Question

Let \( X_{1}, \ldots, X_{n} \stackrel{i i d}{\sim} \operatorname{Ber}(p) \). We want to test \[ \begin{array}{l} H_{0}: \quad p \leq .5 \\ H_{1}: \quad p>.5 \end{array} \] at asymptotic level \( 5 \% \). Let \( \psi \) be the Wald test. 1) Assume that the true \( p \) is equal to 0.2 . What is the limit of \( \mathbf{P}_{0.2}[\psi=1] \) as \( n \rightarrow \infty \) ? \[ \lim _{n \rightarrow \infty} \mathbf{P}_{0.2}[\psi=1]= \] \( \square \) 2) For every \( \lambda>0 \) we can show that \[ \lim _{n \rightarrow \infty} \mathbf{P}_{.5-\frac{\lambda}{\sqrt{n}}}[\psi=1]=\mathbf{P}[Z>1.645+f(\lambda)] \] where \( f(\lambda) \) is a function that depends only on \( \lambda \). What is this function? \[ f(\lambda)= \] \( \square \)

          Let \( X_{1}, \ldots, X_{n} \stackrel{i i d}{\sim} \operatorname{Ber}(p) \). We want to test
\[
\begin{array}{l}
H_{0}: \quad p \leq .5 \\
H_{1}: \quad p>.5
\end{array}
\]
at asymptotic level \( 5 \% \). Let \( \psi \) be the Wald test.
1) Assume that the true \( p \) is equal to 0.2 . What is the limit of \( \mathbf{P}_{0.2}[\psi=1] \) as \( n \rightarrow \infty \) ?
\[
\lim _{n \rightarrow \infty} \mathbf{P}_{0.2}[\psi=1]=
\]
\( \square \)
2) For every \( \lambda>0 \) we can show that
\[
\lim _{n \rightarrow \infty} \mathbf{P}_{.5-\frac{\lambda}{\sqrt{n}}}[\psi=1]=\mathbf{P}[Z>1.645+f(\lambda)]
\]
where \( f(\lambda) \) is a function that depends only on \( \lambda \). What is this function?
\[
f(\lambda)=
\]
\( \square \)

Let X1, …, Xni i d∼Ber(p). We want to test

H0: p ≤ .5

H1: p>.5

at asymptotic level 5 %. Let ψ be the Wald test.
1) Assume that the true p is equal to 0.2 . What is the limit of 𝐏0.2[ψ=1] as n →∞ ?

limn →∞𝐏0.2[ψ=1]=

□
2) For every λ>0 we can show that

limn →∞𝐏.5-(λ)/(√(n))[ψ=1]=𝐏[Z>1.645+f(λ)]

where f(λ) is a function that depends only on λ. What is this function?

f(λ)=

□

Added by Alan R.

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