An approximate method, with decision‐based error‐rates, for testing linear statistical hypotheses (0κi) aboutaindependent binomial parameters (φj) is given which is more general than the usual methods for partitioningXm2from a 2 ×atable. Parameters in Table 2 may be used to choose sample‐size (n) for stated alternatives to 0κiand power (or expected proportion of rejections). The procedure may be used for planned tests (v1 = 1) orpost‐hoctests (v1 =a− 1 orv1 =a) and in the latter case the probability of rejectingrof the 0κi(r= 1, 2, …,v1) can be directly calculated for any statement of what is true in the populations.The results from sampling experiments are used to examine the adequacy of the approximation and compare the method proposed (Rv1) with two others (Gv1 andZv1) Although the latter methods have a power advantage under some circumstances, their experimental results forn≤ 30 are generally less close to ‘theory’ than those ofRv1, and they are always arithmetically and conceptually more complex. The results from these experiments suggest that theRv1 approximation is adequate whenn≥ 10 and the row of smaller expected frequencies (nφ) is greater than or equal to 1. When the φjdiffer and the true φjapproach 0 or 1, “end‐effects” appear but these diminish asnincreases.These methods can be used in a 2×ageneralization of the 2 × 2 median test. If this is interpreted not as a test of population medians, but as a test of proportions of population distributions to the right ofMe(the value of the common sample median), which is φjin