# Test to prove a random selection

An urn has infinite balls of four color, say A, B, C, D. All balls are in equal numbers and they are properly mixed together. A man picks $N$ balls randomly. He picks $w$, $x$, $y$ and $z$ balls of colors A, B, C and D respectively, such that $w+x+y+z=N$. How to test if

1. color A was over represented in the selection?
2. color B was over represented in the selection?
3. color C was over represented in the selection?
4. color D was over represented in the selection?

NB: I found this answer but could not gather much from here. Proper way to test hypothesis of random selection?

• Please consult a textbook about the chi-squared test or search our site for related questions. That will give you a head start by helping you decide whether there is evidence of non-uniform results. Then, if you do decide that, the harder part of your question is deciding which (if any!) of the colors were over-represented. For that, you are likely to find relevant advice by searching for Tukey HSD.
– whuber
May 16 '17 at 13:06
• Can you more clearly identify your aim? Are you after a single test to pick up any deviations from equal probability (the usual approach being a chi-squared test of goodness of fit), or a test for each individual category (individual tests of proportion)? Or perhaps the first followed by post hoc identification of the most discrepant categories? Note that you cannot prove that the categories are selected with equal probability; to come close you may want something like an equivalence test perhaps. May 17 '17 at 2:52
• You may get some value from this thread May 17 '17 at 2:56
• Thanks for the comments. I'll read up on your suggestions, and if I can't figure it out, ask question when I am better informed.
– Mr K
May 17 '17 at 7:15