A population has N people, with ID numbers from 1 to N. Let yj be the value of some

Chapter 4, Problem 49

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A population has N people, with ID numbers from 1 to N. Let yj be the value of some numerical variable for person j, and y = 1 N XN j=1 yj be the population average of the quantity. For example, if yj is the height of person j then y is the average height in the population, and if yj is 1 if person j holds a certain belief and 0 otherwise, then y is the proportion of people in the population who hold that belief. In this problem, y1, y2,...,yn are thought of as constants rather than random variables. A researcher is interested in learning about y, but it is not feasible to measure yj for all j. Instead, the researcher gathers a random sample of size n, by choosing people one at a time, with equal probabilities at each stage and without replacement. Let Wj be the value of the numerical variable (e.g., height) for the jth person in the sample. Even though y1,...,yn are constants, Wj is a random variable because of the random sampling. A natural way to estimate the unknown quantity y is using W = 1 n Xn j=1 Wj . Show that E(W )=y in two dierent ways: (a) by directly evaluating E(Wj ) using symmetry; (b) by showing that W can be expressed as a sum over the population by writing W = 1 n XN j=1 Ij yj , where Ij is the indicator of person j being included in the sample, and then using linearity and the fundamental bridge.