If you flip a coin and get 268 heads and 98 tails, you can calculate the probability that coin is fair several ways. A simple, heuristic observation would have most likely conclude that such a coin is unfair. I've calculated the p-value in R with:
> coin <- pbinom(98, 366, 0.5) > coin*2  2.214369e-19
This value is smaller than .05, ergo we reject the hypothesis that it's a fair coin.
But what if you where told that the same coin landed on its side 676 times during the trial. Heuristically you'll likely come to the same conclusion, but would the typical fair coin tests still be valid?
Here is a graph to illustrate the problem:
What are valid methods to test the hypothesis that there is equal probability that an event occurs in the shaded areas?
NOTE: there are 629 positive moves (413 negative) in the graph illustration.
R code that generates the data:
require("quantmod") ticker <- getSymbols("SLV")[,6] change <- (ticker - lag(ticker, 24)) / lag(ticker, 24) change <- na.locf(change, na.rm=TRUE) # some other calculations dens <- density(change) plot(dens) # some formatting stuff