How to eliminate constant to derive the decision rule in terms of the sufficient statistic $\bar{X}$ for normal distribution means hypothesis test?

Question

Suppose that we have a random sample, of size $n$, from a population that is normally-distributed. Both the mean, $\mu$, and the standard deviation, $\sigma$, of the population are unknown. We want to test whether the mean is equal to a given value, $\mu_{0}$. Thus, our null hypothesis is $H_{0}: \mu=\mu_{0}$ and our alternative hypothesis is $H_{1}: \mu \neq \mu_{0}$. The likelihood function is $$ \mathcal{L}(\mu, \sigma \mid x)=\left(2 \pi \sigma^{2}\right)^{-n / 2} \exp \left(-\sum_{i=1}^{n} \frac{\left(x_{i}-\mu\right)^{2}}{2 \sigma^{2}}\right) $$ With some calculation (omitted here), it can then be shown that $$ \lambda=\left(1+\frac{t^{2}}{n-1}\right)^{-n / 2} $$ where $t$ is the $t$-statistic with $n-1$ degrees of freedom. Hence we may use the known exact distribution of $t_{n-1}$ to draw inferences.

The likelihood-ratio test provides the decision rule as follows: If $\Lambda>c$, do not reject $H_{0}$; If $\Lambda<c$, reject $H_{0}$ Reject with probability $1$ if $\Lambda=c$. Reject with probability 1 if $\Lambda<c$. The values $c$ and $q$ are usually chosen to obtain a specified significance level $\alpha$, via the relation $\cdot \mathrm{P}\left(\Lambda=c \mid H_{0}\right)+\mathrm{P}\left(\Lambda<c \mid H_{0}\right)=\alpha .$

To determine the decision rule/region region for this test

$ P( \lambda = c|\mu=\mu_0) + P(\lambda<c | \mu=\mu_0)= \alpha$

$P(t^2\leq(c^{\frac{-2}{n}}-1)(n-1)| \mu=\mu_{0})=\alpha$

$P(({\dfrac{\bar{X}-\mu_0}{\dfrac{s}{\sqrt{n}}}})^2 \leq c^{\dfrac{-2}{n}}(n-1))=\alpha$

How do I eliminate "$c$" to make the decision rule or rejection region only depend on the sufficient statistic, $\mu_0$, the sample variance $s^2$, and sample-size, $n$?

Reference: https://en.wikipedia.org/wiki/Likelihood-ratio_test

Answer 1

The statistic $t$ (and hence $\lambda$ also) is continuous. The probability of equality is $0$.

That equality term is important for test statistics that are not continuous, where that probability is not zero. You can safely focus on the inequality in this case.

Note that you can simplify your inequality by moving things that are not $t^2$ across to the other side of the "$<$" symbol (take the same power of both sides, multiply both sides by the relevant constant, etc). You'll get some new constant on the right, $c_1$, say, in terms of something with a known distribution. Once you know $c_1$, you can compute $c$ (if you need it at all).