The empirical rule (also known as the 68-95-99.7 rule) helps estimate probabilities for a normal distribution based on how many standard deviations a value is from the mean.

Step-by-step breakdown:

1. Given Information:

   • The average lifespan of the lizards: $\mu = 3.1$ years.
   • The standard deviation: $\sigma = 0.6$ years.
   • We are interested in the probability that a lizard lives longer than $2.5$ years.

2. Determine how far $2.5$ is from the mean: $z = \frac{X - \mu}{\sigma}$ Here $X = 2.5$, so: $z = \frac{2.5 - 3.1}{0.6} = \frac{-0.6}{0.6} = -1$ A $z$-score of $-1$ means that $2.5$ years is 1 standard deviation below the mean.

3. Use the empirical rule:

   • According to the empirical rule, 68% of the data falls within 1 standard deviation from the mean ($\mu \pm 1\sigma$).
   • This means that 34% of the data falls between the mean and 1 standard deviation below the mean, and another 34% falls between the mean and 1 standard deviation above the mean.

4. Calculate the probability of living longer than 2.5 years:

   • Since the normal distribution is symmetric, the area to the right of 1 standard deviation below the mean (i.e., $2.5$ years) includes half of the data (50%) plus the area from 2.5 years to 3.1 years (which is 34%).
   • Thus, the total probability of living longer than 2.5 years is: $50\% + 34\% = 84\%$

Conclusion:

The probability that a lizard will live longer than 2.5 years is approximately 84%.

Would you like more details or clarification on any part of this?
Here are some related questions:

1. How would the probability change if we wanted to find the lifespan longer than 3.7 years?
2. What is the probability of a lizard living between 2.5 and 3.7 years?
3. How would the empirical rule apply to 2 standard deviations above and below the mean?
4. What are the limits of using the empirical rule for more precise probabilities?
5. How does the empirical rule differ from using a standard normal distribution table?

Tip: The empirical rule is a quick way to estimate probabilities for normal distributions, but for more precise answers, using the standard normal table or software is recommended.