Question

the lifespans of lizards in a particular zoo are normally distributed. the average lizard lives 3.1 years; the standard deviation is 0.6 years. use the empirical rule (68 - 95 - 99.7%) to estimate the probability of a lizard living less than 2.5 years.

Explanation:

Step1: Calculate the z - score

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x = 2.5$, $\mu=3.1$, and $\sigma = 0.6$. So $z=\frac{2.5 - 3.1}{0.6}=\frac{- 0.6}{0.6}=-1$.

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, i.e., between $\mu-\sigma$ and $\mu + \sigma$. The area within $z=-1$ and $z = 1$ is 68%. The total area under the normal curve is 100%. The area outside of the interval $(\mu-\sigma,\mu+\sigma)$ is $100 - 68=32\%$. Since the normal distribution is symmetric, the area to the left of $z=-1$ is $\frac{100 - 68}{2}=16\%$.

Answer:

16