The rejection region of the Wald test for testing $H_0: p = 0.5$ against $H_1: p \neq 0.5$ is given by:
$\sqrt{n}\frac{|\bar X_ n - 1/2|}{\sqrt{\bar X_ n (1-\bar X_ n)}}>1.96$
This is the correct rejection region for the test.
Let X_1, \ldots , X_ n \stackrel{iid}{\sim } \mathsf{Ber}(p). We want to test
\begin{array}{rl} H_0: & p=.5 \\ H_1: & p\neq .5\\ \end{array}
at asymptotic level 5%. What is the rejection region of the Wald test? (Choose all that apply.)
\sqrt{n}\frac{\bar X_ n - 1/2}{\sqrt{\bar X_ n (1-\bar X_ n)}}>1.65
|\bar X_ n - 1/2|>1.96\sqrt{\frac{\bar X_ n (1-\bar X_ n)}{n}}
\bar X_ n > 1/2+ 1.96\sqrt{\frac{\bar X_ n (1-\bar X_ n)}{n}}
\frac{|\bar X_ n - 1/2|}{\sqrt{\textsf{Var}(\bar X_ n)}}>1.96
\sqrt{n}\frac{|\bar X_ n - p|}{\sqrt{p(1-p)}}>1.96
\mathcal{I}_{\mathrm{plug-in}}, defined to be the plug-in confidence interval at asymptotic confidence level 95% from Unit 2.
\sqrt{n}\frac{|\bar X_ n - 1/2|}{\sqrt{\bar X_ n (1-\bar X_ n)}}>1.96
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