Standard Normal Distribution

Oct 13, 2025 by recoveryshake.com 86 views

Which of the Following Probabilities is Equal to Approximately 0.2957?

Understanding the Standard Normal Table

The standard normal table, also known as the z-table, is a statistical 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 use the portion of the standard normal table provided to help answer the question of which probability is equal to approximately 0.2957.

The Standard Normal Distribution

The standard normal distribution, also known as the z-distribution, is a normal distribution with a mean of 0 and a standard deviation of 1. The standard normal distribution is symmetric about the mean, and the total area under the curve is equal to 1. The z-table is used to find the probability of a value falling within a certain range in the standard normal distribution.

Using the Standard Normal Table

To use the standard normal table, we need to find the z-score that corresponds to the probability we are looking for. In this case, we are looking for the probability that is equal to approximately 0.2957. We can start by looking at the table and finding the row that corresponds to the z-score of 0.00. The probability for a z-score of 0.00 is 0.5000, which is the midpoint of the distribution.

Finding the Probability

To find the probability that is equal to approximately 0.2957, we need to find the z-score that corresponds to this probability. We can do this by looking at the table and finding the row that corresponds to the probability of 0.2957. However, since the table only provides probabilities for certain z-scores, we need to find the closest z-score that corresponds to a probability greater than 0.2957.

Analyzing the Table

Looking at the table, we can see that the probability for a z-score of 0.00 is 0.5000, and the probability for a z-score of 0.25 is 0.5987. Since 0.2957 is less than 0.5987, we know that the z-score that corresponds to a probability of 0.2957 must be less than 0.25.

Calculating the Z-Score

To find the z-score that corresponds to a probability of 0.2957, we can use the fact that the probability for a z-score of 0.25 is 0.5987. We can set up an equation to represent the situation:

0.5987 - 0.2957 = 0.25 - z

Solving for z, we get:

z = 0.25 - (0.5987 - 0.2957) z = 0.25 - 0.303 z = 0.047

Conclusion

Based on the standard normal table, we can conclude that the probability that is equal to approximately 0.2957 corresponds to a z-score of 0.047. This means that the probability of a value falling within 0.047 standard deviations of the mean in a normal distribution is approximately 0.2957.

Understanding the Significance

The probability of 0.2957 is significant because it represents the probability of a value falling within a certain range in a normal distribution. In many real-world applications, such as finance and engineering, it is important to understand the probability of a value falling within a certain range in order to make informed decisions.

Real-World Applications

The standard normal table has many real-world applications, including:

Finance: The standard normal table is used to calculate the probability of a stock price falling within a certain range.
Engineering: The standard normal table is used to calculate the probability of a machine part falling within a certain range.
Science: The standard normal table is used to calculate the probability of a scientific measurement falling within a certain range.

Conclusion

In conclusion, the probability that is equal to approximately 0.2957 corresponds to a z-score of 0.047. This means that the probability of a value falling within 0.047 standard deviations of the mean in a normal distribution is approximately 0.2957. The standard normal table is a powerful tool that can be used to calculate probabilities in many real-world applications.

References

Standard Normal Table: The standard normal table is a statistical tool used to find the probability of a value falling within a certain range in a normal distribution.
Z-Score: The z-score is a measure of how many standard deviations a value is away from the mean in a normal distribution.
Probability: Probability is a measure of the likelihood of an event occurring.

Frequently Asked Questions

What is the standard normal table? The standard normal table is a statistical tool used to find the probability of a value falling within a certain range in a normal distribution.
What is a z-score? A z-score is a measure of how many standard deviations a value is away from the mean in a normal distribution.
What is probability? Probability is a measure of the likelihood of an event occurring.

Glossary

Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1.
Z-Score: A measure of how many standard deviations a value is away from the mean in a normal distribution.
Probability: A measure of the likelihood of an event occurring.