A data scientist from two retail chain store company is interested in estimating the revenue at the end of a Christmas promotion. Due to short of time in year 2021, 12 simple random sampled store revenues (in millions) from chain store company A and B are listed in the first table below. A summary statistic from company A stores during the Christmas promotion of year 2020 is listed in the second table below. The table consists of mean revenues (in millions), standard deviation in three areas as well as the number of VIP members in their sample. Comp- 2021 2020 Comp- any A any B 1 1 2 2 3 4 5 6 7 8 9 10 11 12 Total 3 4 5 6 7 8 9 10 11 12 Total Year 2020 Area X Area Y Area Z revenue 1 3 3 1 3 6 5 5 4 6 8 7 52 revenue 4 7 1 3 3 4 4 2 3 2 1 5 39 N 20 50 80 Yi - Txi -4.3333 -6.3333 1.6667 -3.0000 -1.0000 0.6667 -0.3333 2.3333 0.0000 3.3333 6.6667 0.3333 Ei=1(vi-rxi) =133.3333 n 5 15 20 VIP member 1 12 7 2021 revenue 5 10 9 3 6 5 5 4 6 8 7 8 76 Mean 1.56 7.00 4.62 2020 revenue 2 5 4 1 4 2 3 1 3 3 4 3 35 Yi - rxi 0.6571 -0.8571 0.3143 0.8286 -2.6857 0.6571 -1.5143 1.8286 -0.5143 1.4857 -1.6857 1.4857 -1(vi - rx;) = 22.7543 Standard Deviation 0.63 2.05 1.88 (a) Estimate the total revenue during Christmas promotion in 2020 for chain store A in all three areas and place a bound on the error of estimation. (b) A prior study shows the strata standard deviation are ox = 0.5, 0y = 3.5, and oz = 2. The data scientist finds that the cost of obtaining a response from area Z is double of obtaining a response in area X or Y. The cost per observation in area X and Y are estimated to be $50. Find the overall sample size n and the stratum sample sizes, nx, ny, and nz, that allow the data scientist to estimate, at minimum cost, the mean revenue from Christmas promotion in 2020 with a bound on the error of estimation equal to $0.5 million. (c) Suppose the data scientist plans to do another study for checking. Let the cost of obtaining an observation from the area Z lower to the same as area X and Y. Using the proportion of VIP member in the table above, estimate the population proportion p with only $9000 to spend on sampling, choose the sample size n that will give the desired bound at minimum variance. A data scientist from two retail chain store company is interested in estimating the revenue at the end of a Christmas promotion. Due to short of time in year 2021, 12 simple random sampled store revenues (in millions) from chain store company A and B are listed in the first table below. A summary statistic from company A stores during the Christmas promotion of year 2020 is listed in the second table below. The table consists of mean revenues (in millions), standard deviation in three areas as well as the number of VIP members in their sample. Comp- 2021 2020 Comp- any A any B 1 1 2 2 3 4 5 6 7 8 9 10 11 12 Total 3 4 5 6 7 8 9 10 11 12 Total Year 2020 Area X Area Y Area Z revenue 1 3 3 1 3 6 5 5 4 6 8 7 52 revenue 4 7 1 3 3 4 4 2 3 2 1 5 39 N 20 50 80 Yi - Txi -4.3333 -6.3333 1.6667 -3.0000 -1.0000 0.6667 -0.3333 2.3333 0.0000 3.3333 6.6667 0.3333 Ei=1(vi-rxi) =133.3333 n 5 15 20 VIP member 1 12 7 2021 revenue 5 10 9 3 6 5 5 4 6 8 7 8 76 Mean 1.56 7.00 4.62 2020 revenue 2 5 4 1 4 2 3 1 3 3 4 3 35 Yi - rxi 0.6571 -0.8571 0.3143 0.8286 -2.6857 0.6571 -1.5143 1.8286 -0.5143 1.4857 -1.6857 1.4857 -1(vi - rx;) = 22.7543 Standard Deviation 0.63 2.05 1.88 (a) Estimate the total revenue during Christmas promotion in 2020 for chain store A in all three areas and place a bound on the error of estimation. (b) A prior study shows the strata standard deviation are ox = 0.5, 0y = 3.5, and oz = 2. The data scientist finds that the cost of obtaining a response from area Z is double of obtaining a response in area X or Y. The cost per observation in area X and Y are estimated to be $50. Find the overall sample size n and the stratum sample sizes, nx, ny, and nz, that allow the data scientist to estimate, at minimum cost, the mean revenue from Christmas promotion in 2020 with a bound on the error of estimation equal to $0.5 million. (c) Suppose the data scientist plans to do another study for checking. Let the cost of obtaining an observation from the area Z lower to the same as area X and Y. Using the proportion of VIP member in the table above, estimate the population proportion p with only $9000 to spend on sampling, choose the sample size n that will give the desired bound at minimum variance.