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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?

kjetil b halvorsen
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Mr K
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    Please consult a textbook about the chi-squared test or [search our site for related questions](https://stats.stackexchange.com/search?q=chi-squared+test). 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](https://stats.stackexchange.com/search?q=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](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test#Discrete_uniform_distribution)), 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. – Glen_b May 17 '17 at 02:52
  • You may get some value from [this thread](https://stats.stackexchange.com/questions/3194/how-can-i-test-the-fairness-of-a-d20) – Glen_b May 17 '17 at 02:56
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    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 07:15

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