CLT with Estimated Variance via Slutsky's Theorem

Let $X_1, \ldots, X_n$ be i.i.d.\ with mean $\mu$ and finite variance $\sigma^2 > 0$. Define the sample variance $S_n^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X}_n)^2$ and the studentized statistic $$T_n = \frac{\sqrt{n}\,(\bar{X}_n - \mu)}{S_n}.$$

(a) Using the LLN and Slutsky's theorem, show that $T_n \xrightarrow{d} N(0,1)$.

(b) In a study with $n = 100$ observations, you find $\bar{X}_{100} = 12.5$ and $S_{100} = 3.0$. Assuming the true mean is $\mu_0 = 12$, approximate $P(\bar{X}_{100} > 12.5)$ using $T_n$.

You may use $\Phi(1.67) \approx 0.9525$.