the lifespans of seals in a particular zoo are normally distributed. the average seal lives 13.8 years; the…

the lifespans of seals in a particular zoo are normally distributed. the average seal lives 13.8 years; the standard deviation is 3.2 years. use the empirical rule (68 - 95 - 99.7%) to estimate the probability of a seal living less than 7.4 years. % show calculator

the lifespans of seals in a particular zoo are normally distributed. the average seal lives 13.8 years; the standard deviation is 3.2 years. use the empirical rule (68 - 95 - 99.7%) to estimate the probability of a seal living less than 7.4 years. % show calculator

Answer

Explanation:

Step1: Calculate number of standard - deviations

Find how many standard deviations 7.4 is from the mean. Let $\mu = 13.8$ (mean) and $\sigma=3.2$ (standard deviation). Calculate $z=\frac{7.4 - 13.8}{3.2}=\frac{- 6.4}{3.2}=- 2$. So 7.4 is 2 standard deviations below the mean.

Step2: Apply empirical rule

The empirical rule states that for a normal distribution: about 95% of the data lies within 2 standard deviations of the mean, i.e., $P(\mu - 2\sigma<X<\mu + 2\sigma)\approx0.95$. The remaining area outside of this interval is $1 - 0.95 = 0.05$. Since the normal distribution is symmetric, the area below $\mu - 2\sigma$ is $\frac{1 - 0.95}{2}=0.025$.

Answer:

2.5%