In the standard normal distribution, we have considered the case where you get the probability when given an interval. However, what about the reverse problem of finding an interval that would result in a given probability? That is, to solve for example the problem

\begin{equation*} \Phi(b) - \Phi(a) = P(a < z < b) = 0.6217. \end{equation*}

To deal with this we need an "inverse function" \(\Phi^{-1}\text{.}\) Toward that end, consider the simpler problem of solving

\begin{equation*} \Phi(z_0) = P(z < z_0) = \text{some given probability value} = \alpha. \end{equation*}

Then, technically the answer would be

\begin{equation*} z_0 = \Phi^{-1}(\alpha). \end{equation*}

Since integrating the normal probability function is impossible you can expect that finding a nice formula for the inverse of that integration might also be challenging and that is certainly the case. However, you have two options:

Guessing \(z_0\) until you get a probabilty that is close enough to \(\alpha\text{.}\)
Use a calculator that has the inverse built in!

For most graphing calculators, there is a function called "invNorm" and that is a way to compute values involving \(\Phi^{-1}\text{.}\) Indeed, for example if you wanted to solve

\begin{equation*} \Phi(z_0) = P(z < z_0) = 0.813 \end{equation*}

then just use invNorm(\(0.813\)) to get \(z_0 \approx -0.2198\text{.}\)

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