Final answer:
Using the empirical rule for a normally distributed data set with a mean of 22.4 years and a standard deviation of 2.7 years, the probability of a tiger living longer than 14.3 years is approximately 99.7%.
Step-by-step explanation:
The empirical rule, also known as the 68-95-99.7 rule, applies to normally distributed data. Since the lifespans of tigers at the zoo are normally distributed with a mean (average) lifespan of 22.4 years and a standard deviation of 2.7 years, we can apply this rule to estimate the probability of a tiger living longer than 14.3 years.
According to the empirical rule:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% falls within two standard deviations.
- Approximately 99.7% falls within three standard deviations.
To determine the number of standard deviations 14.3 is away from the mean, we calculate the z-score:
Z = (X - mean) / standard deviation = (14.3 - 22.4) / 2.7 = -3.00
A z-score of -3 indicates that 14.3 years is three standard deviations below the mean. Since the empirical rule states that 99.7% of the data falls within three standard deviations of the mean, the probability of a tiger living longer than 14.3 years is approximately 100% - 0.3% (the remaining 0.3% fall beyond three standard deviations), which is 99.7%.