If the lot size is 3000, the sample size is 140, and the probability of accepting each lot is 0.798, what is the ATI for this quality level of...

The ATI, or average total inspection, is the average number of items we end up inspecting per lot in the long run, assuming that we inspect a given sample from each lot every time, and then if one of those is bad we infer that the lot is bad and go on to inspect all of them.

Thus the ATI is an expectation value, summing two possible events:

(1) The sample goes well, in which case we inspect 140 items.
(2) One or more items in the sample is defective, in which case we inspect the whole lot, 3000 items.

We are given that the probability of accepting a lot is 0.798, so the probability of rejecting a lot and inspecting all items must be one minus that, or 0.202. From there we can calculate our expectation value, which is the ATI:
`ATI = (0.798)(140) + (0.202)(3000) = 111.72 + 606 = 717.72`

We will, on average, expect to inspect about 717.72 items per lot, so for example if there were 100 lots we'd inspect somewhere around 71,772 items. (The actual figure could be more or less than this, but that is the long-run average we expect to converge toward by the Law of Large Numbers.)