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 $\displaystyle s^2$ = $\displaystyle \frac{n}{n-1}$p(1-p). This is necessary to complete the problem.