Suppose that a city has 90000 dwelling units, of which 35000 are houses, 45000 are apartments, and 10000 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 appartment, and \$16 for condominiums. Suppose that from an earlier study, we know that 47% of house dwellers, 23% of appartment dwellers, and 3% of condominum 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.
(a) (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 stratas, while proportional allocation is used when costs are equal and variances are equal in all stratas. But in this question, neither cost or variance are equal, then how to apply the formula for Neyman and proprotional allocation? Just ignore the cost and variance and set them equal or there is another formula for that?