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Exam 2

STAT 3470 Spring 2022

1. Let x1 . x2 . 7 7 7 . xn be a random sample from a continuous distribution with pdf

f (z) =

where 9 > 1 is an unknown parameter of interest.

(a) Find the mean of x1 . (3 pts)

(b) Find the method of moments estimator 9˜ for 9 . (4 pts)

(hint: start by finding the (joint) log-likelihood for the sample)

(d) The mean of 9ˆ is E(9ˆ) = 9, and its variance is Var(9ˆ) = . Is 9ˆ biased? Determine the mean squared error (MSE) of 9ˆ. (4 pts)

2. Larry’s Loggers LLC (LLLLC) recently acquired a license to harvest lumber from a new location, and want to estimate the lumber yield per tree at this new location. Assume that the lumber yield of trees at this location follows a Normal distribution, and that each tree’s yield is independent (so the yield from each tree in any collection of felled trees represents a random sample from a normal distribution).

(a) From historical data, LLLLC expects the standard deviation of trees’ lumber yield

to be at most 30 board feet. Assuming this is the exact standard deviation, what is the minimum number of trees that would need to be sampled to determine the average lumber yield with a margin of error of at most 2 board feet, at the 98% confidence level? (3 pts)

(b) LLLLC cuts down eight trees from the location and determines the sample mean

lumber yield is 683 board feet. Assuming the same standard deviation as in (a), determine a 98% confidence interval for the mean lumber yield of a tree from this location. Interpret the interval. (5 pts)

(c) The sample standard deviation for these 8 trees’ yield was 21 board feet. Determine a 98% confidence interval for the true standard deviation of the yield of trees in this location. Based on this interval, was the assumption from (a) and (b) reasonable? (5 pts)

(d) Regardless of your result in (c), determine a 98% confidence interval for the mean lumber yield of a tree from this location without assuming the exact standard deviation is known. (based on the same sample of 8 trees used in (b) and (c)) (3 pts)

(e) Compare the margin of error from your interval in (d) to the margin of error from

your interval in (b). Which is smaller? Why? (3 pts)

3. For each statement below, determine whether the statement is true or false. Circle your answer if you are writing your solutions on this document. If you are writing your solutions in a separate document, write TRUE or FALSE for each statement.

(a) TRUE FALSE The sample variance for a normal random sample is an unbiased

estimator of the true variance. (3 pts)

(b) TRUE FALSE If you obtain 95% confidence intervals for different parameters

in each of 100 independent experiments, 5 of those confidence intervals will not contain the true parameter value, on average. (3 pts)

(c) TRUE FALSE Recall that for a random sample y1 . y2 . 7 7 7 . yn Exp(入) from an exponential distribution with rate 入, the maximum likelihood estimator for 入 is given by = 11 Yi = ()− 1 . For large n, the sampling distribution of is approximately normal. (3 pts)

(d) TRUE FALSE Let x1 . x2 . 7 7 7 . xn N(μ . u2 ) be a random sample from a normal distribution with known variance. The margin of error for a 98% confidence interval for μ is greater when n = 87 than when n = 23. (3 pts)