This is starting to head into territory that might be better handled at CrossValidated, but it's interesting here nonetheless.
One concept that's important in statistics is that even if a random variable is not normally distributed (could be uniform, binomial, gamma, who knows), a sample of its means will be normally distributed. Thus, while you cannot assume that a random sample of 50 at-bats from a player will appear normal, you can assume that if you take 50 samples of 50 samples that they will be approximately normally distributed, assuming the basis for the data doesn't change over time.
This, of course, isn't an assumption you can make over a long period of time - 2500 at bats I'm sure a player will change over - but over a shorter period of time you probably can, if the player doesn't get injured and isn't in the very beginning or very end of his career.
This is used by assuming a player's performance that year/etc. is a random sample from the inferential (but imaginary) population of all at-bats he might take, which would define his 'true' ability level. If a player is batting .280, then the odds of his true ability level being .300 or .260 under a naïve assumption would be identically likely (and, based on the sample size, could be determined.) Further, you could use other samples of his batting ability (say, he batted .250 last year, and .270 the year before that, and .230 the year before that) to determine what the likely overall skill level is - again assuming, naïvely, that his skill level is constant. In this example, the .280 sounds like an outlier on the high end - a hot streak, so to speak - since his performance has fluctuated up and down around an average of .260 or so.
However, that's a very naïve assumption, and is certainly well behind where we stand in sports analytics. The normal distribution assumption is still present in certain elements, but various people have used a much more powerful tool to determine what factors have predictive value as to future performance in other areas than simply a mean of batting averages.
Among other things, with just a simple confidence interval, you can't assume batting average has any effect on other elements of performance (including scoring runs, etc.); instead, sports statisticians have used various forms of regression analysis to determine how the different statistics available correlate with important performance elements (particularly, scoring runs, preventing runs from scoring, and winning games, for baseball as an example). That gives you more ability to identify whether a player is likely on a hot streak, or whether his performance has truly improved - separating the Mike Trouts of the world from the Chris Davises.