Suppose one is given a vector X of a finite set of quantities X_i which are independent Poisson random variables. A null hypothesis H_0 about E(X) is to be tested against an alternative hypothesis H_1. A quantity Σ_iw_ix_i is to be computed and used for the test. The optimal values of w_i are calculated for three cases: (1) signal to noise ratio is used in the test, (2) normal approximations with unequal variances to the Poisson distributions are used in the test, and (3) the Poisson distribution itself is used. The above three cases are considered to the situations that are without background noise and with background noise. A comparison is made of the optimal values of w_i in the three cases for both situations.