The Set Of Life Spans Of An Appliance Is Normally Distributed With A Mean Μ = 48 \mu = 48 Μ = 48 Months And A Standard Deviation Σ = 8 \sigma = 8 Σ = 8 Months. What Is The Life Span Of An Appliance That Has A Z Z Z -score Of -3?A. 3 Months B. 24
Understanding the Problem
The problem states that the life spans of an appliance follow a normal distribution with a mean (μ) of 48 months and a standard deviation (σ) of 8 months. We are asked to find the life span of an appliance that has a z-score of -3.
What is a z-score?
A z-score is a measure of how many standard deviations an element is from the mean. It is calculated using the formula:
z = (X - μ) / σ
where X is the value of the element, μ is the mean, and σ is the standard deviation.
Finding the Life Span of an Appliance with a z-score of -3
To find the life span of an appliance with a z-score of -3, we can use the z-score formula and rearrange it to solve for X:
X = μ + (z × σ)
Plugging in the values given in the problem, we get:
X = 48 + (-3 × 8)
X = 48 - 24
X = 24
Conclusion
Therefore, the life span of an appliance with a z-score of -3 is 24 months.
Understanding the Significance of the z-score
The z-score of -3 indicates that the life span of the appliance is 3 standard deviations below the mean. This means that the life span is significantly lower than the average life span of 48 months.
Implications of the z-score
The z-score of -3 has significant implications for the appliance's reliability and maintenance. Appliances with life spans that are significantly lower than the average may require more frequent maintenance or replacement, which can be costly and inconvenient.
Real-World Applications
The concept of z-scores and normal distributions has numerous real-world applications in fields such as engineering, finance, and medicine. Understanding how to calculate and interpret z-scores can help professionals make informed decisions and predict outcomes.
Example Use Case
For example, a manufacturer of appliances may use z-scores to predict the life span of their products and plan for maintenance and replacement. By understanding the distribution of life spans, they can identify areas for improvement and develop strategies to increase the reliability and lifespan of their products.
Conclusion
In conclusion, the life span of an appliance with a z-score of -3 is 24 months. Understanding z-scores and normal distributions is crucial for making informed decisions and predicting outcomes in various fields. By applying this knowledge, professionals can improve the reliability and lifespan of products, reduce costs, and enhance customer satisfaction.
Frequently Asked Questions
Q: What is a z-score?
A: A z-score is a measure of how many standard deviations an element is from the mean.
Q: How is the z-score formula calculated?
A: The z-score formula is calculated using the formula: z = (X - μ) / σ
Q: What does a z-score of -3 indicate?
A: A z-score of -3 indicates that the life span of the appliance is 3 standard deviations below the mean.
Q: What are the implications of a z-score of -3?
A: The z-score of -3 has significant implications for the appliance's reliability and maintenance, requiring more frequent maintenance or replacement.
Q: What are some real-world applications of z-scores and normal distributions?
A: Z-scores and normal distributions have numerous real-world applications in fields such as engineering, finance, and medicine.
Q: How can z-scores be used in practice?
A: Z-scores can be used to predict the life span of products, plan for maintenance and replacement, and identify areas for improvement.
References
- [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
- [2] Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.
- [3] Devore, J. L. (2012). Probability and statistics for engineering and the sciences. Cengage Learning.
Q&A: Z-Scores and Normal Distributions
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 using the formula: z = (X - μ) / σ
Q: How is the z-score formula calculated?
A: The z-score formula is calculated using the formula: z = (X - μ) / σ, where X is the value of the element, μ is the mean, and σ is the standard deviation.
Q: What does a z-score of -3 indicate?
A: A z-score of -3 indicates that the life span of the appliance is 3 standard deviations below the mean.
Q: What are the implications of a z-score of -3?
A: The z-score of -3 has significant implications for the appliance's reliability and maintenance, requiring more frequent maintenance or replacement.
Q: What are some real-world applications of z-scores and normal distributions?
A: Z-scores and normal distributions have numerous real-world applications in fields such as engineering, finance, and medicine.
Q: How can z-scores be used in practice?
A: Z-scores can be used to predict the life span of products, plan for maintenance and replacement, and identify areas for improvement.
Q: What is the difference between a z-score and a standard deviation?
A: A standard deviation is a measure of the spread of a dataset, while a z-score is a measure of how many standard deviations an element is from the mean.
Q: How do I calculate the z-score for a given value?
A: To calculate the z-score for a given value, use the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
Q: What is the significance of a z-score of 0?
A: A z-score of 0 indicates that the value is equal to the mean.
Q: Can a z-score be negative?
A: Yes, a z-score can be negative, indicating that the value is below the mean.
Q: Can a z-score be greater than 3?
A: Yes, a z-score can be greater than 3, indicating that the value is above the mean.
Q: How do I interpret a z-score of 2.5?
A: A z-score of 2.5 indicates that the value is 2.5 standard deviations above the mean.
Q: What is the relationship between z-scores and the normal distribution?
A: Z-scores are used to calculate the probability of a value occurring within a certain range in a normal distribution.
Q: Can z-scores be used with non-normal distributions?
A: No, z-scores are typically used with normal distributions, as they are based on the mean and standard deviation of the distribution.
Q: How do I determine if a distribution is normal?
A: To determine if a distribution is normal, use statistical tests such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test.
Q: What is the significance of the 68-95-99.7 rule?
A: The 68-95-99.7 rule states that approximately 68% of values fall within 1 standard deviation of the mean, 95% fall within 2 standard deviations, and 99.7% fall within 3 standard deviations.
Q: Can z-scores be used to predict future values?
A: Yes, z-scores can be used to predict future values by extrapolating the trend of the distribution.
Q: How do I use z-scores to make predictions?
A: To use z-scores to make predictions, calculate the z-score for the value you want to predict, and then use the z-score to estimate the probability of the value occurring.
Conclusion
In conclusion, z-scores and normal distributions are fundamental concepts in statistics that have numerous real-world applications. By understanding how to calculate and interpret z-scores, you can make informed decisions and predict outcomes in various fields.
References
- [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
- [2] Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.
- [3] Devore, J. L. (2012). Probability and statistics for engineering and the sciences. Cengage Learning.