If both the mean and the standard deviation of 50 observatios `x_(1), x_(2),….x_(50)` are equal to 16, then the mean of `(x_(1)-4)^(2), (x_(2)-4)^(2),….,(x_(50)-4)^(2)` is

Options

A. 525
B. 480
C. 400
D. 380

Answer

To solve the problem step by step, we need to find the mean of the transformed observations \((x_1 - 4)^2, (x_2 - 4)^2, \ldots, (x_{50} - 4)^2\) given that the mean and standard deviation of the original observations \(x_1, x_2, \ldots, x_{50}\) are both equal to 16.

### Step 1: Understand the Given Information
We know:
- Mean of \(x_i\) (denoted as \(\bar{x}\)) = 16
- Standard deviation of \(x_i\) = 16
- Number of observations, \(n = 50\)

### Step 2: Calculate the Variance
The variance (\(\sigma^2\)) is the square of the standard deviation:
\[ \sigma^2 = 16^2 = 256 \]

### Step 3: Use the Variance Formula
The formula for variance is given by:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
Substituting the known values:
\[ 256 = \frac{\sum x_i^2}{50} - 16^2 \]
\[ 256 = \frac{\sum x_i^2}{50} - 256 \]

### Step 4: Solve for \(\sum x_i^2\)
Rearranging the equation gives:
\[ 256 + 256 = \frac{\sum x_i^2}{50} \]
\[ 512 = \frac{\sum x_i^2}{50} \]
Multiplying both sides by 50:
\[ \sum x_i^2 = 512 \times 50 = 25600 \]

### Step 5: Calculate the Mean of the Transformed Observations
We need to find the mean of \((x_i - 4)^2\):
\[ \text{Mean} = \frac{1}{n} \sum (x_i - 4)^2 \]
Expanding the expression:
\[ (x_i - 4)^2 = x_i^2 - 8x_i + 16 \]
Thus,
\[ \sum (x_i - 4)^2 = \sum x_i^2 - 8\sum x_i + \sum 16 \]
Substituting the known values:
\[ \sum (x_i - 4)^2 = \sum x_i^2 - 8 \cdot (n \cdot \bar{x}) + n \cdot 16 \]
\[ = 25600 - 8 \cdot (50 \cdot 16) + 50 \cdot 16 \]
Calculating each term:
\[ = 25600 - 8 \cdot 800 + 800 \]
\[ = 25600 - 6400 + 800 \]
\[ = 25600 - 5600 = 20000 \]

### Step 6: Calculate the Final Mean
Now, we find the mean:
\[ \text{Mean} = \frac{1}{50} \sum (x_i - 4)^2 = \frac{20000}{50} = 400 \]

### Final Answer
The mean of \((x_1 - 4)^2, (x_2 - 4)^2, \ldots, (x_{50} - 4)^2\) is **400**.

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