For samples from normal populations with unknown means and unknown common standard deviation, but where an upper bound to the standard deviation is known, tests based on the normal distribution (when dealing with one population) and on the Chi-square distribution (when dealing with several populations) are suggested as alternatives to, respectively, thet-test and theF-test. The tests suggested use the upper bound to the standard deviation in place of the sample estimate. Although the tests suggested have a smaller first kind of error, their power can, under certain conditions, exceed the power of the standardt-test andF-test.