Question on proving a random test isn't random

#1
Working on a fake case for a legal studies class:

facts:
government can randomly drug test students receiving financial aid (fake law)
SSU enrolls 15,000 students. Of those 15,000 students, 14,500 receive some form of state-supported financial aid. 4,000 receive ES Grants, which are reserved to low income students. SSU randomly selected 250 students to test, 232 of whom were ES Grant recipients. SSU claims this was a statistical anomaly.

How do I prove that the chances of this happening in a random test are preposterous?

Thanks for any help

Note: I am not a statistics student, this would just help me and my group prove the case that our 'client' was wronged.
 

JohnM

TS Contributor
#2
You can compute the likelihood (probability) of obtaining a sample of 232/250 ES Grant recipients (232/250 = 92.8%) when the population at large is only 4000/15000 = 26.6%.

The method to use is the normal approximation to the binomial distribution - there is an example in the Examples section of this site...
 
#3
Well If I did this right at all
x=92.8
np=100*.2759 (i did 4000/14500)=27.59
npq=27.59*.7241=19.98
z=(92.8-27.59)/sqrt19.98
z=65.21/4.47
z=14.588

Now, again, as I have no knowledge of statistics whatsoever, I don't know how to make the number I get from z into a percentage of probability.

Can you help me find that? Did I do the problem right at all?
 

JohnM

TS Contributor
#5
You got it. z=3.3713

Using the normal distribution, the probability that z >= 3.3713 is 0.00037

An easy way to get this is using Excel:

the function =NORMSDIST(z) gives you the probability of getting anything less than z. To get P(>= z), subtract the function result from 1