= Solution
For fixed $n$, the <likelihood ratio> is strictly decreasing as $t^2$ increases. A <likelihood-ratio test> rejects for small $\lambda$, so its rejection rule is equivalently large $|t|$. Under the <null hypothesis>, the standard <normal distribution> and <chi-squared distribution> results give independent quantities
$$
\frac{\sqrt n(\overline Y-\mu_0)}{\sigma}\sim N(0,1),\qquad
\frac{(n-1)s_Y^2}{\sigma^2}\sim\chi^2_{n-1}.
$$
Their ratio therefore has <Student's t-distribution> with $n-1$ <statistical degrees of freedom>. The exact level-$\alpha$ rule is
$$
\boxed{|t|>t_{n-1,\,1-\alpha/2}.}
$$
This is precisely the two-sided <Student's t-test>; choosing the corresponding cutoff for $\lambda$ yields the same rejection region and the same <p-value>.
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