Question

Please show all work, thank you in advance

1. In order to estimate the average age, enoted by H, of all eligible voters in Michigan Congressional District 14, two methods are used to form a point estimate Method 1: randomly sample an eligible voter in the District and use his/her age as the point estimate, denoted by 81 Method 2: randomly sample 10 eligible voters in the District. For each sampled voter i, take his/her age as xi, then let the point estimate be 02 (x+x10)/10 Method 3: Use θ3-(81 + θ2)/2 as the point estimate Which of the following statements are correct (circle out all that are correct)? A. Method 1 produces an unbiased estimator (call it Estimator 1) B. Method 2 produces an unbiased estimator (call it Estimator 2) C Method 3 produces an unbiased estimator (call it Estimator 3) D. The probability distribution of Estimator 1 is normal E. The probability distribution of Estimator 2 is normal F. The variance of Estimator 2 is smaller than the variance of Estimator 1 G. The variance of Estimator 3 is smaller than the variance of Estimator 2 H. If we repeat Method 1 n times and take the average of the point estimates o, thus obtained, i.e., ě-(81 + 7t 2 + + %)/n, this average will be more and more close to as we increase 2. Assume that the standard deviation of the age of eligible voters is known to be 12, if we repeat Method 2 100 times and therefore obtain 100 values for 02, the standard deviation of this list of 100 values is close to A. 12 8. 12/VTO С. 12/v 100 o. 12/VT000

Answer 1

(1) A, B, C , F, and H are correct

E(1)=, here n=1, so A is true and variance(1)= $sigma^2$ ,

E(2)=, here n=10, so B is true and variance(2)= $sigma^2$ /10

E(3)=, here n=2 so C is true and variance(3)= $sigma^2$ /2

since variance of 2 is less than variance of 1, so F is ture

but variance of 3 is not less than 2 so G is not true

here population distribution is not mentioned so it distribution of estimator may not be normal, so D and E are not true

(2)  right choice is

variance(2)= $sigma^2$ /10, and

standard deviation=since this is repeated 100 times so required answer would be