Chapter 6.5, Problem 5Q

Let $k$ denote the number of successes observed in a sequence of $n$ independent Bernoulli trials, where $p=P$(success).

• (a) Show that the critical region of the likelihood ratio test of $H_0:p=\frac{1}{2}$ versus $H_1:p\ne \frac{1}{2}$ can be written in the form

  $k\cdot \ln \left(k\right)+\left(n-k\right)\cdot \ln \left(n-k\right)\ge {\lambda }^{\ast \ast }$

• (b) Use the symmetry of the graph of

  $f\left(k\right)=k\cdot \text{ln}\left(k\right)+\left(n-k\right)\cdot \ln \left(n-k\right)$ to show that the critical region can be written in the form

  $\left|\bar{k}-\frac{1}{2}\right|\ge c$

  where $c$ is a constant determined by $\alpha$.