We are sometimes given the value of the probability, \(P(X \le x)\) and need to find the value \(x\). If we are provided with a probability, \(p\), then the value \(x\) such that
\[ P(X \le x) = p \]is the \(p\)'th quantile of the distribution of \(X\). We now give an example to illustrate the use of quantiles for a normally distributed random variable.
Example
If the weight of a Fuji apple has the following normal distribution
\[ X \;\; \sim \; \; \NormalDistn(\mu=180, \sigma=10) \]what is the apple weight that will be exceeded with 95% probability? In other words, we want to find the apple weight \(x\) such that
\[ P(X \lt x) \;\;= \;\; 0.05 \]In terms of z-scores,
\[ P(X \lt x) \;= \; P\left(Z \lt \frac {x-180} {10}\right) \;= \; 0.05 \]Using the function "=NORM.S.INV(0.05)" in Excel, we can find that
\[ P(Z \lt -1.645) \;\;=\;\; 0.05 \]Translating back to the original units,
\[ x \;=\; 180 - 1.645 \times 10 \;=\; 163.55 \text{ grams} \]