# Formula for Power of Upper Tail Sign Test

## Given

Consider the upper tail $$H_0: \theta \leq 0 \,\,\, \text{vs} \,\,\, H_A: \theta > 0$$ sign test with the test statistic $$B = \sum_{i=1}^n{\psi_i}$$ where $$\psi_i = \mathbb{I}(Z_i > \theta)$$, where $$\mathbb{P}(\psi_i=1)=\mathbb{P}(Z_i > \theta)=p$$. With the assumption that the $$n$$ observations are iid, answer the following questions.

$$B = \sum_{i=1}^{n} \psi_i \hspace{1em} \text{where} \hspace{1em} \psi_i = \left\{\begin{array}{ll} 1 & \texttt{if} \,\, Z_i > \theta \\ 0 & \texttt{otherwise}\end{array}\right.$$

$$B \overset{iid}{\sim} {\tt Binom}(n, p)$$

## Question

Write down the expression of the power of the upper tail sign test when the alternative has $$p_0$$.

• So, I know that the power of a hypothesis test is the probability of correctly rejecting $$H_0$$ when $$H_A$$ is true. i.e.

$${\tt power} = \mathbb{P}(\text{reject } H_0 | H_A \text{ is true})$$

• However, since the question specifically mentions $$p_0$$, I think I need to express power in terms of $$p$$, $$p_0 = 1 - p$$, $$n$$, $$\psi_i$$, $$\theta$$, and/or $$Z_i$$ -- I'm not sure how to begin doing that, as I can't find a concrete formula for the power of a binomial.

## Note

• The $$\mathbb{I}$$ should be a $$1$$ with two lines, but SO wouldn't let me use \mathbbm{1}.