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I want to identify if there is an association between the Median Household Income of a random selection of 30 metropolitan areas and the rate of motor vehicle thefts in those areas. What type of test should I conduct to see if there is an association between the two?
I would guess a chi-squared test of independence/association, which compares two categorical variables. If should use this test, should I split the median household incomes into categories of ranges (image of my chart attached)?
Depending on your software the "chi-squared test of independence" will be implemented using the chi-squared distribution as an approximation. That will not work with a 6 x 3 table and only 30 observations. You should turn that of and use simulation instead. If using R you could use the argument simulate.p.value = TRUE with the chisq.test command. Nonetheless, 30 observations are not much for such a large table so do not expect too much.
My suggestion is a Spearman correlation. If in R, use cor.test(... , method = "spearman"). It will only test for monotoneous relations but that seems reasonable here. If you do not want that, check out the "Kruskal-Wallis-Test".
should I split the median household incomes into categories of ranges
Do not split data unless forced to by good reasons. Splitting age at a value where those below that value may not vote and those above may, that might be a good reason. Just so you can use a $\chi^2$-Test instead of Spearman correlation or Kruskal-Wallis test is not a good reason, because splitting does destroy information.
(Whether median splits have any place anywhere is a debate that you do not want to become involved with. Many think, it rarely or almost never has.)
The problem with the chi-square test of independence is that you'll reject the null if any of the $6 \times 3$ categories differs statistically significantly from what would be expected under the marginal frequencies of the table. You cannot afterwards point to which category, or more importantly what trend emerges (without correcting for multiple testing). Unfocused hypotheses produce weak findings. If the encodings are not part of data collection, then cutting these data make no sense. Or else, you can encode them by their grouped values (i.e. 0, 1, 2 for low, medium, and high rates of theft). And this will provide a test of trend.
