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SummaryIn estimating the mean /~v of one variable in a bivariate normal distribution, the experimenter can use the other variable, ~, as an auxiliary variable to increase precision. In particular, if /~x is known, he can use the regression estimator. When /~= is unknown, a preliminary test can be performed and the estimator will be made to depend on the result of the preliminary test. The bias and mean square error of the preliminary test estimator are obtained and the relative efficiency is are discussed.