Let $x_1,...,x_{10}$ be a sample from a continuous random variable $X$ with the median $\bar m$. Determine the confidence level for the interval $I_{\bar m}=(x_{(2)},x_{(9)})$. $x_{i}$ is the i:th smallest value of the 10 observations.
My attempt
From the very beginning we let $x_{(1)}<...<x_{(10)}$ be the ordered sample, i.e the observations in "size order". Furthermore, we let $E(X)=\bar m$ and forms the confidence interval $I_{\bar m}=(x_{(2)},x_{(9)})$. But now, I do not know how to continue so that i can determine the confidence level for the interval. Should I suppose that the variable is from a distribution, e.g normal distribution?Any hints or useful defintions/statements?
Hint: The interval runs from the second smallest observation to the second largest observation. The confidence level is the probability that this interval captures the median. To obtain this confidence level, first calculate the probability that the interval fails to capture the median. This event can occur in two ways: (a) the number of observations that lie above the median is zero or one, or (b) the number of observations that lie below the median is zero or one.
To calculate the prob of event (a), argue that the number of observations that lie above the median has a binomial distribution (with what parameters?), and argue similarly for (b).