If the mean of the following frequency distribution is 8, find the value of p.
`{:("Variable"(x_(i)),3,5,7,9,11,13),("Frequency"(f_(i)),6,8,15,p,8,4):}`

Answer

To find the value of \( p \) in the given frequency distribution, we will follow these steps:

### Step 1: Write down the given data
We have the following values for the variable \( x \) and their corresponding frequencies \( f \):

| \( x_i \) | 3 | 5 | 7 | 9 | 11 | 13 |
|-----------|----|----|----|----|----|----|
| \( f_i \) | 6 | 8 | 15 | \( p \) | 8 | 4 |

### Step 2: Calculate the total frequency \( \Sigma f \)
The total frequency is the sum of all frequencies:

\[ \Sigma f = 6 + 8 + 15 + p + 8 + 4 = 41 + p \]

### Step 3: Calculate \( \Sigma fx \)
Next, we calculate \( \Sigma fx \) by multiplying each \( x_i \) by its corresponding \( f_i \):

\[ \Sigma fx = (3 \times 6) + (5 \times 8) + (7 \times 15) + (9 \times p) + (11 \times 8) + (13 \times 4) \]

Calculating each term:

- \( 3 \times 6 = 18 \)
- \( 5 \times 8 = 40 \)
- \( 7 \times 15 = 105 \)
- \( 9 \times p = 9p \)
- \( 11 \times 8 = 88 \)
- \( 13 \times 4 = 52 \)

Now, summing these values:

\[ \Sigma fx = 18 + 40 + 105 + 9p + 88 + 52 = 303 + 9p \]

### Step 4: Set up the equation for the mean
The mean is given by the formula:

\[ \text{Mean} = \frac{\Sigma fx}{\Sigma f} \]

Given that the mean is 8, we can set up the equation:

\[ 8 = \frac{303 + 9p}{41 + p} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ 8(41 + p) = 303 + 9p \]

Expanding both sides:

\[ 328 + 8p = 303 + 9p \]

### Step 6: Rearranging the equation
Now, we will rearrange the equation to isolate \( p \):

\[ 328 - 303 = 9p - 8p \]

This simplifies to:

\[ 25 = p \]

### Final Answer
Thus, the value of \( p \) is \( 25 \).

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