To estimate the probability of a lizard living less than $2.5$ years using the empirical rule (also known as the 68-95-99.7 rule), we need to understand how many standard deviations $2.5$ years is away from the mean.

Step 1: Calculate how far $2.5$ years is from the mean in terms of standard deviations.

The mean lifespan ($\mu$) of the lizards is $3.1$ years, and the standard deviation ($\sigma$) is $0.6$ years. We compute the z-score to find how many standard deviations below the mean the value $2.5$ is:

$z = \frac{X - \mu}{\sigma} = \frac{2.5 - 3.1}{0.6} = \frac{-0.6}{0.6} = -1$

So, $2.5$ years is 1 standard deviation below the mean.

Step 2: Use the empirical rule.

According to the empirical rule:

• 68% of the data lies within 1 standard deviation of the mean, which means between $2.5$ and $3.7$ years (since $\mu - \sigma = 2.5$ and $\mu + \sigma = 3.7$).
• This implies that 68% of the lizards live between 2.5 and 3.7 years.
• Since the normal distribution is symmetric, half of this 68% (i.e., 34%) is between the mean ($3.1$) and $2.5$ years.

Step 3: Calculate the probability of a lizard living less than 2.5 years.

The empirical rule also tells us that 68% of the data is within 1 standard deviation (between $2.5$ and $3.7$), so:

• $50\%$ of the lizards live below the mean.
• Subtracting the $34\%$ that live between $2.5$ and $3.1$ years, the remaining portion living less than $2.5$ years is:

$50\% - 34\% = 16\%$

Therefore, the probability that a lizard lives less than $2.5$ years is approximately 16%.

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Would you like more details on the empirical rule or how this probability was derived?

Here are some follow-up questions to explore further:

1. What is the probability of a lizard living between 2.5 and 3.7 years?
2. How can you calculate the probability of a lizard living more than 4.3 years using the empirical rule?
3. What does it mean when a distribution is "normally distributed"?
4. How would you calculate the z-score for a lizard that lives 4 years?
5. What percentage of lizards live between 2.5 and 4.3 years?

Tip: The empirical rule is a quick way to estimate probabilities, but for more precision, using z-scores with a normal distribution table is recommended.