The contents of 100 match boxes were checked to determine the number of matches they contain.
(i) Calculate, correct to one decimal place, the mean number of matches per box.
(ii) Determine, how many extra matches would have to be added to the total contents o the 100 boxes to bring the mean up to exactly 39 matches?

Answer

To solve the problem step by step, we will follow the instructions given in the question.

### Step 1: Create the Frequency Table
We need to create a frequency table with the number of matches (X) and the corresponding number of boxes (F). The data provided is as follows:

| Number of Matches (X) | Number of Boxes (F) |
|-----------------------|---------------------|
| 35 | 6 |
| 36 | 10 |
| 37 | 18 |
| 38 | 25 |
| 39 | 21 |
| 40 | 12 |
| 41 | 8 |

### Step 2: Calculate \( X_i \times F_i \)
Next, we will calculate the product of the number of matches and the number of boxes for each entry:

| Number of Matches (X) | Number of Boxes (F) | \( X_i \times F_i \) |
|-----------------------|---------------------|-----------------------|
| 35 | 6 | 210 |
| 36 | 10 | 360 |
| 37 | 18 | 666 |
| 38 | 25 | 950 |
| 39 | 21 | 819 |
| 40 | 12 | 480 |
| 41 | 8 | 328 |

### Step 3: Calculate the Summation
Now, we need to calculate the summation of \( F_i \) and \( X_i \times F_i \):

- \( \Sigma F_i = 6 + 10 + 18 + 25 + 21 + 12 + 8 = 100 \)
- \( \Sigma (X_i \times F_i) = 210 + 360 + 666 + 950 + 819 + 480 + 328 = 3813 \)

### Step 4: Calculate the Mean
The mean number of matches per box is calculated using the formula:

\[ \text{Mean} = \frac{\Sigma (X_i \times F_i)}{\Sigma F_i} \]

Substituting the values we found:

\[ \text{Mean} = \frac{3813}{100} = 38.13 \]

Rounding to one decimal place, the mean is:

\[ \text{Mean} = 38.1 \]

### Step 5: Determine Extra Matches Needed
To find out how many extra matches need to be added to achieve a mean of 39 matches, we set up the equation:

Let \( E \) be the extra matches needed. The new total number of matches will be \( 3813 + E \) and the total number of boxes remains 100.

Using the mean formula again:

\[ 39 = \frac{3813 + E}{100} \]

Multiplying both sides by 100:

\[ 3900 = 3813 + E \]

Now, solving for \( E \):

\[ E = 3900 - 3813 = 87 \]

### Final Answers
(i) The mean number of matches per box is **38.1**.
(ii) The number of extra matches needed is **87**.