Can you use the same data twice to construct a confidence interval?


New Member
A while ago I was asked the following question:
What is with a 95% probability the maximum number of defects, given a population of size N=800 of which we inspected n=100 and found k=0 defects.

I constructed a confidence interval based on the assumption of a binomial distribution and gave the upper boundary as the answer.

Now we have inspected another 100 parts in which we found 1 defect.
My question is: can I 'update' the same analysis simply with n=200 (100+100) and k=1 (0+1)?
My guts tell me this isn't proper statistics, but I'm not sure why and what the proper way of doing this calculation is.

Thank you for your help!