Assume $X_i$, where $i=1,..,n$, is random sample from normal distribution $N(\mu, \sigma^2)$, where both $\mu$ and $\sigma^2$ are unknown. Suppose we use $C = \frac{\bar X_n}{S_n} \gt k$ as the rejection region under $H_0: \mu=0$ against $H_a: \mu \gt 0$. Try to find the k value if we let $\alpha = 0.1$, and try to verify whether this is LRT.
I only know that I should let $P(\frac{\bar X_n}{S_n} \gt k)=0.1$, but I don't know what to do next. Any help or hint would be helpful