In A Standard Normal Distribution, What Percentage Of Observations Lie Above $z=1.65$?A. 4.95%B. 5.48%C. 6.06%D. 95.05%
The standard normal distribution, also known as the z-distribution, is a probability distribution that is symmetric about the mean, with a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena. In this article, we will explore the standard normal distribution and how to calculate the percentage of observations that lie above 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 by subtracting the mean from the observation and dividing by the standard deviation. The z-score is a dimensionless quantity that can be used to compare observations from different distributions.
The Standard Normal Distribution
The standard normal distribution is a continuous probability distribution that is symmetric about the mean, with a standard deviation of 1. The distribution is characterized by the following properties:
- The mean is 0.
- The standard deviation is 1.
- The distribution is symmetric about the mean.
- The distribution is continuous.
Calculating the Percentage of Observations Above a z-score
To calculate the percentage of observations that lie above a given z-score, we can use a standard normal distribution table or a calculator. The table provides the probability that a z-score is less than or equal to a given value. To find the percentage of observations above a z-score, we can subtract the probability from 1.
Using a Standard Normal Distribution Table
A standard normal distribution table, also known as a z-table, provides the probability that a z-score is less than or equal to a given value. The table is typically organized in a table format, with the z-score on the left-hand side and the probability on the right-hand side.
Example: Calculating the Percentage of Observations Above z=1.65
To calculate the percentage of observations above z=1.65, we can use a standard normal distribution table. The table provides the probability that a z-score is less than or equal to 1.65, which is approximately 0.9495. To find the percentage of observations above z=1.65, we can subtract the probability from 1:
1 - 0.9495 = 0.0505
Therefore, approximately 5.05% of observations lie above z=1.65.
Conclusion
In conclusion, the standard normal distribution is a fundamental concept in statistics that is used to model a wide range of phenomena. The z-score is a measure of how many standard deviations an observation is away from the mean, and can be used to compare observations from different distributions. By using a standard normal distribution table or a calculator, we can calculate the percentage of observations that lie above a given z-score.
Answer
The correct answer is D. 95.05%. However, this is the percentage of observations below z=1.65, not above. The correct percentage of observations above z=1.65 is 5.05%, which is option A. 4.95% is close but not the correct answer.
References
- Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
- Ross, S. M. (2012). Introduction to probability models. Academic Press.
- Johnson, R. A., & Bhattacharyya, G. K. (2012). Statistics: Principles and methods. John Wiley & Sons.
Additional Resources
- Khan Academy: Standard Normal Distribution
- Stat Trek: Standard Normal Distribution
- Math Is Fun: Standard Normal Distribution
Frequently Asked Questions (FAQs) about the Standard Normal Distribution ====================================================================
The standard normal distribution is a fundamental concept in statistics that is used to model a wide range of phenomena. In this article, we will answer some frequently asked questions about the standard normal distribution.
Q: What is the standard normal distribution?
A: The standard normal distribution, also known as the z-distribution, is a probability distribution that is symmetric about the mean, with a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena.
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 by subtracting the mean from the observation and dividing by the standard deviation.
Q: How do I calculate the percentage of observations above a z-score?
A: To calculate the percentage of observations above a given z-score, you can use a standard normal distribution table or a calculator. The table provides the probability that a z-score is less than or equal to a given value. To find the percentage of observations above a z-score, you can subtract the probability from 1.
Q: What is the difference between a z-score and a t-score?
A: A z-score is a measure of how many standard deviations an observation is away from the mean, while a t-score is a measure of how many standard errors an observation is away from the mean. The main difference between the two is that a z-score is used for large samples, while a t-score is used for small samples.
Q: How do I use a standard normal distribution table?
A: A standard normal distribution table, also known as a z-table, provides the probability that a z-score is less than or equal to a given value. The table is typically organized in a table format, with the z-score on the left-hand side and the probability on the right-hand side. To use the table, you can look up the z-score and find the corresponding probability.
Q: What is the relationship between the standard normal distribution and the normal distribution?
A: The standard normal distribution is a special case of the normal distribution, where the mean is 0 and the standard deviation is 1. The normal distribution is a family of distributions that are symmetric about the mean, with a standard deviation of σ. The standard normal distribution is a specific case of the normal distribution, where σ = 1.
Q: How do I calculate the probability of an observation falling within a certain range?
A: To calculate the probability of an observation falling within a certain range, you can use the following formula:
P(a < X < b) = P(Z < (b - μ) / σ) - P(Z < (a - μ) / σ)
where a and b are the lower and upper bounds of the range, μ is the mean, σ is the standard deviation, and Z is the z-score.
Q: What is the significance of the standard normal distribution in real-world applications?
A: The standard normal distribution is used in a wide range of real-world applications, including:
- Finance: The standard normal distribution is used to model stock prices and returns.
- Engineering: The standard normal distribution is used to model the behavior of complex systems.
- Medicine: The standard normal distribution is used to model the behavior of patients and the effectiveness of treatments.
- Social sciences: The standard normal distribution is used to model the behavior of people and the effectiveness of policies.
Conclusion
In conclusion, the standard normal distribution is a fundamental concept in statistics that is used to model a wide range of phenomena. By understanding the standard normal distribution, you can gain insights into the behavior of complex systems and make informed decisions.
References
- Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
- Ross, S. M. (2012). Introduction to probability models. Academic Press.
- Johnson, R. A., & Bhattacharyya, G. K. (2012). Statistics: Principles and methods. John Wiley & Sons.
Additional Resources
- Khan Academy: Standard Normal Distribution
- Stat Trek: Standard Normal Distribution
- Math Is Fun: Standard Normal Distribution