Let X1, . . . , Xm be a random sample from a Poisson distribution with parameter 1, and let Y1, . . . , Yn be a random sample from another Poisson distribution with parameter 2.

We wish to test H0: 1  2
0 against one of the three standard alternatives. Since
 for a Poisson distribution, when m and n are large the large-sample z test of Section 9.1 can be used. However, the fact that V(X)
/n suggests that a different denominator should be used in standardizing X  Y. Develop a large-sample test procedure appropriate to this problem, and then apply it to the following data to test whether the plant densities for a particular species are equal in two different regions (where each observation is the number of plants found in a randomly located square sampling quadrate having area 1 m2

, so for region 1, there were 40 quadrates in which one plant was observed, etc.):

Frequency 0 1 2 3 4 56 7 Region 1 28 40 28 17 8 2 1 1 m
125 Region 2 14 25 30 18 49 2 1 1 n
140