The variance of 10 observations is 16 and their mean is 12. If each observation is multiplied by 4, what is new standard deviation-

Options

A. (a) 4
B. (b) 8
C. c. 16
D. (d) 32.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the given information
We are given:
- Variance of the observations (σ²) = 16
- Mean of the observations = 12 (not needed for this calculation)
- Number of observations (n) = 10
- Each observation is multiplied by 4.

### Step 2: Recall the relationship between variance and standard deviation
The standard deviation (σ) is the square root of the variance (σ²):
\[ σ = \sqrt{σ²} \]

### Step 3: Calculate the original standard deviation
Using the given variance:
\[ σ = \sqrt{16} = 4 \]

### Step 4: Understand how multiplying observations affects variance
When each observation is multiplied by a constant factor (in this case, 4), the new variance (σ'²) is given by:
\[ σ'² = a² \cdot σ² \]
where \(a\) is the constant factor by which the observations are multiplied.

### Step 5: Calculate the new variance
Here, \(a = 4\) and \(σ² = 16\):
\[ σ'² = 4² \cdot 16 = 16 \cdot 16 = 256 \]

### Step 6: Calculate the new standard deviation
Now, we find the new standard deviation (σ'):
\[ σ' = \sqrt{σ'²} = \sqrt{256} = 16 \]

### Final Answer
The new standard deviation after multiplying each observation by 4 is **16**.

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