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
Ans:
1)Use normal distribution with n=10 and p=0.06
2)
P(x=k)=10Ck*0.06k*(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
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