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"This week, we examined the simplest theory of spatial criminology – that crimes are distributed at random. When spatial criminologists want to test observed data for randomness, they test that data’s distribution against the Poisson distribution and look for a statistically significant difference. What is the null hypothesis in this situation? What conclusion would we draw from rejecting the null hypothesis?"

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If the occurrence of events is homogeneous and independent then the number of events per unit of exposure (e.g. area) will have a Poisson distribution, so you could test for a null of that particular kind of randomness by doing a goodness of fit test of a Poisson distribution.

You can then phrase the null either as a null of the process being random (a Poisson process) or as having a Poisson distribution.

If your null is of a non-homogeneous process then things become more complicated than that.

I would add that this seems a highly implausible model and I see no good reason to be testing for it.

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