What is the purpose of the sign test?
To test if the population median equals a given value (e.g. the sample median).
What are the assumptions made?

Independence: Each sample is independent of the others.
 Identical Distributions

Continuity: There are no ties
 Needed for the hypothesis test

Means there is exactly one point x such that F(x) =1/2 and that point is the median ($theta$)
How do you use it?

Setup the null hypothesis:
\[ \begin{aligned}
H_0 &: m = m_0
\end{aligned} \]
$m$ is the true median, $m_0$ is the proposed median.

Setup the alternative hypothesis as one of:
\[\begin{aligned}
H_a &: m \neq m_0\\
H_a &: m > m_0\\
H_a &: m < m_0
\end{aligned}\]
Collect a random sample from the population.

Assign a 1 or 0 to each value in the data:
 $sign = 0 \; if value < m_0$
 $sign = 1 \; if value > m_0$
 remove value from sample if $value = m_0$

Sum all the signs to get $k$:
\[\begin{aligned}
x &= \sum_{i=0}^{n1} sign(value_i)
\end{aligned} \]

Find $k$ on the binomial distribution:
 $n$ is the current sample size
 $p = 0.5$ (If $H_0$ is true, half are above, half are below)

Find the pvalue:
 if $H_a$ uses <, add probabilities for $x \leq k$
 if $H_a$ uses >, add probabilities for $x \geq k$
 if $H_a$ uses $neq$, $p = 2 \times sum(x \geq k)$
Draw a conclusion:
If $pvalue < alpha$ (0.05 for 95%), reject $H_0$. Otherwise don't reject.