Heather MacDonald makes us aware of an interesting mathematical claim being made by President Obama and his happy-go-lucky sidekick Scranton Joe.
The most common statistic thrown out these days by President Obama, Vice President Biden, on down is that one in five women will be the victims of sexual assault during their college careers. Detroit is America’s most violent city. Its violent crime rate for all four violent felonies—that’s rape, murder, aggravated assault, and robbery—is 2%. Its rape rate is 0.05%.
So let’s calculate the probability that Barack Obama is actually being completely honest and accurate. That is, assuming the average American university is as violent and infested with gibbering, pseudo-sapient rapeofiles as America’s most violent city; Detroit, MI. So if we admit a population of 1000 female students to this university, and assume they’ll require a mean time to graduate of 5 years; then 200 of these poor ladies should be raped by the time five years have passed. Let’s run the numbers and see if that passes any sort of statistical bravo sierra test at a significance of alpha=0.05.
We assume vile acts of rape are randomly distributed over a woman’s tenure at the university using a Poisson Distribution with a lambda = 0.0005/ year. In order to encompass every possible chance that President Barack Obama is totally accurate with his claim, we will assume any woman suffering the misfortune of rape will remain enrolled the entire five years regardless of how many times she suffers a rape. Any repeated rape of the same target will hence be counted.*
Thus, in a single year, we can divide the probability by 12 and check for each month as an independent event. There are two possible outcomes for each simulated month, rape or not rape. Sixty months per woman times 1000 women in the sample gives an N=60,000 which is sufficiently large to allow me to use the Poisson distribution as a limiting form of the Binomial Distribution described above without committing any rough violence against the practice of statistical inference. This allows a non-zero probability of one victim being assaulted 60 discreet times and still remaining on campus. The likelihood under these conditions of the woman not being raped at all in 60 months on a college campus as violently criminal as Detroit, MI = 99.750%.
If we raise the value of 99.75% to the exponent equal to the number of women in a sample, we can determine the likelihood that a population of women of size n can go 60 months in our simulated violent college environment without being raped. A population of 10 independently simulated female college careers is 97.53% likely to produce zero rapes. 100 simulations is 77.88% likely to have a result of zero. 1000 simulations would have a 91.79% chance of having at least 1 occurrence of a rape. Thus we establish that despite our president’s laughable exaggerations, women should still proceed with some caution because violent locations have some non-zero threat of sexual predation.
So if we multiply the sample size N (n women times 60 months of attendance each) by the probability p of a rape occurring, we can derive the expected number of rapes given a set of potential risks. Multiplying 60,000 potential risk months by a probability of p=0.0005/12 (an annual crime rate of 0.05%/year evenly split into 12 months of risk) we get an expected value of 2.5 rapes per 60,000 risk months spent on a college campus as violent as Detroit, MI.
To complete our analysis, we now statistically test the proposition that 2.5 rapes /60,000 risk intervals is the same as 200 rapes / 60,000 risk intervals. We will execute this at an alpha = 0.025 since the condition stated above implies a two-tailed test of the sample distributions.
Ho: There is no discernible statistical difference between a rate of incidence equal to 2.5 / 60000 versus 200 / 60000. In essence it could be said loosely that President Obama has accurately described the threat of rape on the average American college campus if we assume all colleges have the same level of violent crime as Detroit, MI.
Ha: There is a big difference between an incidence rate of 2.5/60000 and 200/60000 and we cannot therefore establish that President Obama has accurately spoken on this issue.
Since the Poisson distribution in question is a limiting result of a small p, large n binomial, we can test the proposition that President Obama is making a numerate statement of probability by testing his statement versus the observed value from our simulation. Po (Presidential Likelihood Estimate) = 200/60000 = 0.0033 rapes per risk month. Pa(Simulated Incidence Rate) = 2.5/60000 = 0.000042 rapes per risk month.
Z = (Po-Pa)/Square Root (P*q* 1/2n). If Z> 1.96 or Z< -1.96 we reject Ho at alpha = 0.025. Here p =202.5/120000, q = 1-p, 1/2n = 30000. So Po-Pa = 197.5, and the denominator ends up equaling 50.54. Thus, the Z-Score in question = 3.91 which is high enough to reject Ho and totally question what President Obama is telling us.
We therefore conclude, at a significance of 97.5%; that our Fearless Leader has once more misstated, prevaricated, exaggerated, fertilized the corn, Axelrodded the data, Axelrodded the Poochie, lied etc… with respect to the likelihood of a woman being raped on even the most barbaric university campus in America. It’s time for him and his followers to give in and accept the science.
*-I’m bending over backwards for the C-in-C on this assumption.