Question:Suppose that a specific population of individuals is composed of k different strata (k 鈮� 2), and that for i = 1,...,k, the proportion of individuals in the total population who belong to stratum i is pi, where pi > 0 and\(\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{p}}_{\bf{i}}}{\bf{ = 1}}} \). We are interested in estimating the mean value 渭 of a particular characteristic among the total population. Among the individuals in stratum i, this characteristic has mean\({{\bf{\mu }}_{\bf{i}}}\)and variance\({\bf{\sigma }}_{\bf{i}}^{\bf{2}}\), where the value of\({{\bf{\mu }}_{\bf{i}}}\)is unknown and the value of\({\bf{\sigma }}_{\bf{i}}^{\bf{2}}\)is known. Suppose that a stratified sample is taken from the population as follows: From each stratum i, a random sample of ni individuals is taken, and the characteristic is measured for each individual. The samples from the k strata are taken independently of each other. Let\({{\bf{\bar X}}_{\bf{i}}}\)denote the average of the\({{\bf{n}}_{\bf{i}}}\)measurements in the sample from stratum i.
a. Show that\({\bf{\mu = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{p}}_{\bf{i}}}{{\bf{\mu }}_{\bf{i}}}} \), and show also that\({\bf{\hat \mu = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{p}}_{\bf{i}}}{{{\bf{\bar X}}}_{\bf{i}}}} \)is an unbiased estimator of 渭.
b. Let\({\bf{n = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{n}}_{\bf{i}}}} \)denote the total number of observations in the k samples. For a fixed value of n, find the values for which the variance \({\bf{\hat \mu }}\)will be a minimum.
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