A Simple Test for the Median

It is best to look at examples. We will begin by defining a median of a distribution. If F is a distribution function, we say that m is median for F if $F(m) \geq 1/2$ and $F(m^{-}) \leq 1/2$. If F is continuous and strictly increasing at a meadian m, then the median is unique, and F(m) = 1/2. However, if F has jumps, and m is a median, we can have F(m^-) < 1/2 and F(m) > 1/2, and if F is constant on some intervals, there may be more than one median.

Suppose now that the null hypothesis is a subset of the set of all distribution functions with a unique median and that F is continuous at the median. To be concrete, let us suppose that this median is 1. We could construct a test of this hypothesis against the alternative that the median is 2 in the following way. Collect a random sample of size N, say $(X_1,\dots,X_N)$, and let T_N be the number of observations which are greater than 1. If the null hypothesis is true, then T_N has a binomial (N,1/2) distribution. If the alternative hypothesis is true, then T_N will have a binomial (N,p) distribution where p > 1/2, so we would want to reject the null hypothesis if T_N is too large. How large is too large? Just select a target probability of Type I error, $\alpha$, and proceed as if testing when the null hypothesis is that the distribution is binomial (N,1/2) vs the alternative that it is binomial (N,p) for p > 1/2. You won't be able to compute power for your test without substatially more information!