Quick powerful tests with grouped data

A test of a statistical hypothesis using data grouped into two or more contiguous classes may be preferable to the efficient test if the cost per observation is greatly reduced by grouping, although the reduction in observation cost is offset by the loss in power of the inefficient test. It is shown that in certain cases one may do better by combining the inefficient test with the efficient one. A two-stage procedure for doing this is suggested, and the losses are evaluated for a test on the mean of a normal distribution and for a test of independence in a bivariate normal distribution. While the loss or gain in observation cost and power depends on various parameters such as the test size and the relative costs of observing grouped and ungrouped data, it is shown that for the test on a normal mean a reduction of 80 % in expected observation cost can be achieved with only a 0.3 % loss in power.