Question

the lifespans of gorillas in a particular zoo are normally distributed. the average gorilla lives 16 years; the standard deviation is 1.7 years. use the empirical rule (68 - 95 - 99.7%) to estimate the probability of a gorilla living less than 14.3 years.

Explanation:

Step1: Calculate the z - score

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x = 14.3$, $\mu=16$, and $\sigma = 1.7$. So $z=\frac{14.3 - 16}{1.7}=\frac{-1.7}{1.7}=- 1$.

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 that the area between $z=-1$ and $z = 1$ is 68%. The total area under the normal curve is 100%. The area to the left of $z = 0$ is 50%. Since the area between $z=-1$ and $z = 1$ is 68%, the area between $z=-1$ and $z = 0$ is $\frac{68\%}{2}=34\%$. So the area to the left of $z=-1$ is $50\%-34\% = 16\%$.

Answer:

16