What is the meaning of $Z_{0.3}$ and how do I calculate it?
I know it was calculated this way: $$Z_{0.3} = -Z_{0.7} = -0.52$$
I tried to follow the General Distribution table but I can't seem to find the way to get this.
It's quite hard to look for information about a term you have no clue how it is called.
This is the quantile of the standard normal distribution. In other words, if $\Phi(x)$ is the cdf of a normal distribution with mean 0 and variance 1, then $Z_{0.3}$ simply means the value of $x$ for which $\Phi(x)=0.3$. Essentially, it's the inverse of the cdf.
The equation given follows from the symmetry of the standard normal curve about the mean 0.
If you have access to a table a table of values of the cdf $\Phi$ then you can look for the value closest to 0.7 in the table and see which $x$ gives that $Z$ value.
Hope this helps.