Question

A TV manufacture is supplied with a certain component by a specialist producer. Each incoming consignments is subject to the following quality control procedure. A random sample of 10 components is individually tested. If there are one or more defective components among the 10 tested, the entire consignment is rejected. If there are no defective components in the sample, the consignment is accepted.

i)What are the probabilities of a consignment being rejected if the true proportions of defective components are:

1) 1%

2) 10%

3) 30%

ii) if a sample of 20 components ( instead of 10) were tested,and the consignment rejected if two or more proved defective, calaculate the probabilities of rejecting a consignment for the same proportions of defective components( i.e 1%,10% and 30%)

iii) which quality control procedure do u think is the better

Answer 1

According to the given question, a random sample of 10 components is individually tested. If there are one or more defective components among the 10 tested, the entire consignment is rejected. If there are no defective components in the sample, the consignment is accepted.

Let us define be the event that the number of defective components in the sample.

Therefore follows binomial distribution with the probability mass function of

with pmf as:

where is the true proportions of defective components .

i) The probabilities of a consignment being rejected if the true proportions of defective components are:

1) 1% or true proportions of defective components is determined as:

  

  

  

Therefore the probabilities of a consignment being rejected if the true proportions of defective components is 1% is .

2) 10% or true proportions of defective components

  

  

  

Therefore the probabilities of a consignment being rejected if the true proportions of defective components is 10% is .

3) 30% or true proportions of defective components ​​​​​​​

  

  

  

Therefore the probabilities of a consignment being rejected if the true proportions of defective components is 30% is .

ii) The probabilities of a consignment being rejected if the true proportions of defective components are 1%, 10% and 30% when a sample of 20 components were tested,and the consignment rejected if two or more proved defective,:

1) 1% or true proportions of defective components is determined as:

  

  

  

Therefore the probabilities of a consignment being rejected if the true proportions of defective components is 1% is .

2) 10% or true proportions of defective components ​​​​​​​

  

  

  

Therefore the probabilities of a consignment being rejected if the true proportions of defective components is 10% is .

3) 30% or true proportions of defective components ​​​​​​​

  

  

  

Therefore the probabilities of a consignment being rejected if the true proportions of defective components is 30% is .

c) According to the calculation the required probability are as follows:

true proportions of defective components	the probabilities of a consignment being rejected with sample 10 in problem (i)	the probabilities of a consignment being rejected with sample 20 in problem (ii)
1%	0.096	0.0168
10%	0.6513	0.6082
30%	0.9717	0.9923

Therefore on the basis of quality control procedure, if there are no defective components in the sample, the consignment is accepted, and hence lower the probability of defective item is accepted and hence for 1% and 10% true defective proportion, sample 20 has lower probability of rejecting assignment compare to sample 10 and when 30% true defective proportion, sample 20 has higher probability of rejecting assignment compare to sample 10.

Therefore for 1% and 10% true defective proportion sample 20 is better than sample 10 with their condition and for 30% sample 10 is better than the sample 20 with their condition of defective components.