Solve according to statistics

$$\sum (x_i \cdot f_i) = (1 \cdot 7) + (2 \cdot 8) + (3 \cdot 4) + (4 \cdot 4) + (5 \cdot 3) + (6 \cdot x)$$ $$= 7 + 16 + 12 + 16 + 15 + 6x$$ $$= 66 + 6x$$

$$\sum f_i = 7 + 8 + 4 + 4 + 3 + x$$ $$= 26 + x$$

$$3 = \frac{66 + 6x}{26 + x}$$ $$3(26 + x) = 66 + 6x$$ $$78 + 3x = 66 + 6x$$ $$78 - 66 = 6x - 3x$$ $$12 = 3x$$ $$x = \frac{12}{3}$$ $$x = 4$$

The value of $x$ is $\boxed{\mathbf{4}}$.

Question 1a: What is the mode?
The mode is the value with the highest frequency. With $x=4$, the frequencies are:
Goals scored 1: 7 matches
Goals scored 2: 8 matches
Goals scored 3: 4 matches
Goals scored 4: 4 matches
Goals scored 5: 3 matches
Goals scored 6: 4 matches (since $x=4$)
The highest frequency is 8, which corresponds to 2 goals scored.
The mode is $\boxed{\mathbf{2}}$.

Question 1b: What is the median?
First, find the total number of matches (total frequency):

$$N = \sum f_i = 26 + x = 26 + 4 = 30$$

Since $N=30$ is an even number, the median is the average of the $\frac{N}{2}$-th and $(\frac{N}{2} + 1)$-th values.
The median will be the average of the $15^{th}$ and $16^{th}$ values.

Let's list the cumulative frequencies:

• Goals scored 1: 7 matches (values 1-7 are '1')
• Goals scored 2: $7 + 8 = 15$ matches (values 8-15 are '2')
• Goals scored 3: $15 + 4 = 19$ matches (values 16-19 are '3')

The $15^{th}$ value is 2.
The $16^{th}$ value is 3.

Calculate the median:

$$\text{Median} = \frac{15^{th} \text{ value} + 16^{th} \text{ value}}{2} = \frac{2 + 3}{2} = \frac{5}{2} = 2.5$$

The median is $\boxed{\mathbf{2.5}}$.