Let's tackle the problem step by step as requested.
### a) Work out the range of the marks
The range of a set of numbers is the difference between the highest and the lowest numbers in the set.
From the table, the marks given are:
- 6
- 7
- 8
- 9
- 10
The highest mark is 10 and the lowest mark is 6.
[tex]\[
\text{Range} = \text{Highest mark} - \text{Lowest mark} = 10 - 6 = 4
\][/tex]
So, the range of the marks is 4.
### b) How many students are in the group?
To find out how many students took the test, we simply sum the frequencies given for each mark.
[tex]\[
\text{Total students} = 5 + 4 + 7 + 10 + 4 = 30
\][/tex]
Thus, there are 30 students in the group.
### c) Work out the mean mark of the group
To calculate the mean (average) mark, we need to find the total number of marks and divide it by the total number of students.
First, we calculate the total marks by multiplying each mark by its frequency and then summing these products.
[tex]\[
\begin{aligned}
\text{Total marks} &= (6 \times 5) + (7 \times 4) + (8 \times 7) + (9 \times 10) + (10 \times 4) \\
&= 30 + 28 + 56 + 90 + 40 \\
&= 244
\end{aligned}
\][/tex]
Next, we divide this total by the number of students to find the mean mark.
[tex]\[
\text{Mean mark} = \frac{\text{Total marks}}{\text{Total students}} = \frac{244}{30} \approx 8.133\text{ (rounded to 3 decimal places)}
\][/tex]
So, the mean mark of the group is approximately 8.133.
To summarize:
a) The range of the marks is 4.
b) There are 30 students in the group.
c) The mean mark of the group is approximately 8.133.
