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

a. How would you allocate a sample of 900 observations if you want to estimate the mean electricity consumption for all households in the city?

b. Now suppose that you want to estimate the overall proportion of households in which energy conservation is practiced. You have strong reason to believe that about 45% of house dwellers use some sort of energy conservation and that the corresponding percentages are 25% for apartment dwellers and 3% for condo- minium residents. What gain would proportional allocation offer over simple random sampling?

c. Someone else has taken a small survey, using an SRS, of energy usage in houses. On the basis of the survey, each house is categorized as having electric heating or some other kind of heating. The January electricity consumption in kilowatt-hours for each house is recorded (y1) and the results are given below:

Type of Heating        Number of Houses       Sample Mean       Sample Variance

Electric                            24                        972                     202,396

Nonelectric                       36                        463                       96,721

Total                               60

From other records, it is known that 16,450 of the 35,000 houses have electric heating, and 18,550 have nonelectric heating.

i Using the sample, give an estimate and its standard error of the proportion of houses with electric heating. Does your 95% CI include the true proportion?

ii Give an estimate and its standard error of the average number of kilowatt hours used by houses in the city. What type of estimator did you use, and why did you choose that estimator?