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?
Best Answer
Proportional allocation is exactly that - proportional. Sampling cost does not matter here, the only information you need to calculate $n$ is the stratum weights, a bound, and stratum variances or variance estimate.
The stratum weights are $W_h= \frac{N_h}{N}$ where $h=1,2,3$. $N_h$ is the size of stratum $h$ and $N$ is the population size. In this case, $N= 90,000 $ and $W_1=\frac{35,000}{90,000}$, for example.
Your bound is B=0.1, which is fine.
You don't have a variance or variance estimate of each strata, so use the earlier study's information for this. For example, a variance estimate for stratum 1 would be $\hat{S}_1^2 = \hat{p}_1\hat{q}_1 = (0.47)(0.53)$.
Now, use the formula $$n= \frac{\sum_h W_h\hat{S}^2_h }{\frac{B^2}{4} + \frac{1}{N}\sum_h W_h\hat{S}^2_h}$$ and then calculate each $n_h = nW_h$, rounding where required.
Neyman Allocation ignores costs for these calculations as well because we assume they are equal. The calculation for $n$ is $$n= \frac{\sum_h W^2_h\frac{\hat{S}^2_h}{w_h} }{\frac{B^2}{4} + \frac{1}{N}\sum_h W^2_h\frac{\hat{S}^2_h}{w_h}}$$ Where $$w_h = \frac{W_h\hat{S}_h}{\sum_h W_h\hat{S}_h}$$
Note that the $\hat{S}$'s are not squared. Finally, we can calculate $n_h=nw_h$ and then you're done, since it sounds like you have optimal allocation covered.