Consider the problem of estimating a common mean of two independent normal distributions, each with unknown variances. Note that the problem of recovery of interblock information in balanced incomplete blocks designs is such a problem. Suppose a random sample of size m is drawn from the first population and a random sample of size n is drawn from the second population. We first show that the sample mean of the first population can be improved on (with an unbiased estimator having smaller variance), provided m ≧ 2 and n ≧ 3. The method of proof is applicable to the recovery of information problem. For that problem, it is shown that interblock information could be used provided b ≧ 4. Furthermore for the case b = t = 3, or in the common mean problem, where n = 2, it is shown that the prescribed estimator does not offer improvement. Some of the results for the common mean problem are extended to the case of K means. Results similar to some of those obtained for point estimation, are also obtained for confidence estimation.