\( 2006 / 23 \) Exercise 8.24 Find the variance of \( 2 x, 2 x-1 \) and \( 2 x+1 \) \( \begin{array}{lll}\text { A. } \sqrt{\frac{2}{3}} & \text { B. } 2 & \text { C. } 1\end{array} \)

To find the variance of the values \( 2x, 2x-1, \) and \( 2x+1 \), we start by calculating their mean. The mean \( \mu \) of these three values is: \[ \mu = \frac{(2x) + (2x-1) + (2x+1)}{3} = \frac{6x}{3} = 2x \] Next, we compute the variance. The variance \( \sigma^2 \) is given by \( \sigma^2 = \frac{1}{n} \sum (x_i - \mu)^2 \), where \( x_i \) are the values. 1. For \( 2x \): \( (2x - 2x)^2 = 0 \) 2. For \( 2x-1 \): \( ((2x-1) - 2x)^2 = (-1)^2 = 1 \) 3. For \( 2x+1 \): \( ((2x+1) - 2x)^2 = (1)^2 = 1 \) Now, summing these squared differences gives: \[ 0 + 1 + 1 = 2 \] Dividing by the number of observations (which is 3), we find the variance: \[ \sigma^2 = \frac{2}{3} \] Thus, the answer is \( \sqrt{\frac{2}{3}} \). Hence, option \( A \) is correct. Now, regarding some additional information: Many of the statistical concepts, such as variance, have historical roots dating back to the works of Karl Pearson in the late 19th century. He pioneered the field of statistics and created many foundational methods that we still use today, including the Pearson correlation coefficient, which helps determine the relationship between variables. In real-world application, understanding variance is crucial for fields like finance, where it measures the risk of an asset. A higher variance indicates a wider spread of returns, which can signal higher risk. Investors use this knowledge to balance their portfolios and make informed decisions about where to allocate their resources.