# How Defect Rate Changes After Inspection

A

#### ali2233

Notation :
N lot size
M defects in lot
n sample size
m defects in lot
Pa probability of acceptance
C0 per unit inspection cost
C1 recall cost if items slips through inspection

I am working on acceptance sampling. I was wondering how defect rate changes after inspection. for example , i have lot size N , with M defects. so if i don't do inspection , I will have recall cost give as , total cost = N*M*C1, where C1 is recall cost. M have Binomial distribution.

But if i do inspection , using sampling method , i have inspection cost as well a portion of recall cost , as I use Probability of acceptance.

what defect rate I should use after inspection , let say i took a sample n and got m defects in the sample ,, so how M , m are related,

Is it alway N/M = n/m or what ?

if yes there is no point of inspection.

if I set sampling plan with acceptance number based on M , using Hyper geometric formula i calculate Pa.

so let say now my cost is cost = n*cC0 + (N-n)*M*Pa forget bout reject for now.

what different does it make if I still use M even after inspection.
so my point is how M and m are related , what happens to defect rate after inspection.

any help in this regard much highly be appropriated.

#### Stijloor

Super Moderator
A Quick Bump!

Can someone help?

Thank you very much!!

#### Steve Prevette

##### Deming Disciple
Super Moderator
Notation :
N lot size
M FRACTION defects in lot
n sample size
m FRACTION defects in SAMPLE?
Pa probability of acceptance
C0 per unit inspection cost
C1 recall cost if items slips through inspection

I am working on acceptance sampling. I was wondering how defect rate changes after inspection. for example , i have lot size N , with M fraction defects. so if i don't do inspection , I will have recall cost give as , total cost = N*M*C1, where C1 is recall cost. N*M have Binomial distribution.

You ONLY "know" the expected recall cost if you "know" M. If you don't do any sampling, you don't know what M is. So now you have an unpredicatable cost, except perhaps from past history of recall costs. Another probability in here is - what is the chance the customer will detect the defect?

what defect rate I should use after inspection , let say i took a sample n and got m defects in the sample ,, so how M , m are related,

Is it alway N/M = n/m or what ?

Generally, you will be using m as the estimator for M. You will NEVER know the "true value" of M, even with 100% inspection. Hopefully you are plotting sequential m's on a control chart and understand how good of an estimator it is. M is unchanged by the inspection itself. For your estimate of OUTGOING QUALITY, you could revise the M subtracting off the failures detected and removed but its impact would usually be pretty minimal and you still have to account for probabilities of failures to detect and false alarms.

NOTE: It does look like in your original post that you are cross-mixing M as the number of defective versus M as the fraction defective.

#### Bev D

##### Heretical Statistician
Super Moderator
Steve is correct about the 'other' factors that effect the true cost..let me add to his excellent comments:

you can calculate a theoretical Average Outgoing Quality (AOQ) level that will give you the escapign defect rate. this assumes a lot:
1. The defect rate you use in the calcualtion is a close estimation of the true defct rate - from historical complaints or inspection findings.
You 100% inspect annd remove all defects from any lot that fails the inspection.
2. The actual 'miss' and detect rate for defects in your inspection lots is based solely on the 'AQL' probability levels; in other words that your ability to detect a defect when it is present in your sample is perfect.
3. The return - or warranty - rate is based on the Customer actually detecting and complaining about every defect that is present.
Given that none of these holds in real life, you will none the less get an idea of the defect/warranty rate with and without inspection. Be cautious that the real rate will deviate from the estimate.

Don't confuse the precision of the mathematical calculation with the accuracy of the prediction of the actual rates. It will probably be in the ballpark... Unfortunately, much of statistics isnt' about the formula it is about the complexities of the other factors...

Also understand that the financial cost of a failure in the field - while 10-100 times greater than the same defect detected in house - it is still insignificant compared to the effect on Customer loyalty and trust. ('Google' the sad story of the Intel Pentium...)

You can 'google' Average Outgoing Quality for further resources and the formula. to get you started you may want to start here.