A very large batch of components has arrived at a distributo

A very large batch of components has arrived at a distributor. The batch can be characterized as acceptable only if the proportion of defective components is at most 0.10. The distributor decides to randomly select 10 components and to accept the batch only if the number of defective components in the sample is at most two.

What is the probability that the batch will be accepted when the actual proportion of defectives is

0.01? 0.05? 0.10? 0.20? 0.25?

repeat with replacing \"two\" with \"one\"

Solution

let X denotes the number of defective components in the sample.

so X~Bin(10,p) where p is the proportion of defectives.

we accept the batch only if X<=2.

case 1. [p=0.01]

so P[accepting the batch]=P[X<=2]   where X~Bin(10,0.01)

                                      =0.999886

replacing \"two\" with \"one\" P[X<=1]=0.995734

case 2 p=0.05

so P[accepting the batch]=P[X<=2]   where X~Bin(10,0.05)

                                      =0.988496

replacing \"two\" with \"one\" P[X<=1]=0.913862

case 3 p=0.10

so P[accepting the batch]=P[X<=2]   where X~Bin(10,0.10)

                                      =0.929809

replacing \"two\" with \"one\" P[X<=1]=0.736099

case 4 p=0.20

so P[accepting the batch]=P[X<=2]   where X~Bin(10,0.20)

                                      =0.677800

replacing \"two\" with \"one\" P[X<=1]=0.375810

case 4 p=0.25

so P[accepting the batch]=P[X<=2]   where X~Bin(10,0.25)

                                      =0.525593

replacing \"two\" with \"one\" P[X<=1]=0.244025