Suppose that a city has 90,000 dwelling units, of which 35.000 are houses, 45,000 are apartments, and 10,000 are condominiums. We want to estimate the overall proportion (p) of households in which energy conservation is practiced, with a bound on the error of estimation equal 0.1. The cost for obtaining an observation is \$9 for houses, \$10 for apartment, and \$16 for condominiums. Suppose that from an earlier study, we know that 47% of house dwellers, 23% of apartment dwellers, and 3% of condominium residents practice energy conservation.
(a) (4 marks) Using a proportional allocation, find the strata sample sizes, $n_1$, $n_2$, and $n_3$,
and the sample size n.
(b) (4 marks) Using a optimal allocation, find the strata sample sizes, $n_1$, $n_2$, and $n_3$, and the sample size n.
(c) (4 marks) Using a Neyman allocation, find the strata sample sizes, $n_1$, $n_2$, and $n_3$, and the sample size n.
Neyman allocation is used when the cost of obtaining an observation is the same for all strata, while proportional allocation is used when costs are unequal among strata but variances are equal in all strata. But in this question, neither cost or variance are equal, then how can I apply the formula for Neyman and proportional allocation? Just ignore the cost and variance and set them equal or is there another formula for that?