Working backwards — finding x, μ or σ from a probability

In the last lesson the value $x$ was given and you found an area. Now the area is given and you must find $x$.

The table lists $\Phi(z)=\mathrm{P}(Z<z)$. To reverse it, look inside the table for the probability, then read out to the $z$ value.

The area you search for must be a left-tail area, because that is what the table stores. If the question gives an upper-tail probability, subtract it from 1 first: $\mathrm{P}(X>k)=0.2 \quad\Longrightarrow\quad \mathrm{P}(X<k)=0.8.$

Once you have $z$, convert back to the original scale. Start from $z=\dfrac{x-\mu}{\sigma}$, multiply both sides by $\sigma$ to get $\sigma z = x-\mu$, then add $\mu$ to both sides: $x=\mu+\sigma z.$

Some probabilities appear so often that the formulae booklet prints their z-values in a small separate table of critical values under the main normal table. Here they are:

For a left-tail probability below 0.5, the z-value is negative. For example, the lower 10% boundary has $z=-1.282$, since $\Phi(-1.282)=0.10$.

Always sketch. Shade the area named in the question, decide whether $k$ is above or below the mean, and check that the sign of your z matches the picture.