In a large city, there are 40% people living in the west side, and 60% people living in the east side. A researcher wishes to estimate the average annual family income y P in the city. Based on other sources, the following information is known: (i) the average annual family income on the east side is $2000 below y P and the average annual family income on the west side is $4000 above y P ; (ii) the variance of annual family incomes on the east side is 1410^6, and the variance of annual family incomes on the west side is 1610^6. To plan sampling schemes and determine sample sizes, the researcher needs to estimate certain variances and sampling allocation. Assume the cost of sampling each unit is the same on the east and west sides.

(a) (8 pts) If the researcher plans to obtain a single SRS, what is the estimated/guessed total variance in annual family income in the whole city?

(b) (8 pts) If the researcher plans to obtain a stratified sample from the east side and west side respectively, what is the estimated/guessed (combined from east and west sides) within- stratum variance?

(c) (9 pts) If the researcher plans to obtain a stratified sample with optimal allocation, what are the proportions of sampled units (sample sizes) from east side and west side respectively?

Report your final answers to two decimal places. You must show the key steps of your calculation and define notation used.