If the arithmetic mean of the following distribution is 8.2, then a =
`{:(x_(i) " :",1,3,5,9,11,13),(f_(i)" :",3,2,7,a,4,8):}`

Options

A. 5
B. 6
C. 9
D. none of these

Explanation

To find the value of \( a \) given that the arithmetic mean of the distribution is 8.2, we can follow these steps:

### Step 1: Write down the formula for the arithmetic mean
The arithmetic mean (mean) is given by the formula:

\[ \text{Mean} = \frac{\Sigma (x_i \cdot f_i)}{\Sigma f_i} \]

where \( x_i \) are the values and \( f_i \) are the corresponding frequencies.

### Step 2: Identify the values and frequencies
From the question, we have:

\[ x_i = \{1, 3, 5, 9, 11, 13\} \]
\[ f_i = \{3, 2, 7, a, 4, 8\} \]

### Step 3: Calculate \( \Sigma f_i \)
Now we calculate \( \Sigma f_i \):

\[ \Sigma f_i = 3 + 2 + 7 + a + 4 + 8 = 24 + a \]

### Step 4: Calculate \( \Sigma (x_i \cdot f_i) \)
Next, we calculate \( \Sigma (x_i \cdot f_i) \):

\[ \Sigma (x_i \cdot f_i) = (1 \cdot 3) + (3 \cdot 2) + (5 \cdot 7) + (9 \cdot a) + (11 \cdot 4) + (13 \cdot 8) \]

Calculating each term:

- \( 1 \cdot 3 = 3 \)
- \( 3 \cdot 2 = 6 \)
- \( 5 \cdot 7 = 35 \)
- \( 9 \cdot a = 9a \)
- \( 11 \cdot 4 = 44 \)
- \( 13 \cdot 8 = 104 \)

Now summing these:

\[ \Sigma (x_i \cdot f_i) = 3 + 6 + 35 + 9a + 44 + 104 = 192 + 9a \]

### Step 5: Set up the equation for the mean
We know that the mean is 8.2, so we set up the equation:

\[ 8.2 = \frac{192 + 9a}{24 + a} \]

### Step 6: Cross-multiply to solve for \( a \)
Cross-multiplying gives:

\[ 8.2(24 + a) = 192 + 9a \]

Expanding this:

\[ 196.8 + 8.2a = 192 + 9a \]

### Step 7: Rearrange the equation
Rearranging gives:

\[ 196.8 - 192 = 9a - 8.2a \]

This simplifies to:

\[ 4.8 = 0.8a \]

### Step 8: Solve for \( a \)
Now, dividing both sides by 0.8:

\[ a = \frac{4.8}{0.8} = 6 \]

Thus, the value of \( a \) is:

\[ \boxed{6} \]

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