In experimental designs the quantities investigated are often grouped into blocks as a method of obtaining a higher precision for the experiment. This grouping may result in high correlation among observations within the same block. If the positions of the treatments within a block are assigned at random, the amount of correlation is difficult to determine and may vary greatly between treatments and from block to block. Also for many experiments the variances of the observations may differ substantially between treatments and from block to block. When one or both of these complications arise, thet-statistic is not necessarily applicable for comparing the effects of the treatments under investigation. This paper presents some nonparametric tests of Student's hypothesis which are usually valid for a well-known type of experimental design if there is statistical independence among blocks (number of blocks ≧4). These nonparametric results are reasonably efficient (compared to those based on thet-statistic) for the case where the totality of observations are independent, normally distributed, and have the same variance. High precision can sometimes be obtained by designing the experiment to yield large positive correlation within blocks and then using the non-parametric results.