Question

The variance of 50 observations is 7. Suppose that each observation in this data is multiplied by 6 and then 5 is subtracted from it. Then the variance of that new data is

• A. 37
• B. 42
• C. 247
• D. 252

Answer 1

The Correct Option is D

To solve this problem, we are given 50 observations \( x_1, x_2, ..., x_{50} \) with a variance of 7, and we are tasked with finding the variance of a transformed dataset.

1. Understanding the Variance Formula:
The variance \( \sigma^2 \) is given by the formula:

\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2 \] where \( \mu \) is the mean of the observations. In this case, we are told that \( \sigma^2 = 7 \), so we can write:

\[ \frac{1}{50} \sum_{i=1}^{50} (x_i - \mu)^2 = 7 \]

2. Transformation of Data:
We now transform each observation by multiplying by 6 and subtracting 5. The new observations are given by:

\[ y_i = 6x_i - 5 \] Let \( \mu' \) be the mean of the new observations. Then, we can find \( \mu' \) as:

\[ \mu' = \frac{1}{50} \sum_{i=1}^{50} y_i = \frac{1}{50} \sum_{i=1}^{50} (6x_i - 5) \] \[ = 6 \left( \frac{1}{50} \sum_{i=1}^{50} x_i \right) - \frac{1}{50} \sum_{i=1}^{50} 5 = 6\mu - 5 \]

3. Variance of the New Data:
The variance of the new data, \( \sigma'^2 \), is given by:

\[ \sigma'^2 = \frac{1}{50} \sum_{i=1}^{50} (y_i - \mu')^2 \] Substitute the expression for \( y_i \) and \( \mu' \):

\[ \sigma'^2 = \frac{1}{50} \sum_{i=1}^{50} ((6x_i - 5) - (6\mu - 5))^2 \] \[ = \frac{1}{50} \sum_{i=1}^{50} (6x_i - 6\mu)^2 \] \[ = \frac{1}{50} \sum_{i=1}^{50} 36(x_i - \mu)^2 \] \[ = 36 \left( \frac{1}{50} \sum_{i=1}^{50} (x_i - \mu)^2 \right) = 36 \sigma^2 \] Since \( \sigma^2 = 7 \), we get:

\[ \sigma'^2 = 36 \cdot 7 = 252 \]

Final Answer:
The variance of the new data is \( {252} \).