I'm exploring the likelihood ratio principle in hypothesis testing, specifically within the context of normal distributions, and I've encountered a challenge in deriving a specific likelihood ratio. The principle is typically used to select a suitable statistic for testing hypotheses, where we compare the likelihood of the data under the null hypothesis against an alternative hypothesis.
Consider a scenario where we have a set of samples $x_1, \ldots, x_n$ that are independently and identically distributed (i.i.d.) from a normal distribution $N(\mu, \sigma^2)$ with unknown parameters $\mu$ and $\sigma^2$. We're interested in testing the null hypothesis $H_0: \mu = \mu_0$ against the alternative $H_a: \mu \neq \mu_0$, where $\mu_0$ is a specified value.
Let $L(\widehat{\Omega}_0)$ denote the maximum likelihood of observing the samples given $\mu = \mu_0$, and let $L(\widehat{\Omega})$ denote the maximum likelihood over all possible values of $\mu$ and $\sigma^2$. According to the likelihood ratio principle, the rejection region for $H_0$ is determined by the ratio $\frac{L(\widehat{\Omega}_0)}{L(\widehat{\Omega})}$ being less than or equal to a critical value $c$, which is chosen based on the desired level of statistical significance $\alpha$.
I am trying to show that, for this particular setup, the likelihood ratio simplifies to $\left(1 + \frac{t^2}{n-1}\right)^{-\frac{n}{2}}$, where $t$ is the test statistic defined as $t = \frac{\overline{x} - \mu_0}{s / \sqrt{n}}$, with $\overline{x}$ being the sample mean and $s^2$ the unbiased sample variance.
I've made several attempts to derive this expression from the definition of the likelihood ratio, considering the probability density function of the normal distribution, but I'm not sure how to proceed. Could someone guide me through the derivation or point out any resources that could help with understanding this specific case of the likelihood ratio in hypothesis testing for normal distributions?
$\overline{x}= \frac{1}{n}\Sigma_{i=1}^{n} x_i $ and $s^2 = \frac{1}{n-1}\Sigma_{i=1}^{n}\left(x_i-\overline{x} \right)^2 $