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Magazine Chapter 6: Problem 15
Suppose \(X \sim N(2,6) .\) What value of \(x\) has a \(z\) -score of three?
Short Answer
Expert verified
The value of \(x\) is approximately 9.3485.
Step by step solution
01
Understand the Standard Normal Distribution
A standard normal distribution, represented as \(Z\), has a mean of 0 and a standard deviation of 1. The \(z\)-score indicates how many standard deviations a particular value is away from the mean.
02
Identify the Distribution Parameters
For this problem, the random variable \(X\) is normally distributed with a mean \(\mu = 2\) and a variance \(\sigma^2 = 6\). Hence, the standard deviation \(\sigma = \sqrt{6}\).
03
Use the z-score Formula
The \(z\)-score formula is given by \( z = \frac{x - \mu}{\sigma} \). We need to solve for \(x\) when \(z = 3\).
04
Substitute Known Values
Substitute \(z = 3\), \(\mu = 2\), and \(\sigma = \sqrt{6}\) into the \(z\)-score formula: \(3 = \frac{x - 2}{\sqrt{6}}\).
05
Solve for x
Rearrange the equation to solve for \(x\):\(x - 2 = 3 \times \sqrt{6}\)Calculate \(3 \times \sqrt{6}\) and add 2 to find \(x\).
06
Calculate the Result
Calculate \(3 \times \sqrt{6}\): approximately 3 times 2.4495, which is approximately 7.3485. Thus, \(x = 7.3485 + 2 = 9.3485\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Z-Score Calculation
Z-scores are an essential part of understanding the placement of a data point within a standard normal distribution. The z-score, sometimes referred to as a standard score, helps determine how far away a particular value is from the mean of a distribution in terms of standard deviations.
Here's the basic formula for calculating a z-score:
For instance, if a value has a z-score of 3, this indicates it's 3 standard deviations above the mean.
Here's the basic formula for calculating a z-score:
- \( z = \frac{x - \mu}{\sigma} \)
- \( z \) is the z-score we want to calculate.
- \( x \) is the data point we're examining.
- \( \mu \) is the mean of the distribution.
- \( \sigma \) is the standard deviation of the distribution.
For instance, if a value has a z-score of 3, this indicates it's 3 standard deviations above the mean.
Standard Normal Distribution
The standard normal distribution is a foundational concept in statistics, representing a normal distribution with a mean of 0 and standard deviation of 1. It is a crucial tool for understanding data spread and probabilities.
Key properties include:
Key properties include:
- The area under the curve corresponds to the probability of the data falling within a certain range.
- Approximately 68% of the data lies within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is commonly known as the "68-95-99.7 (empirical) rule".
Variance and Standard Deviation
Variance and standard deviation are two closely related statistics that measure variability within a dataset.
**Variance**: This is the average of the squared differences from the mean. It's expressed as \( \sigma^2 \) for a population or \( s^2 \) for a sample, and it provides a general sense of how spread out the values are.
**Standard Deviation**: This is simply the square root of the variance, expressed as \( \sigma \) for a population or \( s \) for a sample. Because it is in the same unit as the data set, it is often more interpretable than variance.
For a dataset, both measures indicate how much the data varies from the mean. A larger value indicates more spread among the data points.
**Variance**: This is the average of the squared differences from the mean. It's expressed as \( \sigma^2 \) for a population or \( s^2 \) for a sample, and it provides a general sense of how spread out the values are.
**Standard Deviation**: This is simply the square root of the variance, expressed as \( \sigma \) for a population or \( s \) for a sample. Because it is in the same unit as the data set, it is often more interpretable than variance.
For a dataset, both measures indicate how much the data varies from the mean. A larger value indicates more spread among the data points.
- Understanding these measures helps in comparing the distribution of two different datasets.
- Standard deviation is especially useful when it comes to interpreting z-scores and their relative distance from the mean.
