Complete The Table Below And Choose The Probability Corresponding To The Given Z Z Z Value.$[ \begin{array}{|c|c|} \hline z & \text{Probability} \ \hline 0.00 & \text{0.5000} \ \hline 1.00 & \text{0.8413} \ \hline 2.00 & \text{0.9772}
Introduction to the Standard Normal Distribution Table
The standard normal distribution table, also known as the z-table, is a mathematical tool used to find the probability of a value falling within a certain range in a normal distribution. The table provides the probability that a value will be less than or equal to a given z-score. In this article, we will explore the standard normal distribution table, learn how to use it, and complete a table with given z-values.
What is the Standard Normal Distribution?
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 probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. The standard normal distribution is used as a reference distribution in many statistical analyses.
How to Use the Standard Normal Distribution Table
To use the standard normal distribution table, you need to know the z-score of the value you are interested in. The z-score is calculated by subtracting the mean from the value and dividing by the standard deviation. The z-score tells you how many standard deviations away from the mean your value is.
Once you have the z-score, you can look up the probability in the standard normal distribution table. The table provides the probability that a value will be less than or equal to the given z-score. If you want to find the probability that a value will be greater than a given z-score, you can subtract the probability from 1.
Completing the Table
| z | Probability |
|---|---|
| 0.00 | 0.5000 |
| 1.00 | 0.8413 |
| 2.00 | 0.9772 |
Given z-Values
We are given the following z-values: -1.5, 0.5, and 2.5. We need to find the corresponding probabilities.
z = -1.5
To find the probability corresponding to z = -1.5, we need to look up the z-score in the standard normal distribution table. The table shows that the probability corresponding to z = -1.5 is approximately 0.0668.
z = 0.5
To find the probability corresponding to z = 0.5, we need to look up the z-score in the standard normal distribution table. The table shows that the probability corresponding to z = 0.5 is approximately 0.6915.
z = 2.5
To find the probability corresponding to z = 2.5, we need to look up the z-score in the standard normal distribution table. The table shows that the probability corresponding to z = 2.5 is approximately 0.9938.
Completed Table
| z | Probability |
|---|---|
| -1.5 | 0.0668 |
| 0.5 | 0.6915 |
| 2.5 | 0.9938 |
Conclusion
In this article, we learned about the standard normal distribution table and how to use it to find the probability of a value falling within a certain range in a normal distribution. We completed a table with given z-values and found the corresponding probabilities. The standard normal distribution table is a powerful tool in statistics, and understanding how to use it can help you make informed decisions in a variety of fields.
References
- Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
- Ross, S. M. (2014). Introduction to probability models. Academic Press.
Discussion
The standard normal distribution table is a fundamental tool in statistics, and understanding how to use it can help you make informed decisions in a variety of fields. The table provides the probability that a value will be less than or equal to a given z-score, and it can be used to find the probability of a value falling within a certain range in a normal distribution.
The standard normal distribution table is commonly used in fields such as engineering, economics, and finance. It is also used in social sciences, such as psychology and sociology, to analyze data and make predictions.
Q: What is the standard normal distribution table?
A: The standard normal distribution table, also known as the z-table, is a mathematical tool used to find the probability of a value falling within a certain range in a normal distribution. The table provides the probability that a value will be less than or equal to a given z-score.
Q: How do I use the standard normal distribution table?
A: To use the standard normal distribution table, you need to know the z-score of the value you are interested in. The z-score is calculated by subtracting the mean from the value and dividing by the standard deviation. The z-score tells you how many standard deviations away from the mean your value is.
Once you have the z-score, you can look up the probability in the standard normal distribution table. The table provides the probability that a value will be less than or equal to the given z-score. If you want to find the probability that a value will be greater than a given z-score, you can subtract the probability from 1.
Q: What is the difference between the standard normal distribution table and the normal distribution table?
A: The standard normal distribution table and the normal distribution table are both used to find the probability of a value falling within a certain range in a normal distribution. However, the standard normal distribution table is used for a normal distribution with a mean of 0 and a standard deviation of 1, while the normal distribution table is used for a normal distribution with a mean of 0 and a standard deviation of 1, but with a different scale.
Q: How do I calculate the z-score?
A: To calculate the z-score, you need to know the value, the mean, and the standard deviation. The z-score is calculated by subtracting the mean from the value and dividing by the standard deviation.
Q: What is the probability that a value will be greater than a given z-score?
A: To find the probability that a value will be greater than a given z-score, you can subtract the probability from 1. For example, if the probability that a value will be less than or equal to a given z-score is 0.5, then the probability that a value will be greater than the given z-score is 1 - 0.5 = 0.5.
Q: Can I use the standard normal distribution table for non-normal distributions?
A: No, the standard normal distribution table is only used for normal distributions. If you have a non-normal distribution, you will need to use a different type of table or a statistical software package to find the probability.
Q: How do I choose the right z-score?
A: To choose the right z-score, you need to know the value and the standard deviation. The z-score tells you how many standard deviations away from the mean your value is.
Q: Can I use the standard normal distribution table for large or small samples?
A: Yes, the standard normal distribution table can be used for large or small samples. However, if you have a very large or very small sample, you may need to use a different type of table or a statistical software package to find the probability.
Q: How do I interpret the results of the standard normal distribution table?
A: To interpret the results of the standard normal distribution table, you need to understand what the probability means. The probability that a value will be less than or equal to a given z-score tells you how likely it is that the value will be within a certain range.
Conclusion
In this article, we answered some frequently asked questions about the standard normal distribution table. We covered topics such as how to use the table, how to calculate the z-score, and how to interpret the results. We also discussed some common misconceptions about the table and provided examples to illustrate the concepts.