Questions pertaining to the admissibility of fixed sample size tests of hypotheses, when sequential tests are available, are considered. For the normal case with unknown mean, suppose the risk function is a linear combination of probability of error and expected sample size. Then any fixed sample size test, with sample size n ⩾ 2, is inadmissible. On the other hand, suppose the risk function consists of the pair of components, probability of error and expected sample size. Then any optimal fixed sample size test for the one sided hypothesis is admissible. When the variance of the normal distribution is unknown, t-tests are studied. For one-sided hypotheses and componentwise risk functions the fixed sample size t-test is inadmissible if and only if the absolute value of the critical value of the test is greater than or equal to one. This implies that for the most commonly used sizes, the fixed sample size t-test is inadmissible. Other loss functions are discussed. Also an example for a normal mean problem is given where a nonmonotone test cannot be improved on by a monotone test when the risk is componentwise.