How do I find data to show whether a shaved die is really loaded? I'm making a video about dice, so I went online and bought some loaded dice. The ones I bought are "shaved dice", or "flats", these ones in specific: http://www.amazon.com/gp/product/B008QDJ4RI/ref=oh_details_o04_s00_i00?ie=UTF8&psc=1
I've been doing chi-squared tests of 30 trials at a time with these, and it's really hard to see a bias in any direction. I'm trying to get 95% confidence, but the best I've gotten is 90% (out of 4 tests, 2 of them were at 90% and 2 were very low). I understand it may be a weak effect, but how do I tell with more confidence whether there's actually an effect or not? Do I do more trials? That seems to make the effect even murkier. At what point can I just shrug my shoulders and say, "Well, I guess shaving the die doesn't actually do anything?" Or is 90% good enough?
 A: One simple approach is to focus on a single face, say six (from the link it looks like this is one of the "flat" faces so should come up less often than 1/6 of the time). Then, if you roll the die $n$ times, you can test the hypothesis that $p = 1/6$ using the test statistic
$$ Z = \frac{ \hat p - 1/6}{\sqrt{\frac{ (1/6) \times (5/6) }{n} }} $$
where $\hat p$ is the fraction of rolls that come up six.
A quick calculation tells you that if you roll the die 100 times you will get a p-value < 0.05 (often considered "strong evidence" against the null hypothesis that the die is fair) only if your observed fraction of sixes is more than about 0.24 or less than about 0.1. If you roll the die 1000 times, observed fractions > 0.19 or < 0.145 will yield evidence at the 0.05 level to reject the "fair die" hypothesis. 
You can reduce the number of required tosses somewhat (but not much) by counting two shaved faces (e.g., one and six), where the null hypothesis is that these faces will come up 1/6 + 1/6 = 1/3 of the time and the relevant statistic is
$$ Z = \frac{ \hat p - 1/3}{\sqrt{\frac{ (1/3) \times (2/3) }{n} }} $$
I suspect that the amount of shaving which can go undetected by the naked eye does not modify roll probabilities substantially, so you might be looking at a lot of throws to test out your dice!
