The Mean Of A Set Of Credit Scores Is Μ = 690 \mu = 690 Μ = 690 And Σ = 14 \sigma = 14 Σ = 14 . Which Statement Must Be True About Z 694 Z_{694} Z 694 ?A. Z 694 Z_{694} Z 694 Is Within 1 Standard Deviation Of The Mean. B. Z 694 Z_{694} Z 694 Is Between 1 And 2
Understanding the Normal Distribution and z-Scores
The normal distribution is a fundamental concept in statistics, and it is widely used to model real-world data. In this article, we will explore the concept of z-scores and how they relate to the normal distribution.
What are z-Scores?
A z-score is a measure of how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the element and then dividing by the standard deviation. The formula for calculating a z-score is:
z = (X - μ) / σ
where X is the element, μ is the mean, and σ is the standard deviation.
Interpreting z-Scores
z-scores can be used to determine how many standard deviations an element is from the mean. A z-score of 0 indicates that the element is equal to the mean, while a positive z-score indicates that the element is above the mean, and a negative z-score indicates that the element is below the mean.
The 68-95-99.7 Rule
The 68-95-99.7 rule, also known as the empirical rule, states that about 68% of the data falls within 1 standard deviation of the mean, about 95% of the data falls within 2 standard deviations of the mean, and about 99.7% of the data falls within 3 standard deviations of the mean.
Applying the 68-95-99.7 Rule to the Problem
Given that the mean of a set of credit scores is μ = 690 and σ = 14, we can use the 68-95-99.7 rule to determine which statement must be true about z_{694}.
Calculating the z-Score
To calculate the z-score, we need to subtract the mean from the element and then divide by the standard deviation.
z_{694} = (694 - 690) / 14 z_{694} = 4 / 14 z_{694} = 0.286
Interpreting the z-Score
Since the z-score is 0.286, which is less than 1, we can conclude that z_{694} is within 1 standard deviation of the mean.
Conclusion
Based on the 68-95-99.7 rule, we can conclude that statement A is true. z_{694} is within 1 standard deviation of the mean.
The Importance of Understanding z-Scores
Understanding z-scores is crucial in statistics, as it allows us to determine how many standard deviations an element is from the mean. This information can be used to make informed decisions in a variety of fields, including finance, medicine, and social sciences.
Real-World Applications of z-Scores
z-scores have a wide range of real-world applications, including:
- Finance: z-scores can be used to determine the creditworthiness of a borrower.
- Medicine: z-scores can be used to determine the severity of a disease.
- Social Sciences: z-scores can be used to determine the effectiveness of a treatment.
Conclusion
In conclusion, z-scores are a fundamental concept in statistics, and they have a wide range of real-world applications. Understanding z-scores is crucial in making informed decisions in a variety of fields. By applying the 68-95-99.7 rule, we can determine which statement must be true about z_{694}.
Frequently Asked Questions about z-Scores
In this article, we will answer some of the most frequently asked questions about z-scores.
Q: What is a z-score?
A: A z-score is a measure of how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the element and then dividing by the standard deviation.
Q: How do I calculate a z-score?
A: To calculate a z-score, you need to subtract the mean from the element and then divide by the standard deviation. The formula for calculating a z-score is:
z = (X - μ) / σ
where X is the element, μ is the mean, and σ is the standard deviation.
Q: What does a z-score of 0 mean?
A: A z-score of 0 means that the element is equal to the mean.
Q: What does a positive z-score mean?
A: A positive z-score means that the element is above the mean.
Q: What does a negative z-score mean?
A: A negative z-score means that the element is below the mean.
Q: What is the 68-95-99.7 rule?
A: The 68-95-99.7 rule, also known as the empirical rule, states that about 68% of the data falls within 1 standard deviation of the mean, about 95% of the data falls within 2 standard deviations of the mean, and about 99.7% of the data falls within 3 standard deviations of the mean.
Q: How do I use the 68-95-99.7 rule to determine which statement must be true about z_{694}?
A: To use the 68-95-99.7 rule, you need to calculate the z-score of the element and then determine which statement is true based on the z-score.
Q: Can I use z-scores to compare data from different distributions?
A: No, z-scores are only applicable to data that follows a normal distribution. If the data does not follow a normal distribution, you cannot use z-scores to compare it.
Q: Can I use z-scores to determine the probability of an event?
A: Yes, z-scores can be used to determine the probability of an event. By using a z-table or a calculator, you can determine the probability of an event occurring based on the z-score.
Q: What are some real-world applications of z-scores?
A: Some real-world applications of z-scores include:
- Finance: z-scores can be used to determine the creditworthiness of a borrower.
- Medicine: z-scores can be used to determine the severity of a disease.
- Social Sciences: z-scores can be used to determine the effectiveness of a treatment.
Q: How do I interpret a z-score in a real-world context?
A: To interpret a z-score in a real-world context, you need to understand the meaning of the z-score in the context of the data. For example, if you are using z-scores to determine the creditworthiness of a borrower, a z-score of 2 might indicate that the borrower is above average in terms of creditworthiness.
Conclusion
In conclusion, z-scores are a fundamental concept in statistics, and they have a wide range of real-world applications. By understanding how to calculate and interpret z-scores, you can make informed decisions in a variety of fields.