If the standard deviation of `x_(1),x_(2),….,x_(n)` is 3.5 then the standard deviation of `-2x_(1)-3, -2x_(2)-3,….,-2x_(n)-3` is

Options

A. `-7`
B. `-4`
C. 7
D. 1.75

Answer

To find the standard deviation of the transformed data set \(-2x_1 - 3, -2x_2 - 3, \ldots, -2x_n - 3\), given that the standard deviation of \(x_1, x_2, \ldots, x_n\) is 3.5, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Transformation**:
The transformation applied to each data point \(x_i\) is of the form \(y_i = -2x_i - 3\). This transformation consists of a multiplication by -2 and a subtraction of 3.

2. **Effect of Multiplication on Standard Deviation**:
The standard deviation is affected by multiplication. If we multiply a data set by a constant \(a\), the standard deviation of the new data set is given by:
\[ \sigma' = |a| \cdot \sigma \]
where \(\sigma\) is the original standard deviation.

3. **Apply the Multiplication**:
Here, we multiply by -2. Therefore, the new standard deviation after this transformation will be:
\[ \sigma' = |-2| \cdot 3.5 = 2 \cdot 3.5 = 7 \]

4. **Effect of Addition/Subtraction on Standard Deviation**:
The standard deviation is not affected by adding or subtracting a constant from all data points. Thus, subtracting 3 from each data point does not change the standard deviation.

5. **Final Result**:
Therefore, the standard deviation of the transformed data set \(-2x_1 - 3, -2x_2 - 3, \ldots, -2x_n - 3\) is:
\[ \sigma' = 7 \]

### Conclusion:
The standard deviation of the transformed data set is **7**.