Suppose that a city has 90,000 dwelling units, of which 35,000 are houses, 45,000 are apartments, and 10,000 are condominiums.

a. You believe that the mean electricity usage is about twice as much for houses as for apartments or condominiums, and that the standard deviation is proportional to the mean so that S1 = 2S2 = 2S3. How would you allocate a stratified sample of 900 observations if you wanted to estimate the mean electricity consumption for all households in the city?

b. Now suppose 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 house dwellers, 25% of apartment dwellers, and 3% of condomium residents practice energy conservation, what is p for the population? What gain would the stratified sample with proportional allocation offer over an SRS, that is, what is Vprop (Ṕstr) / VSRS (ṔSRS)?

THerefore, we allocate a stratified sample of 900 observations if we wanted to estimate the mean electricity consumption for all households in the city

(b)

p={35000*0.45+45000*0.25+10000*0.12}{90000}=0.3133

V_{prop}({p}_{str})={h=1}^{H}W_{h}^2(1-f_{h})\{p_{h}(1-p_{h})}{n_{h}}

where

H=3, W_{h}={N_{h}}/{N}, f_{h}={n}/{N}

Then

V_{prop}({p}_{str})= ( 1-{n}/{N} \right ){1}{n}/{h=1}^{3}W_{h}p_{h}(1-p_{h})

V_{prop}({p}_{str})= ( 1-{1}{100}\{1}{900} \{7}{18}*0.45*0.55+\{1}{2}*0.25*0.75 +\{1}{9}*0.12*0.88\]

V_{prop}(\{p}_{str})=0.000221907

V_{SRS}(\{p}_SRS)=\{N-n}{N-1}\{p(1-p)}{n}= ({90000-900}/{90000-1} \{0.3133*(1-0.3133)}{900}

V_{SRS}(\hat{p}_SRS)={N-n}/{N-1}\{p(1-p)}{n}=0.0002366

VSRS(pˆSRS)/Vprop(pˆstr) =1.06645