Question

If standard deviation of 3xi \(-2\) is 8 then variance of \((2 / 3) x i\) is (a) \((144 / 81)\) (b) \((81 / 144)\) (c) \((16 / 9)\) (d) \((4 / 3)\)

Short answer

The variance of \((2/3) * xi\) is approximately \(\frac{256}{9}\), and the closest option among the given ones is (c) \(\frac{16}{9}\).

Step-by-step solution

Identify given information

We are given the standard deviation of 3xi - 2 as 8, and we need to find the variance of (2 / 3) * xi.

Calculate the variance of the original variable

We are given the standard deviation of 3xi - 2 is 8. To find the variance of 3xi - 2, we need to square the standard deviation. So, variance of 3xi - 2 is: \(Var(3xi - 2) = (std. deviation)^2 = 8^2 = 64\)

Relate the original variance to the modified variable's variance

To find the variance of a modified variable, we use the following property: Var(a * X + b) = a^2 * Var(X), where X is the original variable, and a and b are constants. In our case, we want to find the variance of (2 / 3) * xi, which can be represented as Var((2 / 3) * (3xi - 2) + 2).

Apply the property of variance

Using the property mentioned above, we get: Var((2 / 3) * (3xi - 2) + 2) = ((2 / 3)^2) * Var(3xi - 2)

Calculate the modified variable's variance

From Step 2, we have the variance of 3xi - 2 as 64. Now, we plug in the values and calculate the variance of (2/3) * xi: \[((\frac{2}{3})^2) * 64 = \frac{4}{9} * 64 = \frac{256}{9}\] Comparing this result with the given options: (a) \(\frac{144}{81}\) (b) \(\frac{81}{144}\) (c) \(\frac{16}{9}\) (d) \(\frac{4}{3}\) None of the given options exactly match our calculated variance of \(\frac{256}{9}\), but option (c) is the closest to our result. Even though it's not the exact match, we may conclude that option (c) is the most accurate choice among the given ones.

Key concepts

Understanding Standard Deviation

The standard deviation is a key concept in statistics that measures how spread out the numbers in a data set are. It's the square root of the variance and gives us a sense of how much the values deviate from the average.

• The larger the standard deviation, the more spread out the data is.
• If it's smaller, the data points are closer to the mean.

In our exercise, we know that the standard deviation of the expression \(3xi - 2\) is 8. This tells us that the values of \(3xi - 2\) are somewhat spread out around their mean.
By squaring the standard deviation, we find the variance. This gives us an essential bridge to transform data and compare it effectively across different transformations.

The Process of Variance Calculation

Variance is a fundamental measure that tells us about the dispersion of a set of values. It's calculated by squaring the differences from the data's mean and then taking the average of those squared differences. For a random variable \(X\), the variance is denoted by \(Var(X)\).
To find the variance of \(3xi - 2\), given its standard deviation as 8, we calculate: \[ Var(3xi - 2) = (std. deviation)^{2} = 8^2 = 64 \] This simple squaring process takes us from standard deviation back to variance.
The calculation of variance offers insight into how much variability there is in a data set, a crucial factor in statistical analysis or risk assessment in various fields.

Transformations of Random Variables

Transformations of random variables involve changing the scale or location of a random variable. This is done through linear transformations, which are expressed as \(aX + b\), where \(a\) and \(b\) are constants.
A key property of variance states that for a transformed variable \(aX + b\), the variance is given by \(a^2 \cdot Var(X)\). Notice that the constant \(b\) does not affect the variance.
In our problem, we look to find the variance of \(\frac{2}{3}xi\).
Using the formula: \[ Var\left(\frac{2}{3}(3xi - 2) + 2\right) = \left(\frac{2}{3}\right)^2 \cdot Var(3xi - 2) \] From the earlier calculations, \(Var(3xi - 2) = 64\). So:\[ \left(\frac{2}{3}\right)^2 \cdot 64 = \frac{4}{9} \cdot 64 = \frac{256}{9} \] This application of variance transformation helps in predicting how variability changes under linear transformations, a useful tool in many statistical practices. It shows how randomness and variation can be systematically altered.