Question

the lifespans of gorillas in a particular zoo are normally distributed. the average gorilla lives 20.8 years; the standard deviation is 3.1 years. use the empirical rule (68 - 95 - 99.7%) to estimate the probability of a gorilla living less than 23.9 years.

Explanation:

Step1: Calculate the z - score

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x = 23.9$, $\mu=20.8$, and $\sigma = 3.1$. So $z=\frac{23.9 - 20.8}{3.1}=\frac{3.1}{3.1}=1$.

Step2: Apply the empirical rule

The empirical rule for a normal distribution states that about 68% of the data lies within 1 standard - deviation of the mean ($\mu\pm\sigma$), which means 34% of the data lies between $\mu$ and $\mu+\sigma$. Also, 50% of the data lies below the mean. Since $x = 23.9$ is 1 standard - deviation above the mean, the percentage of data below $x = 23.9$ is $50\%+34\% = 84\%$.

Answer:

84