In A Standard Normal Distribution, What Proportion Of Observations Are Below $z=0.18$?A. 0.0359 B. 0.4286 C. 0.5714 D. 0.9641
The standard normal distribution, also known as the z-distribution, is a type of normal distribution with a mean of 0 and a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena in various fields, including finance, engineering, and social sciences. In this article, we will explore the concept of the standard normal distribution and how to calculate the proportion of observations below a given z-score.
What is a z-score?
A z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated using the following formula:
z = (X - μ) / σ
where X is the observation, μ is the mean, and σ is the standard deviation.
Interpreting z-scores
Z-scores can be used to determine the probability of an observation falling within a certain range. For example, a z-score of 0.18 means that the observation is 0.18 standard deviations above the mean.
Calculating the proportion of observations below a z-score
To calculate the proportion of observations below a given z-score, we can use a standard normal distribution table or a calculator. The table provides the area to the left of the z-score, which represents the proportion of observations below that z-score.
Using a standard normal distribution table
A standard normal distribution table, also known as a z-table, is a table that provides the area to the left of a given z-score. The table is typically organized in a grid, with the z-score values listed in the left column and the corresponding areas listed in the top row.
Finding the area to the left of a z-score
To find the area to the left of a z-score, we can use the following steps:
- Locate the z-score in the left column of the table.
- Move down to the row that corresponds to the z-score.
- Read the area to the left of the z-score from the top row.
Calculating the proportion of observations below z=0.18
Using a standard normal distribution table, we can find the area to the left of z=0.18. The table shows that the area to the left of z=0.18 is approximately 0.5714.
Conclusion
In conclusion, the proportion of observations below z=0.18 is approximately 0.5714. This means that about 57.14% of the observations in a standard normal distribution will be below z=0.18.
Answer
The correct answer is C. 0.5714.
Additional Resources
For more information on the standard normal distribution and how to calculate the proportion of observations below a given z-score, please refer to the following resources:
- A standard normal distribution table or calculator
- A statistics textbook or online resource
- A online calculator or software package, such as R or Python
References
- Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
- Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.
Discussion
The standard normal distribution, also known as the z-distribution, is a fundamental concept in statistics. However, it can be a bit tricky to understand and work with, especially for those who are new to statistics. In this article, we will answer some frequently asked questions about the standard normal distribution to help you better understand this important concept.
Q: What is the standard normal distribution?
A: The standard normal distribution is a type of normal distribution with a mean of 0 and a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena in various fields, including finance, engineering, and social sciences.
Q: What is a z-score?
A: A z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated using the following formula:
z = (X - μ) / σ
where X is the observation, μ is the mean, and σ is the standard deviation.
Q: How do I calculate the proportion of observations below a given z-score?
A: To calculate the proportion of observations below a given z-score, you can use a standard normal distribution table or a calculator. The table provides the area to the left of the z-score, which represents the proportion of observations below that z-score.
Q: What is the significance of the standard normal distribution in statistics?
A: The standard normal distribution is significant in statistics because it provides a way to model a wide range of phenomena in various fields. It is used to calculate probabilities, test hypotheses, and make predictions. The standard normal distribution is also used in many statistical tests, including the z-test and the t-test.
Q: How do I use a standard normal distribution table?
A: To use a standard normal distribution table, you need to locate the z-score in the left column of the table and move down to the row that corresponds to the z-score. The area to the left of the z-score is listed in the top row.
Q: What are some common mistakes to avoid when working with z-scores and standard normal distribution tables?
A: Some common mistakes to avoid when working with z-scores and standard normal distribution tables include:
- Not using the correct z-score or standard deviation
- Not using the correct table or calculator
- Not understanding the concept of the standard normal distribution
- Not checking the units of measurement
- Not using the correct statistical test or method
Q: How do I choose the right statistical test or method?
A: To choose the right statistical test or method, you need to consider the research question, the data, and the level of measurement. You should also consider the assumptions of the test or method and the level of significance.
Q: What are some real-world applications of the standard normal distribution?
A: The standard normal distribution has many real-world applications, including:
- Finance: The standard normal distribution is used to model stock prices, interest rates, and other financial variables.
- Engineering: The standard normal distribution is used to model the behavior of complex systems, such as bridges and buildings.
- Social sciences: The standard normal distribution is used to model the behavior of people, including their attitudes, behaviors, and opinions.
Q: How do I learn more about the standard normal distribution?
A: To learn more about the standard normal distribution, you can:
- Read statistics textbooks or online resources
- Take online courses or attend workshops
- Practice using statistical software or calculators
- Join online communities or forums
- Consult with a statistician or researcher
Conclusion
The standard normal distribution is a fundamental concept in statistics that has many real-world applications. By understanding the standard normal distribution, you can better model and analyze data, make predictions, and test hypotheses. We hope that this article has helped you better understand the standard normal distribution and its significance in statistics.