Select The Probability Associated With A Z-score Of 1.00 From The Table Below:$[ \begin{array}{|c|c|} \hline z & \text{Probability} \ \hline 0.00 & 0.5000 \ \hline 1.00 & 0.8413 \ \hline 2.00 & 0.9772 \ \hline 3.00 & 0.9987
Introduction
In statistics, a z-score is a measure of how many standard deviations an observation is away from the mean. It is a crucial concept in understanding probability distributions and is often used in hypothesis testing and confidence intervals. A z-score table, also known as a standard normal distribution table, provides the probability of observing a value less than or equal to a given z-score. In this article, we will explore how to use a z-score table to find the probability associated with a given z-score.
What is a Z-Score Table?
A z-score table is a table that lists the z-scores and their corresponding probabilities. The table is typically organized in a way that the z-scores are listed in the left column, and the probabilities are listed in the top row. The table is usually symmetric around the mean, which is 0. The probabilities listed in the table are the probabilities of observing a value less than or equal to the given z-score.
How to Use a Z-Score Table
To use a z-score table, you need to find the z-score in the left column and the probability in the top row that corresponds to the z-score. The probability listed in the table is the probability of observing a value less than or equal to the given z-score. For example, if you want to find the probability associated with a z-score of 1.00, you would look for the row that corresponds to a z-score of 1.00 and the column that corresponds to a probability of 0.00.
Selecting the Probability Associated with a Z-Score of 1.00
In this example, we are given a z-score table with the following values:
| z | Probability |
|---|---|
| 0.00 | 0.5000 |
| 1.00 | 0.8413 |
| 2.00 | 0.9772 |
| 3.00 | 0.9987 |
To select the probability associated with a z-score of 1.00, we need to look for the row that corresponds to a z-score of 1.00. In this case, the row is already listed in the table, and the probability associated with a z-score of 1.00 is 0.8413.
Interpreting the Probability
The probability associated with a z-score of 1.00 is 0.8413. This means that there is a 84.13% chance of observing a value less than or equal to 1.00 standard deviation away from the mean. In other words, if we were to randomly select a value from a normal distribution with a mean of 0 and a standard deviation of 1, there is a 84.13% chance that the value would be less than or equal to 1.00.
Conclusion
In conclusion, a z-score table is a useful tool for finding the probability associated with a given z-score. By using a z-score table, we can quickly and easily find the probability associated with a given z-score. In this example, we used a z-score table to find the probability associated with a z-score of 1.00, which was 0.8413.
Common Applications of Z-Score Tables
Z-score tables have many common applications in statistics and data analysis. Some of the most common applications include:
- Hypothesis testing: Z-score tables are often used in hypothesis testing to determine the probability of observing a value less than or equal to a given z-score.
- Confidence intervals: Z-score tables are often used in confidence intervals to determine the probability of observing a value within a given range.
- Data analysis: Z-score tables are often used in data analysis to determine the probability of observing a value less than or equal to a given z-score.
Limitations of Z-Score Tables
While z-score tables are a useful tool for finding the probability associated with a given z-score, they do have some limitations. Some of the most common limitations include:
- Assumes normal distribution: Z-score tables assume that the data follows a normal distribution. If the data does not follow a normal distribution, the z-score table may not be accurate.
- Only for continuous data: Z-score tables are only for continuous data. If the data is discrete, the z-score table may not be accurate.
- Only for symmetric distributions: Z-score tables are only for symmetric distributions. If the distribution is not symmetric, the z-score table may not be accurate.
Conclusion
In conclusion, z-score tables are a useful tool for finding the probability associated with a given z-score. While they have some limitations, they are a powerful tool for data analysis and hypothesis testing. By understanding how to use a z-score table, we can quickly and easily find the probability associated with a given z-score.
References
- Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
- Ross, S. M. (2014). Introduction to probability models. Academic Press.
- Walpole, R. E., Myers, R. H., & Myers, S. L. (2015). Probability and statistics for engineers and scientists. Pearson Education.
Frequently Asked Questions (FAQs) about Z-Score Tables =====================================================
Q: What is a z-score table?
A: A z-score table, also known as a standard normal distribution table, is a table that lists the z-scores and their corresponding probabilities. The table is typically organized in a way that the z-scores are listed in the left column, and the probabilities are listed in the top row.
Q: How do I use a z-score table?
A: To use a z-score table, you need to find the z-score in the left column and the probability in the top row that corresponds to the z-score. The probability listed in the table is the probability of observing a value less than or equal to the given z-score.
Q: What is the difference between a z-score and a probability?
A: A z-score is a measure of how many standard deviations an observation is away from the mean. A probability, on the other hand, is a measure of the likelihood of observing a value less than or equal to a given z-score.
Q: Can I use a z-score table for non-normal distributions?
A: No, z-score tables are only for normal distributions. If the data does not follow a normal distribution, the z-score table may not be accurate.
Q: Can I use a z-score table for discrete data?
A: No, z-score tables are only for continuous data. If the data is discrete, the z-score table may not be accurate.
Q: Can I use a z-score table for asymmetric distributions?
A: No, z-score tables are only for symmetric distributions. If the distribution is not symmetric, the z-score table may not be accurate.
Q: How do I find the probability associated with a given z-score?
A: To find the probability associated with a given z-score, you need to look for the row that corresponds to the z-score in the left column and the column that corresponds to the probability in the top row.
Q: What is the probability associated with a z-score of 1.00?
A: The probability associated with a z-score of 1.00 is 0.8413. This means that there is a 84.13% chance of observing a value less than or equal to 1.00 standard deviation away from the mean.
Q: Can I use a z-score table to find the z-score associated with a given probability?
A: No, z-score tables are only for finding the probability associated with a given z-score. If you want to find the z-score associated with a given probability, you need to use a different type of table or calculator.
Q: Are there any limitations to using a z-score table?
A: Yes, there are several limitations to using a z-score table. These include:
- Assumes normal distribution
- Only for continuous data
- Only for symmetric distributions
Q: Can I use a z-score table for hypothesis testing?
A: Yes, z-score tables are often used in hypothesis testing to determine the probability of observing a value less than or equal to a given z-score.
Q: Can I use a z-score table for confidence intervals?
A: Yes, z-score tables are often used in confidence intervals to determine the probability of observing a value within a given range.
Q: Where can I find a z-score table?
A: Z-score tables can be found in many statistics textbooks and online resources. Some popular online resources include:
- Wikipedia
- Khan Academy
- Stat Trek
- Mathway
Conclusion
In conclusion, z-score tables are a useful tool for finding the probability associated with a given z-score. While they have some limitations, they are a powerful tool for data analysis and hypothesis testing. By understanding how to use a z-score table, we can quickly and easily find the probability associated with a given z-score.