Question

the patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.9 days and a standard deviation of 2.1 days. What is the 85th percentile for recovery times?

Answer 1

The 85th percentile is the cutoff time $t$ such that

$\mathbb P(X$

In other words, the 85th percentile refers to the time needed to belong to the top 15% of the distribution; more generally, the $n$ percentile is the top $(100-n)\%$ of the distribution.

Anyway, to find this value of $t$, transform $X$ to a random variable $Z$ with the standard normal distribution using

$Z=\dfrac{X-\mu}\sigma$

where $\mu$ is the mean of $X$ and $\sigma$ is the standard deviation of $X$.

$\mathbb P(X$

Here $t^*$ is used to denote the z-score corresponding to the cutoff time $t$. Referring to a z-score table, you find that this occurs for $t^*\approx1.036$. So,

$\dfrac{t-5.9}{2.1}=t^*\implies t=5.9+2.1t^*\approx8.076$