An interesting and practical use of the χ² test comes about in testing for segregation of species of plants or animals. Suppose that two species of plants, A and B, are growing on a test plot. To assess whether the species tend to segregate, a researcher randomly samples n plants from the plot; the species of each sampled plant, and the species of its nearest neighbor are recorded. The data are then arranged in a table, as shown here.

If a and d are large relative to b and c, we would be inclined to say that the species tend to segregate. (Most of A’s neighbors are of type A, and most of B’s neighbors are of type B.) If b and c are large compared to a and d, we would say that the species tend to be overly mixed. In either of these cases (segregation or overmixing), a χ² test should yield a large value, and the hypothesis of random mixing would be rejected. For each of the following cases, test the hypothesis of random mixing (or, equivalently, the hypothesis that the species of a sample plant is independent of the species of its nearest neighbor). Use α = .05 in each case.

1. a a = 20, b = 4, c = 8, d = 18.
2. b a = 4, b = 20, c = 18, d = 8.
3. c a = 20, b = 4, c = 18, d = 8.