First let me say I am not a student asking for homework help. I have been perusing a old book of mine 'Introduction to the Theory of Statistics' (Mood 1950) and have come across the following problem at the end of one of the chapters. It is stated exactly as follows:"Suppose intelligence quotients for students in a particular age group are normally distruibuted about a mean of 100 with standard deviation of 15. The I.Q.,say x , of a particular student is to be estimated by a test on which he scores 130. It is further given that test scores are normally distributed about the true I.Q. as a mean with standard deviation 5. What is the maximum-likelihood estimate of the student's I.Q.? (The answer is not 130)."
Unless the answer is 110, I haven't a clue as to how to proceed.

I'm not much of a stats person, but your answer seems to make sense to me.

He is two full deviations away from the true IQ, and we find out that the deviations of the test he took are actually 1/3 the size. It would seem that in this case he is still two deviations away, but the test has a curve, making it 110.

I've never really understood MLE and barely the standard deviation curve. But that's my take on things.