On Governorship and Violent Crime

On a somewhat-recent episode of the Daily Show, an interesting claim was made:
You have a higher chance of going to prison after serving as governor of Illinois than if you were to murder someone.

The statistics cited are:

• 4 of the last 8 governors (50%) have faced prison sentences after holding office (counting Blago as a near certainty)
• Only 48% of murderers actually serve time for their crime

Now, what about these figures?  First off, without actually checking into the history of Illinois, it seems like the first number was probably cherry-picked; that is, governors being convicted of crimes is likely a more recent phenomenon, so if we go back a few more governors we'll probably find mostly non-convicts, which will drop the percentage.

Okay, so that's just an assumption.  Let's actually check Wikipedia.

Well, checking all the way back to Henry Horner back in the 1930s, there aren't any additional convictions other than the four already mentioned (that I found, anyway).  So while that statistic is true, it is indeed a bit cherry-picked.

So now the crime statistics.  This was a bit more difficult to find, and I still haven't found anything backing up the claim made on the Daily Show.  Best I found was this, which reports that the FBI's homicide case closure rate is about 61%, but this is a bit different from the proportion of individuals serving time for the crime: the FBI considers a case cleared when an arrest is made for the crime, not by whether the suspect is convicted or not (it's a tad more complicated than that, apparently, but that's the basic idea).  There's a few other issues with this, as well, but I think they would likely have less of an impact; for example, what about those who murder several individuals but are only charged with some of the murders?  For the sake of this blog post, though, let's just assume the 48% figure is correct.

From the world of statistics comes the idea of a hypothesis test, which allows us to decide with a certain degree of accuracy whether or not a claim about a an attribute of a population.  In this case, we're dealing with the proportion of governors of Illinois that eventually become convicted felons.  With p^ (pronounced "pee hat") representing the given sample proportion of felonious governors, and p representing the population proportion (which is unknown), the claim made by the Daily Show is essentially that p > m, where m is the proportion of murderers who serve a sentence for their crime.  From the above, we know p^ = .5 and m = .48.  Using a significance level of .05, we get a critical value of z_α/2 = 1.645 and thus find the test statistic to be z = .080064.  Now, the null hypothesis in this case is p = m (which I should have mentioned earlier), and clearly the test statistic falls outside the rejection region.  Thus there is not sufficient evidence to reject the claim that p = .48, that is, 48% of Illinois governors either have been or will be convicted of a crime.  Basically, this means, no, we can't say with a good degree of certainty that p will be greater than m.

In conclusion, the claim made by the Daily Show, while interesting, has little evidence to back it up.  To remedy this, I suggest we either just let it go, or dig some dirt up on long-dead former governors, as dead people are easier targets.  Also, I have too much time on my hands.

Disclaimer:  most of the above is probably wrong, and numerous egregious statistical errors were made in the writing of this post.