2. What is the average salary offered to a Stony Brook college graduate? To study this...

    

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2. What is the average salary offered to a Stony Brook college 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, py. You assumed that the sample of Y's is iid. First you calculate the following estimate of py: 1 = 1 N Y. i=1 You collected of the data and your friend collected of the data. Thus, you collected Y,..., YN and your friend collected Y2N+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) = o, i = 1, 7 2N 3 then your friends data have the variance, var (Y) = o(1 + 3c), 2N i = +1,..., N, 3 for some constant c 0. (a) Isp an unbiased estimator of uy? [Prove it is, or prove it is not]. (b) What is the variance of ? (c) Is a consistent estimator of py? [Prove it is, or prove it is not]. (d) Your friend is sorry that of the data are not as precise as they could have been, and suggest that you discard the noise data, and simply use N/3 2N/3 Y, as your estimator for y. Which estimator is most efficient (has the smallest variance) 1 or 2? Does your answer depend on c? = 2 , (. - ) (e) Suppose now that c = 0 such that var(Y) = for i have N 100 observation and calculate s 50,000,000 and = $45,300. Before collecting the data, your friend argues mean salary, py, is $46, ZZZ, using a 5% significance level. In answering the question, replace ZZZ with the last three digits of your student ID number. (If my id number is 123456789, I would have to suppose mean salary, py, as $46, 789, and solve the problem based on that.) Write down the confidence interval at 5% significance level and decide whether you will accept your friend's argument. 1,..., N. You 2. What is the average salary offered to a Stony Brook college 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, py. You assumed that the sample of Y's is iid. First you calculate the following estimate of py: 1 = 1 N Y. i=1 You collected of the data and your friend collected of the data. Thus, you collected Y,..., YN and your friend collected Y2N+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) = o, i = 1, 7 2N 3 then your friends data have the variance, var (Y) = o(1 + 3c), 2N i = +1,..., N, 3 for some constant c 0. (a) Isp an unbiased estimator of uy? [Prove it is, or prove it is not]. (b) What is the variance of ? (c) Is a consistent estimator of py? [Prove it is, or prove it is not]. (d) Your friend is sorry that of the data are not as precise as they could have been, and suggest that you discard the noise data, and simply use N/3 2N/3 Y, as your estimator for y. Which estimator is most efficient (has the smallest variance) 1 or 2? Does your answer depend on c? = 2 , (. - ) (e) Suppose now that c = 0 such that var(Y) = for i have N 100 observation and calculate s 50,000,000 and = $45,300. Before collecting the data, your friend argues mean salary, py, is $46, ZZZ, using a 5% significance level. In answering the question, replace ZZZ with the last three digits of your student ID number. (If my id number is 123456789, I would have to suppose mean salary, py, as $46, 789, and solve the problem based on that.) Write down the confidence interval at 5% significance level and decide whether you will accept your friend's argument. 1,..., N. You

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