Let X be a single (i.e. n=1 ) Gaussian random variable with unknown mean μ and variance 1 . Cons

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Question

Let \( X \) be a single (i.e. \( n=1 \) ) Gaussian random variable with unknown mean \( \mu \) and variance 1 . Consider the following hypotheses: \[ H_{0}: \mu=0 \quad \text { vs } \quad H_{1}: \mu \neq 0 . \] (a) 1 point possible (graded) Define a test \( \psi_{\alpha}: \mathbb{R} \rightarrow\{0,1\} \) with level \( \alpha \) that is of the form \[ \psi_{\alpha}=\mathbf{1}\left\{f_{\alpha}(X)>0\right\}, \] for some function \( f_{\alpha}: \mathbb{R} \rightarrow \mathbb{R} \). We want our test \( \psi \) above to satisfy the following: - Symmetric in the value of \( X \) about 0 , so \( f(X)=f(-X) \); - Its "acceptance region" is an interval. (The acceptance region of a test is the region in which the null hypothesis is not rejected, i.e. the complement of its rejection region.) Specify the function \( f_{\alpha}(X) \) in terms of \( \alpha \) below.

          Let \( X \) be a single (i.e. \( n=1 \) ) Gaussian random variable with unknown mean \( \mu \) and variance 1 . Consider the following hypotheses:
\[
H_{0}: \mu=0 \quad \text { vs } \quad H_{1}: \mu \neq 0 .
\]
(a)
1 point possible (graded)
Define a test \( \psi_{\alpha}: \mathbb{R} \rightarrow\{0,1\} \) with level \( \alpha \) that is of the form
\[
\psi_{\alpha}=\mathbf{1}\left\{f_{\alpha}(X)>0\right\},
\]
for some function \( f_{\alpha}: \mathbb{R} \rightarrow \mathbb{R} \).
We want our test \( \psi \) above to satisfy the following:
- Symmetric in the value of \( X \) about 0 , so \( f(X)=f(-X) \);
- Its "acceptance region" is an interval. (The acceptance region of a test is the region in which the null hypothesis is not rejected, i.e. the complement of its rejection region.)

Specify the function \( f_{\alpha}(X) \) in terms of \( \alpha \) below.

Let X be a single (i.e. n=1 ) Gaussian random variable with unknown mean μ and variance 1 . Consider the following hypotheses:

H0: μ=0 vs H1: μ≠ 0 .

(a)
1 point possible (graded)
Define a test ψα: ℝ→{0,1} with level α that is of the form

ψα=1{fα(X)>0},

for some function fα: ℝ→ℝ.
We want our test ψ above to satisfy the following:
- Symmetric in the value of X about 0 , so f(X)=f(-X);
- Its "acceptance region" is an interval. (The acceptance region of a test is the region in which the null hypothesis is not rejected, i.e. the complement of its rejection region.)

Specify the function fα(X) in terms of α below.

Added by Adri-N B.

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