Question

Let \( X_{1}, \ldots, X_{n} \) be i.i.d. Bernoulli random variables with unknown parameter \( p \in(0,1) \). Suppose we want to test \[ H_{0}: p \in[0.48,0.51] \quad \text { vs } \quad H_{1}: p \notin[0.48,0.51] \] We want to construct an asymptotic test \( \psi \) for these hypotheses using \( \bar{X}_{n} \). For this problem, we specifically consider the family of tests \( \psi_{c_{1}, c_{2}} \) where we reject the null hypothesis if either \( \bar{X}_{n}<c_{1} \leq 0.48 \) or \( \bar{X}_{n}>c_{2} \geq 0.51 \) for some \( c_{1} \) and \( c_{2} \) that may depend on \( n \), i.e. \[ \psi_{c_{1}, c_{2}}=\mathbf{1}\left(\left(\bar{X}_{n}<c_{1}\right) \cup\left(\bar{X}_{n}>c_{2}\right)\right) \quad \text { where } c_{1} \leq 0.48<0.51 \leq c_{2} . \] Throughout this problem, we will discuss possible choices for constants \( c_{1} \) and \( c_{2} \), and their impact to both the asymptotic and non-asymptotic level of the test. Use the central limit theorem and the approximation \( \sqrt{p(1-p)} \approx \frac{1}{2} \) for \( p \in[0.48,0.51] \) to approximate \( \mathbf{P}_{p}\left(\bar{X}_{n}<c_{1}\right) \) and \( \mathbf{P}_{p}\left(\bar{X}_{n}>c_{2}\right) \) for large \( n \). Express your answers as a formula in terms of \( c_{1}, c_{2}, n \) and \( p \) (Write Phi for the cdf of a Normal distribution, \( \mathbf{c}_{\mathbf{2}} \mathbf{1} \) for \( c_{1} \), and \( \mathbf{c}_{-} \mathbf{2} \) for \( c_{2} \).) \[ \mathbf{P}_{p}\left(\bar{X}_{n}<c_{1}\right) \approx \] For what value of \( p \in[0.48,0.51] \) is the expression above for \( \mathbf{P}_{p}\left(\bar{X}_{n}<c_{1}\right) \) maximized? \( \mathbf{P}_{p}\left(\bar{X}_{n}<c_{1}\right) \) is \( \max \) at \( p= \) \[ \mathbf{P}_{p}\left(\bar{X}_{n}>c_{2}\right) \approx \] For what value of \( p \in[0.48,0.51] \) is the expression above for \( \mathbf{P}_{p}\left(\bar{X}_{n}>c_{2}\right) \) maximized? \[ \mathbf{P}_{p}\left(\bar{X}_{n}>c_{2}\right) \text { is } \max \text { at } p= \]

          Let \( X_{1}, \ldots, X_{n} \) be i.i.d. Bernoulli random variables with unknown parameter \( p \in(0,1) \). Suppose we want to test
\[
H_{0}: p \in[0.48,0.51] \quad \text { vs } \quad H_{1}: p \notin[0.48,0.51]
\]

We want to construct an asymptotic test \( \psi \) for these hypotheses using \( \bar{X}_{n} \). For this problem, we specifically consider the family of tests \( \psi_{c_{1}, c_{2}} \) where we reject the null hypothesis if either \( \bar{X}_{n}<c_{1} \leq 0.48 \) or \( \bar{X}_{n}>c_{2} \geq 0.51 \) for some \( c_{1} \) and \( c_{2} \) that may depend on \( n \), i.e.
\[
\psi_{c_{1}, c_{2}}=\mathbf{1}\left(\left(\bar{X}_{n}<c_{1}\right) \cup\left(\bar{X}_{n}>c_{2}\right)\right) \quad \text { where } c_{1} \leq 0.48<0.51 \leq c_{2} .
\]

Throughout this problem, we will discuss possible choices for constants \( c_{1} \) and \( c_{2} \), and their impact to both the asymptotic and non-asymptotic level of the test.
Use the central limit theorem and the approximation \( \sqrt{p(1-p)} \approx \frac{1}{2} \) for \( p \in[0.48,0.51] \) to approximate \( \mathbf{P}_{p}\left(\bar{X}_{n}<c_{1}\right) \) and \( \mathbf{P}_{p}\left(\bar{X}_{n}>c_{2}\right) \) for large \( n \). Express your answers as a formula in terms of \( c_{1}, c_{2}, n \) and \( p \)
(Write Phi for the cdf of a Normal distribution, \( \mathbf{c}_{\mathbf{2}} \mathbf{1} \) for \( c_{1} \), and \( \mathbf{c}_{-} \mathbf{2} \) for \( c_{2} \).)
\[
\mathbf{P}_{p}\left(\bar{X}_{n}<c_{1}\right) \approx
\]

For what value of \( p \in[0.48,0.51] \) is the expression above for \( \mathbf{P}_{p}\left(\bar{X}_{n}<c_{1}\right) \) maximized?
\( \mathbf{P}_{p}\left(\bar{X}_{n}<c_{1}\right) \) is \( \max \) at \( p= \)
\[
\mathbf{P}_{p}\left(\bar{X}_{n}>c_{2}\right) \approx
\]

For what value of \( p \in[0.48,0.51] \) is the expression above for \( \mathbf{P}_{p}\left(\bar{X}_{n}>c_{2}\right) \) maximized?
\[
\mathbf{P}_{p}\left(\bar{X}_{n}>c_{2}\right) \text { is } \max \text { at } p=
\]

Let X1, …, Xn be i.i.d. Bernoulli random variables with unknown parameter p ∈(0,1). Suppose we want to test

H0: p ∈[0.48,0.51] vs H1: p ∉[0.48,0.51]

We want to construct an asymptotic test ψ for these hypotheses using X̅n. For this problem, we specifically consider the family of tests ψc1, c2 where we reject the null hypothesis if either X̅n<c1≤ 0.48 or X̅n>c2≥ 0.51 for some c1 and c2 that may depend on n, i.e.

ψc1, c2=1((X̅n<c1) ∪(X̅n>c2)) where c1≤ 0.48<0.51 ≤ c2 .

Throughout this problem, we will discuss possible choices for constants c1 and c2, and their impact to both the asymptotic and non-asymptotic level of the test.
Use the central limit theorem and the approximation √(p(1-p))≈(1)/(2) for p ∈[0.48,0.51] to approximate 𝐏p(X̅n<c1) and 𝐏p(X̅n>c2) for large n. Express your answers as a formula in terms of c1, c2, n and p
(Write Phi for the cdf of a Normal distribution, 𝐜21 for c1, and 𝐜-2 for c2.)

𝐏p(X̅n<c1) ≈

For what value of p ∈[0.48,0.51] is the expression above for 𝐏p(X̅n<c1) maximized?
𝐏p(X̅n<c1) is max at p=

𝐏p(X̅n>c2) ≈

For what value of p ∈[0.48,0.51] is the expression above for 𝐏p(X̅n>c2) maximized?

𝐏p(X̅n>c2) is max at p=

Added by Tina F.

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