Consider a population of size N = 20 such that yz- = a + bi, i = 1, . . . ,20, for some positive constants a and b. Such a population is said to have a linear trend. We consider systematic samples of size n = 4. a Show that the variance of 1? under systematic sampling is equal to 800132. 2; (b) Show that the variance of t; under simple random sampling Without replacement is equal to 2800b2. Hint: Start with the expression fo the variance of it; under both sampling designs. Remark: This exercise sugests that, in the case of a linear trend, systematic sampling is more efcient than simple random sampling without replacement