Question

If the standard deviation of \(2 \mathrm{x}_{\mathrm{i}}+3\) is 8, then the variance of \(\frac{3 \mathrm{x}_{\mathrm{i}}}{2}\) is (1) 24 (2) 36 (3) 6 (4) 18

Short answer

Based on the given information, the correct answer for the variance of the expression \(\frac{3x_i}{2}\) is \(\frac{1024}{9}\); however, this answer is not among the given options. This indicates a possible error in the problem or the given options.

Step-by-step solution

Find the variance of \(2x_i + 3\)

We are given the standard deviation of \(2x_i + 3\), which is equal to 8. To find the variance, we need to square the standard deviation. The variance of \(2x_i + 3\) is \(8^2\).

Use the property of variance to find the variance of \(\frac{3x_i}{2}\)

The property of variance states that the variance of a constant times a random variable is equal to the constant squared times the variance of the random variable. So, the variance of \(\frac{3}{2} x_i\) is \(\left(\frac{3}{2}\right)^2\) times the variance of \(x_i\). Now, we need to find the relationship between the variance of \(2x_i+3\) and the variance of \(x_i\). Since \(2x_i + 3 = \frac{4}{3} \cdot \frac{3}{2}x_i + 3\), the relationship between the two variances is: Variance of \(\frac{3}{2} x_i\) = \(\left(\frac{4}{3}\right)^2\) * Variance of \(2x_i + 3\) Now substitute the value of the variance of \(2x_i + 3\), which is \(8^2\): Variance of \(\frac{3}{2} x_i\) = \(\left(\frac{4}{3}\right)^2\) * \(8^2\)

Calculate the variance of \(\frac{3x_i}{2}\)

Variance of \(\frac{3}{2} x_i\) = \(\left(\frac{4}{3}\right)^2\) * \(8^2\) = \(\frac{16}{9}\) * \(8^2\) = \(\frac{16}{9}\) * 64 = \(\frac{16 * 64}{9}\) = \(\frac{1024}{9}\) Thus, the variance of \(\frac{3x_i}{2}\) is \(\frac{1024}{9}\), which is not among the given options. Therefore, there might be an error in the problem or the given options.

Key concepts

Standard Deviation

The concept of standard deviation is foundational in statistics. It measures the spread or dispersion of a set of values. In simpler terms, it tells us how much the individual numbers in a data set tend to differ from the mean (average) of the set.

For example, consider scores on a test. A low standard deviation implies that most students scored close to the average, whereas a high standard deviation indicates a wide range of scores. Mathematically, to calculate standard deviation, you first find the variance (the average of squared differences from the mean) and then take the square root of that number. This exercise gave the standard deviation of a linear transformation of a random variable, which allows us to explore the properties of variance and standard deviation in depth. Understanding this concept is crucial, as it’s used in nearly every field involving data analysis.

Variance Properties

Variance properties are rules that describe how the variance behaves under different mathematical operations. Knowing these properties is essential for correctly manipulating and interpreting variances of random variables.

• If you add or subtract a constant from a random variable, the variance remains unchanged because the spread of the data has not been altered.
• If you multiply a random variable by a constant, the variance is multiplied by the square of that constant.
• The variance of a sum of independent random variables is equal to the sum of their variances.

These rules are applied in the given exercise where the first step involves squaring the standard deviation to find the variance, and the second step applies the scaling property to find the variance of a scaled random variable.

Random Variable Statistics

In statistics, a random variable represents a numerical outcome of a random phenomenon. Random variable statistics include descriptive measures such as the mean (expected value), variance, and standard deviation. These measures help us understand the behavior of random variables.

For instance, the mean provides a measure of the central tendency, while the variance and standard deviation tell us how dispersed the values of the random variable are around the mean. The exercise involves a random variable that is altered through scaling and translation, both operations that are common when analyzing random variables. Here, we see the practical application of how a random variable's transformation affects its variance.

Mathematical Statistics

Mathematical statistics involves the study of data using mathematical concepts and tools to make inferences or predictions. It harnesses the power of probability theory to interpret and analyze random data. This area of study is crucial to making scientifically grounded decisions in various fields such as economics, biology, and engineering.

In the context of our exercise, mathematical statistics is used to derive properties of the variance that result from linear transformations of random variables. This is a small piece of the larger puzzle, where mathematical theory guides us in making sense of real-world phenomena by understanding the structure and behavior of the data derived from them.