The empirical rule (68-95-99.7%) helps approximate probabilities in a normal distribution based on the number of standard deviations a data point lies from the mean. Here's the breakdown:

Step 1: Identify the given values

• Mean ($\mu$) = $3.1$ years
• Standard deviation ($\sigma$) = $0.6$ years

Step 2: Standard deviations for the given range

1. Lower bound: $2.5$ $z = \frac{2.5 - 3.1}{0.6} = \frac{-0.6}{0.6} = -1$ This is $1$ standard deviation below the mean.

2. Upper bound: $4.3$ $z = \frac{4.3 - 3.1}{0.6} = \frac{1.2}{0.6} = 2$ This is $2$ standard deviations above the mean.

Step 3: Apply the empirical rule

• Within $1$ standard deviation ($\mu \pm \sigma$): 68% of the data is included.
• Within $2$ standard deviations ($\mu \pm 2\sigma$): 95% of the data is included.

The range $2.5$ to $4.3$ spans:

• From $1$ standard deviation below the mean ($-1$) to $2$ standard deviations above the mean ($+2$).

Step 4: Probability estimation

1. The total probability within $2$ standard deviations ($\pm 2\sigma$) is $95\%$.

2. The range from $-1\sigma$ to $+2\sigma$ excludes the lower tail beyond $-1\sigma$. Half of the remaining $5\%$ lies below $-2\sigma$, and the other half ($2.5\%$) lies below $-1\sigma$.

   Therefore, the range from $-1\sigma$ to $+2\sigma$ covers: $95\% - 2.5\% = 92.5\%.$

Final Answer:

The probability of a lizard living between $2.5$ and $4.3$ years is approximately 92.5%.

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Would you like more details or have any questions?

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Related Questions:

1. How do we calculate probabilities beyond $3$ standard deviations using the empirical rule?
2. What is the probability of a lizard living less than $2.5$ years?
3. Can you calculate probabilities using exact z-scores instead of the empirical rule?
4. How does the empirical rule differ from other methods like the z-table?
5. What is the probability of a lizard living more than $4.3$ years?

Tip:

When using the empirical rule, remember that it only provides approximate probabilities. For precise values, you can use the standard normal distribution table or software tools.