1. Quality control at a manufacturing plant consists of testing batches of 100 u

Question

1. Quality control at a manufacturing plant consists of testing batches of 100 units, randomly selected as units come off the assembly line. If 6 or more units in the batch are defective, production is halted and the machinery adjusted. We could define the defect threshold as the maximum independent probability of individual units being defective such that the expected number of defective units in the batch is 6. What is the defect threshold of this quality control process?

2. Production will continue if 5 or fewer units in the batch are defective. If we assume that units are defective with probability equal to the defect threshold found above, what is the probability that production will continue?

3. Production is halted if 6 or more units in the batch are defective. If we assume that units are defective with probability equal to the defect threshold found above, what is the probability production will be halted?

4. Noticing that stopping production is more likely than not, an enterprising quality control worker suggests changing the process. For every lot of 1000 units produced, he proposes randomly selecting a batch of 30. If 6 or more of the 30 units are defective, then the whole lot should be discarded. Assume that the percentage of defective units in the lot is equal to the defect threshold above, and find the probability that a lot will be discarded.

5. We claimed that the binomial distribution is a good model for the hypergeometric distribution, in the right circumstances. Suppose samples of size 50 are being tested. Consider units defective with probability p = 0.06, so that if the size of the population is N, M = N p units are defective. Use R to find the minimum population size N for which no corresponding values in the binomial and hypergeometric PMFs differ by more than 0.01. (In other words, find N for which |P(Xbinom = x) P(Xhyper = x)| 0.01, for all values of x in the sample space.)

6. Does your answer for #5 change if the defect probability p is changed?

Already have answerd 1-4, can someone solve 5 and 6??

Explanation / Answer

Solution:

5) Suppose the population size be N. A sample of size n=50 is taken from the population. The probability of defective, p=0.06.

Now pmf of Binomial distribution, f_binom=choose(n,x)*p^x*(1-p)^(n-x); x=0,1,2,...,n.

pmf of Hypergeometric distribution, f_hyper=choose(N*p,x)*(N(1-p),(n-x))/choose(N,n); x=0,1,2,...,n.

We need to find N such that |f_binom(x)-f_hyper(x)| is less than or equal to 0.01 for all x=0,1,2,...,n=50.

The answer is N=613.

The R-code is given below:

p=0.06; n=50
fbinom<-function(x){
return(choose(n,x)*p^x*(1-p)^(n-x))}

fhyper<-function(x,N){
return(choose(N*p,x)*choose(N*(1-p),(n-x))/choose(N,n))}

error<-function(N){
diff<-NULL
for(x in 0:n){
diff[x]=abs(fbinom(x)-fhyper(x,N))
}
return(max(diff))
}

N=51
while(error(N)>0.01){
N=N+1}
N

6) The answer for above question changes if the probability of defective "p" is changed.

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