3. (12pts) For management purposes, it is important to estimate the total catch by recreational fishers. Unfortunately, there is no central registry of fishers, nor is there a central reporting station. Conse- quently, surveys are often used to estimate the total catch. An access survey was conducted to estimate the total catch at a lake in British Columbia. Fortunately, access to the lake takes place at a single landing site and most anglers use boats in the fishery. An observer was stationed at the landing site, but because of time constraints, could only interview a portion of the anglers returning, but was able to get a total count of the number of fishing parties on that day. A total of 168 fishing parties arrived at the landing during the day, of which 30 were sampled. The decision to sample an fishing party was made using a random number table as the boat returned to the dock. The objectives are to estimate the total number of catch and to estimate the proportion of boat trips (fishing parties) that had sufficient life-jackets for the members on the trip. Here is the raw data - each line is the results for a fishing party.. Sufficient Life Jackets? yes yes Party Catch 1 1 2 2 2 1 0 yes no no yes no 0 no 1 yes Number Anglers 1 3 1 1 3 3 1 1 1 1 2 1 2 1 3 1 1 2 3 1 2 1 0 0 1 0 2 3 0 0 0 1 0 0 yes yes yes yes yes yes no yes yes yes yes yes yes yes yes no yes no 1 0 1 1 0 0 0 1 1 2 2 1 1 1 no 1 0 yes yes 0 The following R output may help answer the questions below. > # Read in the data > creel head (creel) Anglers Catch Suff. Jackets 1 1 1 yes 2 3 1 yes 3 1 2 yes 4 1 2 no 5 3 2 no 6 3 1 yes > sum (creel$Anglers); var (creel$Anglers) [1] 46 [1] 0.6022989 > sum (creel$Catch); var (creel$Catch) [1] 20 [1] 0.7126437 > sum (creel $Suff.Jackets == "yes"); var (creel $Suff. Jackets == [1] 22 [1] 0.2022989 "yes") (a) (4pts) What is the population of interest? (b) (8pts) In simple random sampling, E] = 0. (1) Construct an unbiased estimator total for the total number of catch, and prove that it is indeed unbiased. You may NOT use (1) in your proof. Instead, use first principles (e.g. construct Zi as in the lecture notes). Obtain an unbiased estimate of the total number of catch using your estimator. Unbiased estimator for the total number of catch: total = Proof that it is indeed unbiased: Unbiased estimate of the total number of catch: total = 3. (12pts) For management purposes, it is important to estimate the total catch by recreational fishers. Unfortunately, there is no central registry of fishers, nor is there a central reporting station. Conse- quently, surveys are often used to estimate the total catch. An access survey was conducted to estimate the total catch at a lake in British Columbia. Fortunately, access to the lake takes place at a single landing site and most anglers use boats in the fishery. An observer was stationed at the landing site, but because of time constraints, could only interview a portion of the anglers returning, but was able to get a total count of the number of fishing parties on that day. A total of 168 fishing parties arrived at the landing during the day, of which 30 were sampled. The decision to sample an fishing party was made using a random number table as the boat returned to the dock. The objectives are to estimate the total number of catch and to estimate the proportion of boat trips (fishing parties) that had sufficient life-jackets for the members on the trip. Here is the raw data - each line is the results for a fishing party.. Sufficient Life Jackets? yes yes Party Catch 1 1 2 2 2 1 0 yes no no yes no 0 no 1 yes Number Anglers 1 3 1 1 3 3 1 1 1 1 2 1 2 1 3 1 1 2 3 1 2 1 0 0 1 0 2 3 0 0 0 1 0 0 yes yes yes yes yes yes no yes yes yes yes yes yes yes yes no yes no 1 0 1 1 0 0 0 1 1 2 2 1 1 1 no 1 0 yes yes 0 The following R output may help answer the questions below. > # Read in the data > creel head (creel) Anglers Catch Suff. Jackets 1 1 1 yes 2 3 1 yes 3 1 2 yes 4 1 2 no 5 3 2 no 6 3 1 yes > sum (creel$Anglers); var (creel$Anglers) [1] 46 [1] 0.6022989 > sum (creel$Catch); var (creel$Catch) [1] 20 [1] 0.7126437 > sum (creel $Suff.Jackets == "yes"); var (creel $Suff. Jackets == [1] 22 [1] 0.2022989 "yes") (a) (4pts) What is the population of interest? (b) (8pts) In simple random sampling, E] = 0. (1) Construct an unbiased estimator total for the total number of catch, and prove that it is indeed unbiased. You may NOT use (1) in your proof. Instead, use first principles (e.g. construct Zi as in the lecture notes). Obtain an unbiased estimate of the total number of catch using your estimator. Unbiased estimator for the total number of catch: total = Proof that it is indeed unbiased: Unbiased estimate of the total number of catch: total =