The following marks were obtained by pupils in a test.

[tex]$
\begin{array}{lllll}
8 & 4 & 8 & 2 & 8 \\
6 & 8 & 8 & 8 & 10 \\
8 & 9 & 8 & 6 & 10 \\
2 & 2 & 8 & 6 & 6
\end{array}
$[/tex]

a) Construct a frequency distribution table for the data.

b) What is the modal mark?

c) Calculate the mean mark.

d) How many pupils scored more than 7 marks?

e) What is the probability that a student chosen at random obtained 2 marks?

\begin{tabular}{|c|c|c|c|}
\hline Mark & Tally & Frequency & [tex]$fx$[/tex] \\
\hline 2 & ||| & 3 & 6 \\
\hline 4 & | & 1 & 4 \\
\hline 6 & |||| & 5 & 30 \\
\hline 8 & |||||||| & 8 & 64 \\
\hline 9 & | & 1 & 9 \\
\hline 10 & || & 2 & 20 \\
\hline
\end{tabular}



Answer :

Given the marks obtained by pupils in a test:

[tex]\[ \begin{array}{lllll} 8 & 4 & 8 & 2 & 8 \\ 6 & 8 & 8 & 8 & 10 \\ 8 & 9 & 8 & 6 & 10 \\ 2 & 2 & 8 & 6 & 6 \end{array} \][/tex]

Let's address each part of the problem step by step.

### a) Construct a frequency distribution table for the data.

To construct the frequency distribution table, we need to count the occurrences of each mark:

[tex]\[ \begin{array}{|c|c|c|} \hline \text{Mark} & \text{Tally} & \text{Frequency} \\ \hline 2 & ||| & 3 \\ \hline 4 & | & 1 \\ \hline 6 & |||| & 4 \\ \hline 8 & ||||||||| & 9 \\ \hline 9 & | & 1 \\ \hline 10 & || & 2 \\ \hline \end{array} \][/tex]

### b) What is the modal mark?

The modal mark is the mark that appears most frequently in the data set. From the frequency distribution table, we see that the mark 8 appears 9 times, which is more frequent than any other mark.

Thus, the modal mark is [tex]\(8\)[/tex].

### c) Calculate the mean mark.

The mean mark can be calculated using the formula:

[tex]\[ \text{Mean} = \frac{\sum (\text{Mark} \times \text{Frequency})}{\sum (\text{Frequency})} \][/tex]

From the frequency distribution, the total of the marks times their respective frequencies is:

[tex]\[ 2 \times 3 + 4 \times 1 + 6 \times 4 + 8 \times 9 + 9 \times 1 + 10 \times 2 = 6 + 4 + 24 + 72 + 9 + 20 = 135 \][/tex]

The total number of students (sum of frequencies) is:

[tex]\[ 3 + 1 + 4 + 9 + 1 + 2 = 20 \][/tex]

Therefore, the mean mark is:

[tex]\[ \text{Mean} = \frac{135}{20} = 6.75 \][/tex]

### d) How many pupils score more than 7 marks?

We need to count the number of pupils who scored more than 7 marks. From the frequency table:

- 8 marks: 9 pupils
- 9 marks: 1 pupil
- 10 marks: 2 pupils

Thus, the total number of pupils who scored more than 7 marks is:

[tex]\[ 9 + 1 + 2 = 12 \][/tex]

### e) What is the probability that a student chosen at random obtained 2 marks?

The probability can be calculated as the ratio of the number of students who obtained 2 marks to the total number of students.

From the frequency table, the number of students who obtained 2 marks is 3, and the total number of students is 20.

Therefore, the probability that a student chosen at random obtained 2 marks is:

[tex]\[ \frac{3}{20} = 0.15 \][/tex]

### Summary Table:

[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Mark} & \text{Tally} & \text{Frequency} & f \times x \\ \hline 2 & ||| & 3 & 6 \\ \hline 4 & | & 1 & 4 \\ \hline 6 & |||| & 4 & 24 \\ \hline 8 & ||||||||| & 9 & 72 \\ \hline 9 & | & 1 & 9 \\ \hline 10 & || & 2 & 20 \\ \hline \text{Total} & & 20 & 135 \\ \hline \end{array} \][/tex]