# Please help me win my argument! What is an appropriate statistical test to use on Proportional Data?

I recently got into a debate with a friend of mine because I claimed that the 9th circuit court tends to have its rulings overturned at a greater rate than other courts (this is about statistics trust me!)

Looking at the data, from face value he rejected my claim because "the ninth circuit is only reversed at 10% higher than the rest of the courts".

To keep it short, things got personal and my intelligence was insulted. He asked to see a t-test and ANOVA w/ post hoc tests (even though we both acknowledge its not best suited here) in order for me to make my case.

In my pain I wrote it all up in a document (I wasn't able to attach a file here) but I wanted to share it with all of you for some opinions on the subject.

We also both agreed that "Reversal_Rate" (# of reversals / # of hearings) is the variable of interest here, not the quantity of reversals (since courts with higher number of hearings will have higher reversals by design).

The Problem:

1) So my question is, what are some tests I can use to see if ninth court's Reversal Rate is higher than the rest? In other words, what kind of statistical test is best suited to test proportional data such as Reversal_rt?

2) If someone would be so kind, can you check out my results so far (especially in the second half) and let me know their thoughts?

My results: https://i.sstatic.net/ni4pr.jpg

• In order to do a proper test you'd need the numbers of decisions and of reversals for the 9th circuit court and also numbers of decisions and reversals for comparison court(s). Percentages alone are almost useless for testing: 10 reversals for 50 decisions and 100 reversals for 500 decisions carry very different amounts of information. // Answer of @jros is a good place to start. Commented May 14, 2020 at 7:06
• Is independence of the rulings a reasonable assumption here? Commented May 21, 2020 at 21:19

$$\hat{p_1}$$ = proportion for group 1, $$n_1$$ is sample size of group 1
$$\hat{p_2}$$ = proportion for group 2, $$n_2$$ is sample size of group 2
$$z^*$$ is our critical value, ie for a 95% Confidence Interval, $$z^*$$ = 1.96
$$(\hat{p_1}-\hat{p_2}) \pm z_{1-\frac{\alpha}{2}}^*\sqrt{\frac{\hat{p_1}*(1-\hat{p_1})}{n_1}+\frac{\hat{p_2}*(1-\hat{p_2)}}{n_2}}$$