The lifespans of lizards in a particular zoo are normally distributed. The average lizard lives \[3.1\] years; the standard deviation is \[0.6\] years.

**Use the empirical rule \[(68 - 95 - 99.7\%)\] to estimate the probability of a lizard living longer than \[2.5\] years.**

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.

The lifespans of lizards in a particular zoo are normally distributed. The average lizard lives 3.1 years; the standard deviation is 0.6 years. Use the empirical rule (68 - 95 - 99.7%) to estimate the probability of a lizard living longer than 2.5 years.

Learn how to estimate the probability of a lizard living longer than 2.5 years in a normally distributed population. Using the empirical rule (68-95-99.7%), this problem involves calculating the z-score and understanding standard deviations. Suitable for students in Grades 10-12.

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