If the mean of the following data is 5.5, then x =
`{:(x_(i),2,4,6,8),(f_(i)"",3,5,6,x):}`

Options

A. 6
B. 8
C. 15
D. 11

Explanation

To find the value of \( x \) given that the mean of the data is 5.5, we can follow these steps:

### Step 1: Write the formula for the mean
The mean (\( \bar{x} \)) is calculated using the formula:
\[ \bar{x} = \frac{\Sigma (x_i \cdot f_i)}{\Sigma f_i} \]
where \( x_i \) are the data points and \( f_i \) are the corresponding frequencies.

### Step 2: Identify the data points and frequencies
From the question, we have:
- Data points (\( x_i \)): 2, 4, 6, 8
- Frequencies (\( f_i \)): 3, 5, 6, \( x \)

### Step 3: Substitute the values into the mean formula
Given that the mean is 5.5, we can set up the equation:
\[ 5.5 = \frac{(2 \cdot 3) + (4 \cdot 5) + (6 \cdot 6) + (8 \cdot x)}{3 + 5 + 6 + x} \]

### Step 4: Calculate \( \Sigma (x_i \cdot f_i) \)
Calculating each term:
- \( 2 \cdot 3 = 6 \)
- \( 4 \cdot 5 = 20 \)
- \( 6 \cdot 6 = 36 \)
- \( 8 \cdot x = 8x \)

So,
\[ \Sigma (x_i \cdot f_i) = 6 + 20 + 36 + 8x = 62 + 8x \]

### Step 5: Calculate \( \Sigma f_i \)
Calculating the sum of frequencies:
\[ \Sigma f_i = 3 + 5 + 6 + x = 14 + x \]

### Step 6: Set up the equation
Now substituting back into the mean formula:
\[ 5.5 = \frac{62 + 8x}{14 + x} \]

### Step 7: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 5.5(14 + x) = 62 + 8x \]

### Step 8: Distribute on the left side
Distributing \( 5.5 \):
\[ 77 + 5.5x = 62 + 8x \]

### Step 9: Rearrange the equation
Rearranging gives:
\[ 77 - 62 = 8x - 5.5x \]
\[ 15 = 2.5x \]

### Step 10: Solve for \( x \)
Dividing both sides by 2.5:
\[ x = \frac{15}{2.5} = 6 \]

### Final Answer
The value of \( x \) is \( 6 \).

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