A Double Sample Test Procedure

Abstract

Three different testing procedures which involve a minimum of modification of the usual single sample tests of the hypotheses considered are given here. Tests are made by taking samples at two stages for testing the mean of a normal distribution. A known standard deviation is assumed, but an extension to the case where the standard deviation is unknown is also given. Special examples show that tests can be chosen so that the expected number of observations is less than the number required for the ordinary single sample test and indeed can give considerable savings. The tests in Sections 3 and 4 give the greater savings, but the powers are more difficult to evaluate than the power for the test of Section 2. Also, it is a little more work to apply the test in Section 4. Wald in [9] has discussed a sequential test where the observations are taken in groups. The tests given here could be considered very special cases of this where the number of observations is truncated after two groups. Roming in [7] has set up a double sampling procedure for sampling from a finite population that is approximately normal where the rejection points are determined by preassigned engineering or specifications limits and not by the normal distribution itself as is done for the first sample of the double sample tests given below. Bowker and Goode in [1] give tests similar to those given by Romig. Chapman in [2] and Stein in [8] have discussed two sample tests where the object is to obtain tests with the power independent of an unknown variance and where there is no upper limit on the number of observations required. There is a definite ceiling on the number of observations required for the tests presented here and they have many interesting properties that make them very desirable from the standpoint of saving of observations and simplicity.