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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) 6= 30 532 OF = 1- 38 . 700 :. 02 Zigg: 2, 326 n= ( 2. 326 * 30 )2 = 1217.31 2 1 : 1218 (b) LLLLC cuts down eight trees from the location and determines the sample mean lumber yield is 692 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 23 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 from (d) to the margin of error from your interval from (b). Which is smaller? Why? (4 pts) (f) LLLLC has determined that harvesting lumber at this location will be profitable if the mean lumber yield of trees in the area exceeds 675 board feet, Determine an appropriate null and alternative hypothesis to test whether harvesting lumber at the new location would be profitable. (4 pts)