Assume that at the average age of a population of wild turtles is normally distributed with mean age 15 years, and standard deviation 3 years. You see one of the turtles in the park. What is the age of a turtle who is at the 54 percentile? Round to two decimal places.
Assume that at the average age of a population of wild turtles is normally distributed with mean age 15 years, and standard deviation 3 years. You see one of the turtles in the park. What is the age of a turtle who is at the 54 percentile? Round to two decimal places.
MATLAB: An Introduction with Applications
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![**Understanding Percentiles in a Normally Distributed Population**
In this educational example, we are given a scenario about the ages of a population of wild turtles. The age distribution is normally distributed with:
- **Mean (average) age**: 15 years
- **Standard deviation**: 3 years
The problem is to determine the age of a turtle that falls at the 54th percentile.
To solve this problem, you need to understand percentiles in the context of a normal distribution:
1. **Normal Distribution**: This is a probability function that describes how the values of a variable are distributed. It's symmetric around the mean, with most values clustering around a central region and the probabilities for values tapering off equally on both sides.
2. **Percentile**: The nth percentile is a value below which n percent of data falls. For example, the 54th percentile is the value below which 54% of the observations may be found.
3. **Finding the Value at the 54th Percentile**:
- You use the standard normal distribution or a Z-table to find a z-score corresponding to the 54th percentile.
- Then, apply the z-score formula:
\[
X = \mu + (Z \times \sigma)
\]
Where:
- \(X\) is the value at the specific percentile.
- \(\mu\) is the mean value.
- \(Z\) is the z-score from the standard normal distribution for the given percentile.
- \(\sigma\) is the standard deviation.
Finally, solve for \(X\) to find the age of the turtle at the 54th percentile and round your answer to two decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97b5df25-1667-438a-b46e-c2a0d1ae7c86%2F8c79579c-8a2f-4f81-800c-163d7236b023%2Fulcthba_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Percentiles in a Normally Distributed Population**
In this educational example, we are given a scenario about the ages of a population of wild turtles. The age distribution is normally distributed with:
- **Mean (average) age**: 15 years
- **Standard deviation**: 3 years
The problem is to determine the age of a turtle that falls at the 54th percentile.
To solve this problem, you need to understand percentiles in the context of a normal distribution:
1. **Normal Distribution**: This is a probability function that describes how the values of a variable are distributed. It's symmetric around the mean, with most values clustering around a central region and the probabilities for values tapering off equally on both sides.
2. **Percentile**: The nth percentile is a value below which n percent of data falls. For example, the 54th percentile is the value below which 54% of the observations may be found.
3. **Finding the Value at the 54th Percentile**:
- You use the standard normal distribution or a Z-table to find a z-score corresponding to the 54th percentile.
- Then, apply the z-score formula:
\[
X = \mu + (Z \times \sigma)
\]
Where:
- \(X\) is the value at the specific percentile.
- \(\mu\) is the mean value.
- \(Z\) is the z-score from the standard normal distribution for the given percentile.
- \(\sigma\) is the standard deviation.
Finally, solve for \(X\) to find the age of the turtle at the 54th percentile and round your answer to two decimal places.
Expert Solution
Step 1
In question,
Let X be random variable denoting the age of a population of wild turtles.
Given, X ~ N(15, 3)
Then We'll find the 54th percentile.
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