An data set has a mean of 12 and a standard deviation of 2. If 5 is added to twice each data value, the new data set has

Options

A. a mean of 24 and a standard deviation of 10
B. a mean of 24 and a standard deviation of 15
C. a mean of 29 and a standard deviation of 15
D. a mean of 29 and a standard deviation of 4

Explanation

To solve the problem step by step, we will follow the mathematical transformations applied to the original data set.

### Step 1: Understand the original data set
The original data set has:
- Mean (μ) = 12
- Standard Deviation (σ) = 2

### Step 2: Apply the transformation to the data set
We are transforming each data value \( x \) in the original data set by the following operation:
\[ y = 2x + 5 \]

### Step 3: Calculate the new mean
To find the new mean after the transformation, we can use the formula for the mean of the transformed data set:
\[ \text{New Mean} = 2 \times \text{Original Mean} + 5 \]

Substituting the original mean:
\[ \text{New Mean} = 2 \times 12 + 5 \]
\[ \text{New Mean} = 24 + 5 \]
\[ \text{New Mean} = 29 \]

### Step 4: Calculate the new standard deviation
The standard deviation is affected by multiplication but not by addition. Since we are multiplying each data value by 2, the new standard deviation will be:
\[ \text{New Standard Deviation} = 2 \times \text{Original Standard Deviation} \]

Substituting the original standard deviation:
\[ \text{New Standard Deviation} = 2 \times 2 \]
\[ \text{New Standard Deviation} = 4 \]

### Final Answer
The new data set has:
- Mean = 29
- Standard Deviation = 4