Compare the value of computed $\chi^2$ and chi-squared distribution table to confirm “will the event happen”.
If $\chi^2_{0.975} < \chi^2 < \chi^2_{0.025}$, we can convince the event will happen.
$X=Z^2_1+Z^2_2+\cdots+Z^2_n$ follows a chi-squared distribution with $n$ DOF, denoted by $X\sim\chi^2_n$.
If $X_1\sim\chi^2_{n_1}$ and $X_2\sim\chi^2_{n_2}$,
then $X_1 + X_2\sim\chi^2_{n_1+n_2}$.
$\chi^2\sim(n-1)\frac{S^2}{\sigma^2}$ has a chi-squared distribution with DOF $v=n-1$. Note: DOF is $n-1$ rather than $n$.
The target is to confirm whether the statement is creditable, which means we should find the $\mathcal X^2$ value.
$\mu=3,\sigma=1$ is given.
$n=5$ is given with specified values, which means $S^2$ can be calculated.