-0.7734=0.2147