Section A
Q1.
(a) Consider a population of size N. Let S1 be a simple random sample of size n1 drawn without replacement. Another simple random sample S2 of size n2 was also drawn without replacement from the remaining population. (i) Find the probability of obtaining the combined sample S1 cup S2 from the population. (ii) Define hat{overline{Y}}_{lpha} = lpha hat{overline{Y}}_1 + (1 - lpha) hat{overline{Y}}_2, 0 < lpha < 1. Show that hat{overline{Y}}_{lpha} is an unbiased estimator for the population mean. Here hat{overline{Y}}_i is the mean of sample Si, i = 1, 2.
(b) Consider the multiple regression model with a set of linear equality restrictions binding the regression coefficients. (i) Derive the restricted regression estimator by minimizing the residual sum of squares under the set of restrictions. (ii) Obtain the bias of the restricted regression estimator when the restrictions may not be true. Show that the estimator is unbiased and satisfies linear restrictions provided the restrictions are true.
(c) (i) Explain the steps in constructing a consumer price index and discuss a method of its construction. (ii) Indicate the precautions required while using the consumer price index numbers.
