
Stratified Sampling
A city has 90000 dwellings: 35000 are homes, 45000 are apartments, 10,000 are condos.
i) You know the mean electricity use is roughly twice as much for homes as
for apartments or condos, and the standard deviation is proportional
to the mean so $\displaystyle S_1$ = $\displaystyle 2S_2$ = $\displaystyle 2S_2$. How would you distribute a stratiﬁed sample
of 900 observations if you wanted to approximate the mean electricity usage
for all homes in the city?
ii)Now imagine that you take a stratiﬁed random sample with proportional allocation
and want to estimate the overall proportion of households in which energy
conservation is practiced. If 45% of homes, 25% of apartment,
and 3% of condos practice conserving energy, what is p for the
population? What gain would the stratiﬁed sample with proportional allocation
offer over an SRS, that is, what is $\displaystyle \frac{V_{prop}({\hat{p_{str}}})}{V_{SRS}{(\hat{p_{ SRS}}})}$?
Solutions:
i. Not sure how to allocate the sample.
ii. I found p = .303333, however, I am unable to solve for $\displaystyle s^2$ = $\displaystyle \frac{n}{n1}$p(1p). This is necessary to complete the problem.

Re: Stratified Sampling
Sorry, should read $\displaystyle S_1 = 2S_2 = 2S_3$