Answer:
97.35%
Step-by-step explanation:
To estimate the probability of a gorilla living between 11.5 and 27 years using the empirical rule, we first need to determine how many standard deviations away from the mean 11.5 and 27 years are.
Given values:
- Mean (μ) = 20.8 years
- Standard deviation (σ) = 3.1 years
- Target lifespans = 11.5 and 27 years
Let n be the number of standard deviations.
[tex]\mu + n\sigma = 11.5\\\\20.8 + 3.1n = 11.5\\\\3.1n=-9.3\\\\n=-3[/tex]
Therefore, 11.5 years is 3 standard deviations below the mean.
[tex]\mu + n\sigma = 27\\\\20.8 + 3.1n = 27\\\\3.1n=6.2\\\\n=2[/tex]
Therefore, 27 years is 2 standard deviations above the mean.
According to the empirical rule, approximately 95% of the data falls within 2 standard deviations of the mean, and 99.7% of the data falls within 3 standard deviations of the mean.
Therefore, to estimate the probability of a gorilla living between 11.5 and 27 years, we need to add the area of 3 standard deviations to the left of the mean to the area of 2 standard deviations to the right of the mean:
[tex]P(11.5 < X < 27)=\dfrac{99.7\%}{2}+\dfrac{95\%}{2}\\\\\\P(11.5 < X < 27)=49.85\%+47.5\%\\\\\\P(11.5 < X < 27)=97.35\%[/tex]
Therefore, the probability of a gorilla living between 11.5 and 27 years is approximately 97.35%.
