The usual ratio estimates for a population mean use information on an auxiliary variable to improve the estimation. The standard randomization analysis, however, does not take enough account of the observations on the auxiliary variable, so estimates and estimated variates can be conditionally biased unless the sample is balanced with respect to the auxiliary variable. I propose a method of adjustment for the bias of the estimate and its variance based only on the assumption of simple random sampling, which uses an approximate conditional distribution of the estimate given the mean of the auxiliary variable. Roy all and Cumberland (1981) illustrated empirically that the usual random sampling estimates and estimated variances could be badly conditionally biased when samples were obtained that were unbalanced in the auxiliary variate. So if the sample mean and population mean, both of which are known at the time of analysis, are not close, it is known that the results are biased, but the usual random sampling formulas, which are purely unconditional, do not take account of this. They suggested that prediction models were necessary to adjust these. If the joint distribution of the sample means of the variate and covariate are considered together, then using the fact that these are asymptotically bivariate normal, it is possible to adjust approximately for the linear regression of the variate mean on the mean of the auxiliary variate. In this way a bias-adjusted estimate is suggested and formulas for the conditional variances of the estimates are obtained. A simulation study of two populations, one where the model on which the ratio estimate is based holds and one where it does not, is carried out to illustrate the improvements given by the results and to compare them with results based on the formulas obtained by Royall and Cumberland (1981) by using prediction models, showing that results based on the assumption of simple random sampling can provide adequate theoretical results.