Entomologists study the distribution of insects across agricultural fields. A study of fire ant hills across pasture lands is conducted by dividing pastures into 50- meter squares and count-ing the number of fire ant hills in each square. The null hypothesis of a Poisson distribution for the counts is equivalent to a random distribution of the fire ant hills over the pasture. Rejection of the hypothesis of randomness may occur due to one of two possible alternatives. The distribution of fire ant hills may be uniform€” that is, the same number of hills per 50- meter square€” or the distribution of fire ants may be clustered across the pasture. A random distribution would have the variance in counts equal to the mean count, σ2 = μ. If the distribution is more uniform than random, then the distribution is said to be underdispersed, σ2, μ. If the distribution is more clustered than random, then the distribution is said to be overdispersed, σ2. μ. The number of fire ant hills was recorded on one hundred 50- meter squares. In the data set, yi is the number of fire ant hills per square, and ni denotes the number of 50- meter squares with yi ant hills.

a. Estimate the mean and variance of the number of fire ant hills per 50- meter square; that is, compute y and σ2 using the formulas from Chapter 3.
b. Do the fire ant hills appear to be randomly distributed across the pastures? Use a chi-square test of the adequacy of the Poisson distribution to fit the data using a = .05.
c. If you reject the Poisson distribution as a model for the distribution of fire ant hills, does it appear that fire ant hills are more clustered or uniformly distributed across the pastures?

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