The police have made arrests at the following locations:
Find the variance and the mean of the number of arrests per cell, and use the variance-mean ratio to test the hypothesis that the arrests are distributed randomly over the area (against the two-tailed hypothesis that they are not).
First we find the mean number of points per cell. There are 35 cells and 22 points, so that xmean = 22 / 35 = 0.629.
Next we find the variance of the number of points per cell:
s2 = (1/34) * ( (1 - 0.628)2 + (1 - 0.628)2 + (1 - 0.628)2 + (0 - 0.628)2 + (1 - 0.628)2 …)
s2 = (0.029) * (10.171) = 0.299
Next we calculate the VMR:
VMR = s2 / xmean = 0.299 / 0.629 = 0.476
Finally, we find the z value:
z = sqrt((m - 1) /2) (VMR - 1)
z = sqrt((35 - 1) /2) (0.476 - 1) = -2.161
With alpha equal to 0.05, the critical values are -1.96 and +1.96. The z-score is less than the critical value, so that we reject the null hypothesis of no pattern. The data imply spatial dispersion.
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