"In a factory we produce rods the length of which is planned to be $80$ cm. We assume that the length of the rods varies randomly having the normal distribution. A sample of $100$ rods is taken in which the mean was measured to be $82$ cm and the variance to be $0.025$ cm$^2$. Test the null hypothesis that the length of the rods produced in the factory have the desired value of $80$ cm. The alternative hypothesis is that the length differs from the desired value of $80$ cm."
Now, obtaining $\mu = 82$, $\mu_{0} = 80$, $n = 100$ and $S^{2} = 0.025$, we can compute the t value:
$$ t = \frac{\mu - \mu_{0}}{\sqrt{\frac{S^2}{n}}} = \frac{82-80}{\sqrt{\frac{0.025}{100}}} \approx 126.49 \ . $$
From the assumptions, it is also clear that
$$ \frac{\mu - \mu_{0}}{\sqrt{\frac{S^2}{n}}} \sim t(n-1) \ . $$
But since we're not going for any specific significance level, how should we proceed i.e. how does one compute $t(99)$ without the significance level from charts for example? I'm also confused we're looking for a very specific value on a continuous distribution which is impossible is it not? How do we reason to answer the question?