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 less than 
\[2.5\] years.

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%**.

---

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.

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 less than 2.5 years.

Explore how to estimate the probability of a lizard living less than 2.5 years using the empirical rule (68-95-99.7%) and normal distribution. Given the average lifespan of 3.1 years and a standard deviation of 0.6 years, learn how to calculate z-scores and apply the empirical rule for accurate probability estimation. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation