6-Ratio Estimation - statisticalanalysis RatioEstimation 1...

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19/2/2013 1 Ratio Estimation 1 Use of auxiliary variable is common in statistical analysis Auxiliary variable can improve the efficiency of the estimate of the main interest 2 Example Wholesale price paid for oranges in large shipments is based on the sugar content of the load. The exact sugar content can be estimated by taking a random sample of n oranges. Let y i be the sugar content of the i th selected orange. Then, would estimate the total sugar content for the load. Unfortunately, it is time consuming and costly to determine N . 3 y N We could estimate the total by using the fact that the sugar content of an individual orange Y is closely related to its weight X The ratio of the total sugar content to the total weight of the truck load is equal to the ratio of the mean sugar content per orange to 4 the mean weight per orange. Note that and thus X Y T T X Y = Y where and are, respectively the population totals of variables Y and X . X Y T X T = Y T X T 5 We can estimate population means of Y and X by the sample mean of Y and X from the sample. Also we can measure the total weight of the oranges on the trucks. Thus we have X Y T x y t = 6
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