Question

the lifespans of meerkats in a particular zoo are normally distributed. the average meerkat lives 13.1 years; the standard deviation is 1.5 years. use the empirical rule (68 - 95 - 99.7%) to estimate the probability of a meerkat living between 16.1 and 17.6 years.

Explanation:

Step1: Calculate z - scores

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 13.1$ (mean), $\sigma=1.5$ (standard deviation).
For $x = 16.1$, $z_1=\frac{16.1 - 13.1}{1.5}=\frac{3}{1.5}=2$.
For $x = 17.6$, $z_2=\frac{17.6 - 13.1}{1.5}=\frac{4.5}{1.5}=3$.

Step2: Apply empirical rule

The empirical rule states that about 95% of the data lies within $z=\pm2$ (i.e., between $\mu - 2\sigma$ and $\mu+2\sigma$) and about 99.7% of the data lies within $z = \pm3$ (i.e., between $\mu-3\sigma$ and $\mu + 3\sigma$).
The percentage of data between $z = 2$ and $z = 3$ is $\frac{99.7 - 95}{2}=2.35$.

Answer:

$2.35$