In a water pollution study, a sample of lakes is selected from the 320 lakes in a study region by the following procedure. A rectangle of length l and width w was drawn around the study region on a map. Pairs of random numbers between 0 and 1 were generated from the uniform distribution using a random number generator on a computer. The first random number of a pair was multiplied by l and the second by w to give location coordinates within the study region. If the location was in a lake, that lake was selected. This process was continued until four of the points had fallen on lakes. The first lake in the sample was selected twice by this process, while each of the other two was selected just once. The pollutant concentrations (in parts per million) for the three lakes in the sample were 2, 5, and 10. The respective sizes of the three lakes (in km2) were 1.2, 0.2, and 0.5. In all, 80 km2 of the study region was covered by lakes.
(a) Describe concisely the type of design used.
(b) Give an unbiased estimate of the mean pollution concentration per lake in the population.
(c) Estimate the variance of the estimator above.