Gosh! This is hard, without any context. I don't understand the topic really. But I can rewrite the last sentence, and see if it helps you: "*the need for each such source is extrapolated backwards from the use to which the test is to be put.*" => "if you find out what the test is going to be used for, you can then work out whether each source is needed". Does that help?!

Your sentence is perfect: "When he arrived, his wife was asleep." You can also say: "When he arrived, his wife was sleeping." To my British ears, this one sounds a little more literary, a bit more formal. But they are both correct. You can't say "When he arrived, his wife was slept". This is grammatically wrong. You CAN say "When he arrived, his wife slept." It's a bit weird though. It sounds like the moment he came back through the door, she fell asleep immediately, at that moment. If you want to say that she was asleep BEFORE he got home, you need to say, "When he arrived, his wife was asleep" or "When he arrived, his wife was sleeping". Hope that helps!

Hi Mohamed, 'a big deal' means an important matter. We wouldn't normally say 'a big deal of coffee'. But you can say 'a great deal of coffee' = 'a lot of coffee'.