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 1: Understand the effect of transformations on standard deviation
The standard deviation of a dataset is affected by two types of transformations:
1. **Addition or subtraction of a constant**: This does not change the standard deviation.
2. **Multiplication or division by a constant**: This scales the standard deviation by the absolute value of that constant.

### Step 2: Identify the transformations applied
In our case, the transformation applied to each element \(x_i\) is:
\[ y_i = -2x_i - 3 \]
Here, we can see two transformations:
- Multiplication by \(-2\)
- Subtraction of \(3\)

### Step 3: Apply the transformations to the standard deviation
1. **Subtraction of 3**: As mentioned, subtracting a constant does not change the standard deviation. Therefore, the standard deviation remains \(3.5\) after this step.

2. **Multiplication by -2**: The standard deviation will be scaled by the absolute value of \(-2\), which is \(2\). Thus, we calculate the new standard deviation as follows:
\[ \text{New Standard Deviation} = 2 \times \text{Original Standard Deviation} = 2 \times 3.5 = 7 \]

### Conclusion
The standard deviation of the transformed data set \(-2x_1 - 3, -2x_2 - 3, \ldots, -2x_n - 3\) is \(7\).

### Final Answer
The standard deviation of \(-2x_1 - 3, -2x_2 - 3, \ldots, -2x_n - 3\) is **7**.

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