A question on False positives, statistical power, heterogeneity of the treatment effect. see below for etails please

Suppose that we have a model of earnings (outcome y) on some human capital Variables X (eg: education, experience) and some variables representing true individual ability Z. y = XB + Zo+ & The modeling challenge lies in the fact that the ability variable is unobserved and we have to omit it from our empirical model. (a) What are the implications of the omitted Z variable on the OLS sample Estimator b? Specifically, derive the expression for E[b]. (5 marks) (b) The expression you derived can be broken up into two components. Point these components out and explain their meaning in the context of our earnings-education-ability example. (5 marks) (c) Denote education as X1, experience as x2 and unobserved ability as x3. Suppose we have a proxy for unobserved ability in the form of an IQ variable, denoted as X3. Explain what happens when we plug IQ into our empirical model. In particular, explain which parameter estimates are unbiased and which ones are biased. (4 marks)We have discussed the difference between the "population average treatment effect", the "average treatment effect on the treated" and the "local average treatment effect". Please, explain which one of these concepts we identify when our empirical strategy is (i) a randomized control trial, (ii) instrumental variables, (iii) differences-in-differences, and (iv) a regression discontinuity design. (10 marks)