If the mean of the following distribution is 13, then p =
`{:(x_(i) :,5,10,12,17,16,20),(f_(i) :,9,3,P,8,7,5):}`

Options

A. 6
B. 7
C. 10
D. 4

Answer

To find the value of \( p \) given that the mean of the distribution is 13, we can follow these steps:

### Step 1: Write down the formula for the mean
The mean \( \bar{x} \) of a frequency distribution can be calculated using the formula:

\[ \bar{x} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \]

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

### Step 2: Set up the equation with the given mean
We know the mean is given as 13. Therefore, we can set up the equation:

\[ 13 = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \]

### Step 3: Calculate \( \sum (x_i \cdot f_i) \)
Now, we will calculate \( \sum (x_i \cdot f_i) \) using the provided values and frequencies:

- For \( x_1 = 5 \), \( f_1 = 9 \): \( 5 \times 9 = 45 \)
- For \( x_2 = 10 \), \( f_2 = 3 \): \( 10 \times 3 = 30 \)
- For \( x_3 = 12 \), \( f_3 = p \): \( 12 \times p = 12p \)
- For \( x_4 = 17 \), \( f_4 = 8 \): \( 17 \times 8 = 136 \)
- For \( x_5 = 16 \), \( f_5 = 7 \): \( 16 \times 7 = 112 \)
- For \( x_6 = 20 \), \( f_6 = 5 \): \( 20 \times 5 = 100 \)

Now, summing these values gives:

\[ \sum (x_i \cdot f_i) = 45 + 30 + 12p + 136 + 112 + 100 \]

Calculating the constant terms:

\[ 45 + 30 + 136 + 112 + 100 = 423 \]

So,

\[ \sum (x_i \cdot f_i) = 423 + 12p \]

### Step 4: Calculate \( \sum f_i \)
Next, we calculate \( \sum f_i \):

\[ \sum f_i = 9 + 3 + p + 8 + 7 + 5 = 32 + p \]

### Step 5: Substitute into the mean formula
Now we substitute back into the mean formula:

\[ 13 = \frac{423 + 12p}{32 + p} \]

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

\[ 13(32 + p) = 423 + 12p \]

Expanding this:

\[ 416 + 13p = 423 + 12p \]

### Step 7: Rearranging the equation
Now, we rearrange the equation to isolate \( p \):

\[ 13p - 12p = 423 - 416 \]

This simplifies to:

\[ p = 7 \]

### Conclusion
Thus, the value of \( p \) is \( 7 \).

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