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 between 11.5 and 27 years.

Option 1: Approximately 34% of gorillas have lifespans between 11.5 and 27 years.
Option 2: Approximately 68% of gorillas have lifespans between 11.5 and 27 years.
Option 3: Approximately 95% of gorillas have lifespans between 11.5 and 27 years.
Option 4: Approximately 99.7% of gorillas have lifespans between 11.5 and 27 years.

Answer 1

Approximately 95% of gorillas have lifespans between 11.5 and 27 years. Option 3

Step-by-step explanation:

The empirical rule, also known as the 68-95-99.7% rule, states that in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Given that the average lifespan of gorillas in the zoo is 20.8 years with a standard deviation of 3.1 years, we can apply this rule to estimate the probability of a gorilla living between 11.5 and 27 years.

First, calculate the distance from the mean to each endpoint of the range: for 11.5 years, it is (11.5 - 20.8) / 3.1 ≈ -3, and for 27 years, it is (27 - 20.8) / 3.1 ≈ 2. Standardizing these values, we find that the range between -3 and 2 standard deviations from the mean covers approximately 95% of the distribution. Therefore, approximately 95% of gorillas are expected to have lifespans between 11.5 and 27 years.

In conclusion, the correct option is Option 3, as it aligns with the empirical rule and accurately represents the estimated probability based on the normal distribution characteristics of the gorillas' lifespans in the zoo.Option 3