True or False? In Exercises 7–10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


It is impossible to have a z-score of 0.

Verified step by step guidance

1. Understand the concept of a z-score: A z-score represents the number of standard deviations a data point is from the mean. It is calculated using the formula: $z=\frac{x-\mu }{\sigma }$, where $x$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation.
2. Analyze the meaning of a z-score of 0: A z-score of 0 indicates that the data point is exactly equal to the mean of the distribution.
3. Determine if it is possible to have a z-score of 0: Since a z-score of 0 simply means the data point equals the mean, it is indeed possible to have a z-score of 0.
4. Rewrite the statement if it is false: The original statement, 'It is impossible to have a z-score of 0,' is false. A true statement would be: 'It is possible to have a z-score of 0 when the data point equals the mean.'
5. Conclude the reasoning: The statement is false because a z-score of 0 is a valid and meaningful value in statistics, representing a data point that is exactly at the mean.

Key Concepts

Z-Score

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean. A z-score of 0 means the value is exactly equal to the mean, which is a common occurrence in a normal distribution.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. In this distribution, z-scores can take any real number value, including 0, which represents the mean of the distribution. Understanding this concept is crucial for interpreting z-scores correctly.

True/False Statements in Statistics

In statistics, true/false statements often require a clear understanding of definitions and properties of statistical measures. Evaluating such statements involves critical thinking and knowledge of statistical concepts, such as z-scores, to determine their validity. If a statement is false, it is important to rephrase it accurately based on statistical principles.