Answer :
- Calculate the locator $L$ using the formula $L = 0.81
\times 117 = 94.77$.
- Since $L$ is not an integer, round it up to the nearest integer: $\lceil 94.77 \rceil = 95$.
- The 81st percentile, $P_{81}$, is the 95th value in the sorted data set.
- Therefore, $L = 94.77$ and $P_{81} = \boxed{61.2}$.
### Explanation
1. Problem Setup and Goal
We are given a sorted data set with $n = 117$ values and are asked to find the 81st percentile, $P_{81}$. To do this, we first need to calculate the locator $L$.
2. Calculate the Locator
The locator $L$ is calculated using the formula $L = P \times n$, where $P$ is the percentile expressed as a decimal and $n$ is the number of data points. In this case, $P = 0.81$ and $n = 117$. Therefore, we have:
$$L = 0.81 \times 117 = 94.77$$
3. Round the Locator
Since $L = 94.77$ is not an integer, we round it up to the nearest integer to find the position of the 81st percentile in the data set. The ceiling of $94.77$ is $95$. That is, $\lceil 94.77 \rceil = 95$.
4. Determine the Percentile Value
The 81st percentile, $P_{81}$, is the 95th value in the sorted data set. Looking at the data set, the 95th value is $61.2$.
5. Final Answer
Therefore, the locator is $L = 94.77$, and the 81st percentile is $P_{81} = 61.2$.
### Examples
Percentiles are useful in many real-world situations. For example, if you score in the 81st percentile on a standardized test, it means you scored better than 81% of the other test takers. In finance, percentiles can be used to understand the distribution of returns for a particular investment. In healthcare, they help track the growth of children by comparing their height and weight to others in their age group.
\times 117 = 94.77$.
- Since $L$ is not an integer, round it up to the nearest integer: $\lceil 94.77 \rceil = 95$.
- The 81st percentile, $P_{81}$, is the 95th value in the sorted data set.
- Therefore, $L = 94.77$ and $P_{81} = \boxed{61.2}$.
### Explanation
1. Problem Setup and Goal
We are given a sorted data set with $n = 117$ values and are asked to find the 81st percentile, $P_{81}$. To do this, we first need to calculate the locator $L$.
2. Calculate the Locator
The locator $L$ is calculated using the formula $L = P \times n$, where $P$ is the percentile expressed as a decimal and $n$ is the number of data points. In this case, $P = 0.81$ and $n = 117$. Therefore, we have:
$$L = 0.81 \times 117 = 94.77$$
3. Round the Locator
Since $L = 94.77$ is not an integer, we round it up to the nearest integer to find the position of the 81st percentile in the data set. The ceiling of $94.77$ is $95$. That is, $\lceil 94.77 \rceil = 95$.
4. Determine the Percentile Value
The 81st percentile, $P_{81}$, is the 95th value in the sorted data set. Looking at the data set, the 95th value is $61.2$.
5. Final Answer
Therefore, the locator is $L = 94.77$, and the 81st percentile is $P_{81} = 61.2$.
### Examples
Percentiles are useful in many real-world situations. For example, if you score in the 81st percentile on a standardized test, it means you scored better than 81% of the other test takers. In finance, percentiles can be used to understand the distribution of returns for a particular investment. In healthcare, they help track the growth of children by comparing their height and weight to others in their age group.
