The Distribution Of Heights Of Adult Men Is Approximately Normal With Μ = 69 \mu = 69 Μ = 69 Inches And Σ = 2.5 \sigma = 2.5 Σ = 2.5 Inches. Find The Standardized ( Z Z Z -score) Heights Of The Three Men Listed: A) Homer 5'10 B) Bartholomew 6'2
Introduction
The distribution of heights of adult men is a classic example of a normal distribution, with a mean (μ) of 69 inches and a standard deviation (σ) of 2.5 inches. In this article, we will explore the standardized heights of three men, Homer and Bartholomew, using the z-score formula. The z-score is a measure of how many standard deviations an element is from the mean.
Understanding the z-Score Formula
The z-score formula is given by:
z = (X - μ) / σ
where X is the value of the element, μ is the mean, and σ is the standard deviation.
Calculating the z-Score for Homer
Homer's height is 5'10", which is equivalent to 70 inches. To calculate his z-score, we will use the z-score formula:
z = (70 - 69) / 2.5 = 1 / 2.5 = 0.4
Calculating the z-Score for Bartholomew
Bartholomew's height is 6'2", which is equivalent to 74 inches. To calculate his z-score, we will use the z-score formula:
z = (74 - 69) / 2.5 = 5 / 2.5 = 2
Interpretation of z-Scores
A z-score of 0 indicates that the value is equal to the mean. A positive z-score indicates that the value is above the mean, while a negative z-score indicates that the value is below the mean.
In the case of Homer, his z-score of 0.4 indicates that he is 0.4 standard deviations above the mean. This means that he is slightly taller than the average adult man.
In the case of Bartholomew, his z-score of 2 indicates that he is 2 standard deviations above the mean. This means that he is significantly taller than the average adult man.
Conclusion
In conclusion, the z-score formula is a useful tool for understanding the distribution of heights of adult men. By calculating the z-score for Homer and Bartholomew, we were able to determine how many standard deviations above the mean they are. This information can be useful in a variety of applications, such as determining the likelihood of a person being above or below a certain height.
Real-World Applications
The z-score formula has a wide range of real-world applications, including:
- Medical Research: Z-scores can be used to determine the likelihood of a person having a certain medical condition based on their height.
- Sports: Z-scores can be used to determine the likelihood of a person being above or below a certain height for a particular sport.
- Insurance: Z-scores can be used to determine the likelihood of a person being above or below a certain height for insurance purposes.
Limitations of the z-Score Formula
While the z-score formula is a useful tool for understanding the distribution of heights of adult men, it has some limitations. These include:
- Assumes Normal Distribution: The z-score formula assumes that the distribution of heights is normal, which may not always be the case.
- Does Not Account for Variability: The z-score formula does not account for variability in the data, which can affect the accuracy of the results.
Future Research Directions
Future research directions for the z-score formula include:
- Developing More Accurate Models: Developing more accurate models of the distribution of heights of adult men, such as using non-normal distributions.
- Accounting for Variability: Accounting for variability in the data, such as using robust statistical methods.
References
- National Center for Health Statistics. (2020). Height and Weight for Adults.
- World Health Organization. (2019). Growth Reference Data for 5-19 Years.
Appendix
The following is a list of the data used in this article:
| Name | Height (inches) |
|---|---|
| Homer | 70 |
| Bartholomew | 74 |
Introduction
In our previous article, we explored the distribution of heights of adult men and calculated the z-scores for two men, Homer and Bartholomew. In this article, we will answer some frequently asked questions about the distribution of heights of adult men and the z-score formula.
Q: What is the normal distribution of heights of adult men?
A: The normal distribution of heights of adult men is a bell-shaped curve with a mean (μ) of 69 inches and a standard deviation (σ) of 2.5 inches.
Q: How do I calculate the z-score for a given height?
A: To calculate the z-score for a given height, you can use the z-score formula:
z = (X - μ) / σ
where X is the value of the height, μ is the mean, and σ is the standard deviation.
Q: What does a positive z-score indicate?
A: A positive z-score indicates that the value is above the mean. For example, if a person has a z-score of 1, it means that they are 1 standard deviation above the mean.
Q: What does a negative z-score indicate?
A: A negative z-score indicates that the value is below the mean. For example, if a person has a z-score of -1, it means that they are 1 standard deviation below the mean.
Q: Can I use the z-score formula for other types of data?
A: Yes, the z-score formula can be used for other types of data that follow a normal distribution. However, you will need to adjust the mean and standard deviation to match the specific data set.
Q: How do I interpret the z-score?
A: To interpret the z-score, you can use the following guidelines:
- A z-score of 0 indicates that the value is equal to the mean.
- A z-score between 0 and 1 indicates that the value is above the mean, but not significantly so.
- A z-score between 1 and 2 indicates that the value is above the mean, and is significantly so.
- A z-score greater than 2 indicates that the value is above the mean, and is very significantly so.
- A z-score less than -1 indicates that the value is below the mean, but not significantly so.
- A z-score less than -2 indicates that the value is below the mean, and is significantly so.
- A z-score less than -3 indicates that the value is below the mean, and is very significantly so.
Q: What are some real-world applications of the z-score formula?
A: The z-score formula has a wide range of real-world applications, including:
- Medical research: Z-scores can be used to determine the likelihood of a person having a certain medical condition based on their height.
- Sports: Z-scores can be used to determine the likelihood of a person being above or below a certain height for a particular sport.
- Insurance: Z-scores can be used to determine the likelihood of a person being above or below a certain height for insurance purposes.
Q: What are some limitations of the z-score formula?
A: The z-score formula has some limitations, including:
- Assumes normal distribution: The z-score formula assumes that the distribution of heights is normal, which may not always be the case.
- Does not account for variability: The z-score formula does not account for variability in the data, which can affect the accuracy of the results.
Conclusion
In conclusion, the z-score formula is a useful tool for understanding the distribution of heights of adult men. By answering some frequently asked questions about the z-score formula, we hope to have provided a better understanding of this important statistical concept.
References
- National Center for Health Statistics. (2020). Height and Weight for Adults.
- World Health Organization. (2019). Growth Reference Data for 5-19 Years.
Appendix
The following is a list of the data used in this article:
| Name | Height (inches) | Z-Score |
|---|---|---|
| Homer | 70 | 0.4 |
| Bartholomew | 74 | 2 |
Note: The data used in this article is fictional and for illustrative purposes only.