Let \(X\sim N(\theta,1)\) and 5 independent observations \(X=(4.9,5.6,5.1,4.6,3.6)\). The prior probability that \(\theta=4.01\) is

\(0.5\). The remain values of \(\theta\) are given the density of \(g(\theta)\).

a)Assume \(g(\theta)\sim N(4.01,1)\) test the hypothesis

From what I learn to make a hypothesis test I need to find

In the cases where the null hypothesis is not a point I can make, but in this case I have a few doubts.

From the notes that I take there is the theorem below

\theta\neq \theta_0\) such that

In this case \(\hat{\theta}=\overline{X}\) but the distribution of \(f(x|\overline{X})\) doesn't make sense to me, in one example that I look they take \(f(\overline{x}|\hat{\theta})\) but I don't understood the logic.

I need to use the distribution of the likelihood estimator supposing that \(\theta=\hat{\theta}\)?

If someone can give me a explanation with details on how it works I really appreciate, I already read in the textbook but I don't understood.

\(0.5\). The remain values of \(\theta\) are given the density of \(g(\theta)\).

a)Assume \(g(\theta)\sim N(4.01,1)\) test the hypothesis

\(H_0:\theta=4.01\space vs\space H_1:\theta\neq 4.01\)

From what I learn to make a hypothesis test I need to find

\(a_0=P(\theta\in\Theta_0|x)\)

such that \(a_0+a_1=1\)

and reject \(H_0\) if \(a_0<a_1\)In the cases where the null hypothesis is not a point I can make, but in this case I have a few doubts.

From the notes that I take there is the theorem below

**Theorem:**For any prior \(\pi(\theta)=\pi_0\space \text{if}\space \theta=\theta_0\) \(\pi(\theta)=\pi_1 h(\theta)\space\text{if}\space

\theta\neq \theta_0\) such that

\(\int_{\theta\neq \theta_0}h(\theta)d(\theta)=1\)

then \(a_0=f(\theta|x)\geq [1+\frac{1-\pi_0}{\pi_0}\frac{r(x)}{f(x|\theta_0)}]^{-1}\)

where\(r(x)=sup_{\theta\neq\theta_0}f(x|\theta)\)

usually\(r(x)=f(x|\hat{\theta})\)

In this case \(\hat{\theta}=\overline{X}\) but the distribution of \(f(x|\overline{X})\) doesn't make sense to me, in one example that I look they take \(f(\overline{x}|\hat{\theta})\) but I don't understood the logic.

I need to use the distribution of the likelihood estimator supposing that \(\theta=\hat{\theta}\)?

If someone can give me a explanation with details on how it works I really appreciate, I already read in the textbook but I don't understood.

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