How many keyed in items must be checked to assume all remaining items were keyed in correctly? If I am keying in test responses which may be numbers 1-4 on a 344 item test, how many items must I check to be at least 99.9% sure that the remainder of the items were entered correclty?
 A: You have not actually specified a probability model. Without some probability model you cannot actually ask what is the probability of any event. 
The most immediate assumption is that a worker has a probability $p \in (0,1)$ of making a mistake. Each unit they complete can be modeled as a Bernoulli random variable with success probability $p \in (0,1)$ of incorrectly entering the information. Let $\alpha \in (0,1)$ be the desired level of accuracy. 
Given that I know $p$, the number of incorrectly entered items follows a Binomial distribution of $\text{Binom}(n,p)$. I can easily calculate the probability of no mistakes being made.
There are two cases:


*

*$p > \alpha$. Then even if you check 364 of the items, $P(X_{365} = 1) > \alpha$. So, you can never set a threshold better than what your worker can do on an individual question. 

*$p \leq \alpha$. Since each Bernoulli trial is independent if you check any of them it does not tell you about the others. Therefore, you would have to check the smallest $N$ of them such that $P(Y_N = 0) \geq \alpha$ where $Y_N \sim \text{Binom}(N,p)$. Since $p \leq \alpha$ I know that if I check $364$ I am guaranteed to have $P(X_{365} = 1) \leq \alpha$ so I will never have to check all of them. 


I hope this helps. Without some kind of probability model specified it is actually not possible to ask questions about probability. 
