Question

Math

Posted 4 months ago

```
The lifespans of meerkats in a particular zoo are normally distributed. The average meerkat lives 13.1 years; the standard deviation is 1.5 years.
Use the empirical rule $(68-95-99.7 \%)$ to estimate the probability of a meerkat living less than 14.6 years.
$\square$ $\%$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Identify the mean $\mu$ and standard deviation $\sigma$ of the normal distribution for meerkat lifespans

step 2

Calculate the z-score for the value 14.6 years using the formula $z = \frac{x - \mu}{\sigma}$

step 3

Use the empirical rule to estimate the probability of a meerkat living less than 14.6 years

step 4

Since 14.6 is within one standard deviation above the mean, approximately 68% of the data lies within one standard deviation of the mean

step 5

To find the probability of a meerkat living less than 14.6 years, we take half of 68% (which accounts for the lower side of the mean) and add it to the 50% (which accounts for the lower half of the distribution)

step 6

The calculation is $50\% + \frac{68\%}{2} = 50\% + 34\% = 84\%$

[1] Answer

84%

Key Concept

Empirical Rule

Explanation

The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. To find the probability of a value being less than a certain number within one standard deviation above the mean, we add half of 68% to 50%.

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