Question

Suppose the local Childhood Lead Poisoning Prevention Council in a

metropolitan area in a western United State undertakes the responsibility

of determining the proportion of homes in a certain development of 120

homes with unsafe lead levels. Because of the great expense involved in

performing spectrometric testing of interior walls, ceilings, floors,

baseboards, cabinets, and other obvious lead hazards such as crib bars,

as well as of exterior sidings, porches and porch rails, it was decided to

select a sample of homes for a study. A good up-to-date frame exists for

sampling purposes. This frame is a street listing containing the address

and owner of each home for each of the streets in the target area. It was

decided to select a one-in-three (1-in-3) sample of homes. Let us assume

that the only houses with serious lead hazard problems are the 26th, 27th,

28th, and 29th on the list.

a. Suppose the random number 2 was chosen to start the sequence.

Estimate the proportion of homes with lead hazards from the

sample.

b. Obtain a 95% confidence interval for the proportion of homes with

lead hazards. What assumptions did you make?

c. What is the true variance of the distribution of the estimated

proportion of homes with lead hazards? How does this compare

with the variance estimated in part (b)?

d. Suppose that a simple random sample of 40 homes had been

selected instead. What is the variance of the distribution of the

estimated proportion of lead hazardous homes in this case? How

does this value compare with the variance from a 1-in-3 systematic

sample?