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Let's solve part (a) step-by-step first.

### Part (a): Finding the value of \( k \)

We are given the table of goals and their corresponding frequencies:

\[
\begin{array}{|c|c|}
\hline
\text{Goals} & \text{Frequency} \\
\hline
0 & 4 \\
1 & k \\
2 & 3 \\
3 & 2 \\
4 & 3 \\
8 & 1 \\
\hline
\end{array}
\]

The mean number of goals is given as \( 1.95 \), and we need to find \( k \).

The formula for the mean is:

\[
\text{Mean} = \frac{\sum (\text{Goals} \times \text{Frequency})}{\sum \text{Frequency}}
\]

### Step 1: Calculate \( \sum \text{Frequency} \)
From the table, the total frequency (including \( k \)) is:

\[
4 + k + 3 + 2 + 3 + 1 = k + 13
\]

So, \( \sum \text{Frequency} = k + 13 \).

### Step 2: Calculate \( \sum (\text{Goals} \times \text{Frequency}) \)
Now, let's calculate the sum of the goals multiplied by their frequencies:

\[
0 \times 4 + 1 \times k + 2 \times 3 + 3 \times 2 + 4 \times 3 + 8 \times 1
\]

\[
= 0 + k + 6 + 6 + 12 + 8 = k + 32
\]

So, \( \sum (\text{Goals} \times \text{Frequency}) = k + 32 \).

### Step 3: Set up the equation using the mean
We are told that the mean is \( 1.95 \):

\[
1.95 = \frac{k + 32}{k + 13}
\]

### Step 4: Solve for \( k \)
Multiply both sides by \( k + 13 \):

\[
1.95(k + 13) = k + 32
\]

Expand the left side:

\[
1.95k + 25.35 = k + 32
\]

Move all terms involving \( k \) to one side and constants to the other side:

\[
1.95k - k = 32 - 25.35
\]

\[
0.95k = 6.65
\]

Solve for \( k \):

\[
k = \frac{6.65}{0.95} = 7
\]

So, the value of \( k \) is 7.

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Now we can move to part (b) if you'd like. Would you like me to proceed with that, or explain anything further in part (a)?

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Here are 5 related questions:
1. How do we compute the total frequency in a grouped data set?
2. What would happen if the total frequency didn't include all possible data points?
3. How does including outliers, like a player scoring 22 goals, affect the mean?
4. What is the interpretation of the mean in this context?
5. Can you explain how \( k \) influences the distribution of scores?

**Tip:** Always check the frequencies and the total data points to ensure no information is missing, as it can significantly affect statistical results!

Given the data of goals scored by players in a football club during a season, find the value of k if the mean number of goals per player is 1.95. Then, find the correct mean and standard deviation when a top scorer who scored 22 goals is added.

This problem helps you find the value of k given the mean number of goals scored by players in a football club. Additionally, it covers how to recalculate the mean and standard deviation when the top scorer is added to the data set. Suitable for Grades 10-12.

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