Final answer:
Using the empirical rule, the probability of a meerkat living longer than 16.1 years can be estimated as 0.3%, since 16.1 years is three standard deviations away from the mean and approximately 99.7% of values fall within three standard deviations according to the rule.
Step-by-step explanation:
The student's question involves calculating the probability of a meerkat in a particular zoo living longer than 16.1 years given that meerkat lifespans are normally distributed with a mean of 10.4 years and a standard deviation of 1.9 years. To solve this question, we can use the empirical rule sometimes referred to as the 68-95-99.7 rule, which states that for a normal distribution:
- About 68% of the data falls within one standard deviation of the mean.
- About 95% falls within two standard deviations of the mean.
- About 99.7% falls within three standard deviations of the mean.
To find the probability of a meerkat living longer than 16.1 years, we first calculate how many standard deviations 16.1 is away from the mean. This is done by the formula:
Z = (X - Mean) / Standard Deviation
Where X is 16.1, the Mean is 10.4, and the Standard Deviation is 1.9. This calculation gives us a Z-score, which represents the number of standard deviations X is from the mean.
The calculation would be: Z = (16.1 - 10.4) / 1.9 = 3. We see that 16.1 years is three standard deviations away from the mean.
According to the empirical rule, 99.7% of values would fall within three standard deviations, but since we're looking at the probability of living longer, we're interested in the remaining 0.3%.
Therefore, it can be approximated that the probability of a meerkat living longer than 16.1 years is 0.3% or 0.003 in decimal form.