thank you in advance!

2. What is the average salary offered to a graduate? To study this question you and a friend interview N students that graduated last year, and ask them what they earn. Student i's response was recorded as Y. You are interested in the average, MY. You assumed that the sample of Y's is iid. First you calculate the following estimate of MY: N Yi . 1=1 You and your friend each collected half the data. Thus you collected Y1, . .., YN/2 and your friend collected YN/2+1, . .., YN. Unfortunately, it turns out that your friend collected the data at a wild alumnae party, and you suspect that these data may not be as precise as your data. So whereas the variance of your data is , var(Y.) = 02, i = 1, . . . , N/2. then your friends data have the variance, var(Y;) = 02(1 + 3c), i= N/2 +1,. . ., N, for some constant c 2 0. (a) Is j an unbiased estimator of My? [Prove it is, or prove it is not]. (b) What is the variance of ful? (c) Is 1 a consistent estimator of My? [Prove it is, or prove it is not]. (d) Your friend is sorry that half the data are not as precise as they could have been, and suggest that you discard the noise data, and simply use 12 = N/Z ZLIY MY, as your estimator for My. Which estimator is most efficient (has the smallest variance) /1 or /12? Does your answer depend on c? (e) Suppose now that c = 0 such that var()() = 0 for i = 1, . .., N. You have N = 300 observation and calculate s? = N, DM, (Y. - 1) 20, 000, 000 and 1 = $48, 000. Before collecting the data, your friend argues mean salary, Ay, is s $50, 000, using a 1% significance level. Write down the confidence interval at 1% significance level and decide whether you will accept your friend's argument