The data set below represents the ages of 36 executives. Find the percentile that corresponds to an age of 64 years old. 29 29 30 32 32 32 34 36 41 41 43 43 43 49 49 51 52 53 53 54 55 55 55 59 60 61 61 62 63 63 63 63 64 65 65 66 Percentile of 64 = (Round to the nearest integer as needed.)
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![### Understanding Percentiles in a Dataset
**Dataset Analysis**
The data set below represents the ages of 36 executives. The task is to find the percentile that corresponds to an age of 64 years old.
#### Age Data
```
29, 29, 30, 32, 32, 32, 34, 36, 41, 41,
43, 43, 43, 49, 49, 51, 52, 53, 53, 54,
55, 55, 55, 59, 60, 61, 61, 62, 63, 63,
63, 64, 65, 65, 65, 66
```
#### Calculating the Percentile
To determine the percentile corresponding to an age of 64:
1. **Arrange the data in ascending order**: This step is already done as the data is sorted.
2. **Locate the position of the age 64** within the dataset. Since this dataset is sorted, we find the exact position of 64.
3. **Use the formula for the percentile rank**:
\[
P = \left( \frac{\text{Number of values below } 64 + 0.5 \times \text{Number of values equal to } 64}{\text{Total number of values}} \right) \times 100
\]
Breaking it down:
- There are 31 values below 64 (up to the third 63).
- Three values are equal to 64.
- The total number of values is 36.
\[
P = \left( \frac{31 + 0.5 \times 3}{36} \right) \times 100
\]
Simplifying further:
\[
P = \left( \frac{31 + 1.5}{36} \right) \times 100 = \left( \frac{32.5}{36} \right) \times 100 \approx 90.28
\]
So, the percentile that corresponds to an age of 64 years old is approximately 90 (rounded to the nearest integer).
#### Conclusion
In this dataset of 36 executives, an age of 64 years falls approximately at the 90th percentile. This means that 64-year-old executives are older than approximately 90% of the other executives in](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d7dd878-68eb-4c6b-9d67-0903dd8963d8%2F03b2cb9f-f389-4a04-8c66-5a60e92ddd7c%2F7awdhjm_processed.jpeg&w=3840&q=75)
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