An Age Of 30 Is 0.38 Standard Deviation(s) Above The Mean. Determine Whether The Age Is Unusual. Choose The Correct Answer Below.A. No, This Value Is Not Unusual. A Z-score Outside Of The Range From -2 To 2 Is Not Unusual.B. Yes, This Value Is Unusual.
What is a Z-Score?
A z-score is a statistical measure that calculates how many standard deviations an element is from the mean. It is a way to compare the value of an element to the mean of a dataset. The z-score formula is:
z = (X - μ) / σ
Where:
- X is the value of the element
- μ is the mean of the dataset
- σ is the standard deviation of the dataset
Interpreting Z-Scores
A z-score can be positive, negative, or zero. A positive z-score indicates that the value is above the mean, while a negative z-score indicates that the value is below the mean. A z-score of zero indicates that the value is equal to the mean.
Determining Unusual Values
A value is considered unusual if its z-score is outside of the range from -2 to 2. This is because about 95% of the values in a normal distribution fall within this range. Therefore, a z-score outside of this range is likely to be unusual.
The Case of a 30-Year-Old
Let's consider the case of a 30-year-old. We are given that this age is 0.38 standard deviations above the mean. We need to determine whether this value is unusual.
Calculating the Z-Score
To calculate the z-score, we need to know the mean and standard deviation of the dataset. Let's assume that the mean age is 25 years and the standard deviation is 5 years. We can then calculate the z-score as follows:
z = (30 - 25) / 5 z = 5 / 5 z = 1
Is the Value Unusual?
Since the z-score is 1, which is within the range of -2 to 2, we can conclude that the value is not unusual.
Conclusion
In conclusion, a z-score of 0.38 is not unusual. This is because it falls within the range of -2 to 2, which is the range of values that are considered typical in a normal distribution.
Answer
The correct answer is:
A. No, this value is not unusual.
Why is this value not unusual?
This value is not unusual because its z-score is within the range of -2 to 2. This range includes about 95% of the values in a normal distribution, making it the range of typical values.
What does this mean?
This means that a 30-year-old is not unusual. This is because their age is within the range of typical ages, which is defined by the z-score range of -2 to 2.
Implications
This has implications for various fields, such as medicine, psychology, and sociology. For example, in medicine, a 30-year-old patient is not unusual, and their age should not be a factor in their diagnosis or treatment.
Limitations
This analysis has limitations. For example, it assumes that the dataset is normally distributed, which may not always be the case. Additionally, it assumes that the mean and standard deviation are known, which may not always be the case.
Future Research
Future research could explore the implications of this analysis in various fields. For example, researchers could investigate how the z-score range affects diagnosis and treatment in medicine.
Conclusion
Q: What is a z-score, and how is it calculated?
A: A z-score is a statistical measure that calculates 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 of the dataset
- σ is the standard deviation of the dataset
Q: What does a positive z-score indicate?
A: A positive z-score indicates that the value is above the mean.
Q: What does a negative z-score indicate?
A: A negative z-score indicates that the value is below the mean.
Q: What does a z-score of zero indicate?
A: A z-score of zero indicates that the value is equal to the mean.
Q: How do I determine if a value is unusual?
A: A value is considered unusual if its z-score is outside of the range from -2 to 2. This is because about 95% of the values in a normal distribution fall within this range.
Q: What is the significance of the z-score range of -2 to 2?
A: The z-score range of -2 to 2 includes about 95% of the values in a normal distribution, making it the range of typical values.
Q: Can a z-score be greater than 2 or less than -2?
A: Yes, a z-score can be greater than 2 or less than -2. However, these values are considered unusual and are likely to be outside of the typical range.
Q: How do I interpret a z-score of 0.38?
A: A z-score of 0.38 indicates that the value is 0.38 standard deviations above the mean. This is not unusual, as it falls within the range of -2 to 2.
Q: Can I use z-scores to compare values from different datasets?
A: No, z-scores are specific to a particular dataset and cannot be used to compare values from different datasets.
Q: What are some common applications of z-scores?
A: Z-scores are commonly used in various fields, such as medicine, psychology, and sociology, to compare values and determine if they are unusual.
Q: Can I use z-scores to predict future values?
A: No, z-scores are not used to predict future values. They are used to compare values and determine if they are unusual.
Q: What are some limitations of z-scores?
A: Some limitations of z-scores include the assumption that the dataset is normally distributed and that the mean and standard deviation are known.
Q: Can I use z-scores with non-normal distributions?
A: No, z-scores are typically used with normal distributions. If the distribution is non-normal, other statistical measures may be more appropriate.
Q: How do I calculate the z-score for a specific value?
A: To calculate the z-score for a specific value, you need to know the mean and standard deviation of the dataset. You can then use the formula:
z = (X - μ) / σ
Where:
- X is the value of the element
- μ is the mean of the dataset
- σ is the standard deviation of the dataset
Q: What is the difference between a z-score and a standard score?
A: A z-score and a standard score are the same thing. They are both measures of how many standard deviations an element is from the mean.
Q: Can I use z-scores to compare values from different populations?
A: No, z-scores are specific to a particular population and cannot be used to compare values from different populations.
Q: What are some common mistakes to avoid when using z-scores?
A: Some common mistakes to avoid when using z-scores include assuming that the dataset is normally distributed, assuming that the mean and standard deviation are known, and using z-scores to predict future values.