The empirical rule (or 68-95-99.7 rule) helps estimate the probability within a normal distribution for intervals from the mean, defined in terms of standard deviations:

• 68% of values lie within 1 standard deviation of the mean.
• 95% of values lie within 2 standard deviations of the mean.
• 99.7% of values lie within 3 standard deviations of the mean.

Here, we are given:

• Mean lifespan ($\mu$) = $20.5$ years
• Standard deviation ($\sigma$) = $3.9$ years
• We want the probability of a zebra living longer than $32.2$ years.

Step 1: Determine How Many Standard Deviations Above the Mean $32.2$ Is

Calculate the distance of $32.2$ from the mean in terms of standard deviations:

$\frac{32.2 - 20.5}{3.9} = \frac{11.7}{3.9} \approx 3$

This shows that $32.2$ years is 3 standard deviations above the mean.

Step 2: Use the Empirical Rule to Estimate Probability

According to the empirical rule:

• 99.7% of values lie within 3 standard deviations of the mean.
• This means that only 0.3% of values lie outside this range, equally divided in both tails (above and below 3 standard deviations from the mean).

So, the probability of a zebra living longer than $32.2$ years (more than 3 standard deviations above the mean) is approximately:

$\frac{0.3\%}{2} = 0.15\%$

Final Answer

The probability of a zebra living longer than $32.2$ years is approximately 0.15%.

Would you like more details, or do you have any questions?

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1. What is the empirical rule and how is it used in probability estimation?
2. How do we interpret standard deviation in the context of a normal distribution?
3. What would the probability be for a zebra to live less than 8.8 years, given the same distribution?
4. How can we calculate probabilities more accurately without the empirical rule?
5. What are the characteristics of a normal distribution?

Tip: The empirical rule is most effective when data closely follows a normal distribution, providing a quick approximation without calculations.