Answer :
### 1.1.1 Identify the modal mark
To identify the modal mark, we need to find the mark that has the highest frequency. By examining the frequency table, we have:
[tex]\[ \begin{tabular}{|c|c|} \hline \text{Mark} & \text{Frequency} \\ \hline 16 & 1 \\ \hline 19 & 3 \\ \hline 21 & 2 \\ \hline 22 & 5 \\ \hline 23 & 2 \\ \hline 25 & 2 \\ \hline 37 & 2 \\ \hline 38 & 3 \\ \hline 39 & 2 \\ \hline 40 & 2 \\ \hline 43 & 1 \\ \hline 44 & 2 \\ \hline \end{tabular} \][/tex]
From this, we see that the highest frequency is 5, which corresponds to the mark of 22. Hence,
Modal Mark = 22
### 1.1.2 Ms. Sibeko's Claim
Ms. Sibeko claims that only 64.5% of her learners obtained marks higher than the lower quartile. Let's verify this claim:
Firstly, we need to find the lower quartile (Q1). We start by determining the total number of learners.
- Total number of learners (N) including [tex]\(A\)[/tex] is given by the sum of the frequencies:
[tex]\[ N = 29 + 2 = 31 \][/tex]
To find Q1, use the formula to find its position:
[tex]\[ \text{Position of Q1} = \frac{N + 1}{4} = \frac{31 + 1}{4} = \frac{32}{4} = 8 \][/tex]
Since we're looking for the value of the 8th learner in the cumulative frequency:
[tex]\[ \begin{tabular}{|c|c|c|} \hline \text{Mark} & \text{Frequency} & \text{Cumulative Frequency} \\ \hline 16 & 1 & 1 \\ \hline 19 & 3 & 4 \\ \hline 21 & 2 & 6 \\ \hline 22 & 5 & 11 \\ \hline 23 & 2 & 13 \\ \hline 25 & 2 & 15 \\ \hline 37 & 2 & 17 \\ \hline 38 & 3 & 20 \\ \hline 39 & 2 & 22 \\ \hline 40 & 2 & 24 \\ \hline 43 & 1 & 25 \\ \hline 44 & 2 & 27 \\ \hline A & 1 & 28 \\ \hline \end{tabular} \][/tex]
The 8th rank occurs between the first 6 plus 2 more, which lands in mark 22. Therefore, the lower quartile (Q1) is 22.
Next, we need to determine the number of learners who scored higher than this value.
- Those scoring higher rated as (position of 8th + 1 to the total learners):
Number of people above Q1 = 31 - 8 = 23.
- Percentage:
[tex]\[ \frac{23}{31} \times 100 \approx 74.19\% \][/tex]
Given that 74.19% > 64.5%,
Ms. Sibeko's claim is correct.
### 1.1.3 Determine the value of [tex]\(A\)[/tex]
To find the value of [tex]\(A\)[/tex], consider the given average mark is 45:
[tex]\[ \text{Total sum of marks} = \text{Average mark} \times \text{Total frequency} = 45 \times 31 \][/tex]
Count the current sum without [tex]\(A's\)[/tex] influence,
[tex]\[ \text{Sum of marks excluding A} = 16 \times 1 + 19 \times 3 + 21 \times 2 + 22 \times 5 + 23 \times 2 + 25 \times 2 + 37 \times 2 + 38 \times 3 + 39 \times 2 + 40 \times 2 + 43 \times 1 + 44 \times 2 \][/tex]
Sum with contributions:
Sum underneath including A:
Given average total (1457)
Calculate [tex]\(A\)[/tex] value
[tex]\[ \text{A value} = 31 \times 45 - \text{sum individual} = 1395 ... therefore A = 507 \][/tex]
### Answers Summary:
1.1.1 The modal mark is 22
1.1.2 Ms Sibeko's claim that only [tex]\(64.5\%\)[/tex] of the learners obtained marks higher than the lower quartile is accurate.
1.1.3 The value of A is 507 in the table.
To identify the modal mark, we need to find the mark that has the highest frequency. By examining the frequency table, we have:
[tex]\[ \begin{tabular}{|c|c|} \hline \text{Mark} & \text{Frequency} \\ \hline 16 & 1 \\ \hline 19 & 3 \\ \hline 21 & 2 \\ \hline 22 & 5 \\ \hline 23 & 2 \\ \hline 25 & 2 \\ \hline 37 & 2 \\ \hline 38 & 3 \\ \hline 39 & 2 \\ \hline 40 & 2 \\ \hline 43 & 1 \\ \hline 44 & 2 \\ \hline \end{tabular} \][/tex]
From this, we see that the highest frequency is 5, which corresponds to the mark of 22. Hence,
Modal Mark = 22
### 1.1.2 Ms. Sibeko's Claim
Ms. Sibeko claims that only 64.5% of her learners obtained marks higher than the lower quartile. Let's verify this claim:
Firstly, we need to find the lower quartile (Q1). We start by determining the total number of learners.
- Total number of learners (N) including [tex]\(A\)[/tex] is given by the sum of the frequencies:
[tex]\[ N = 29 + 2 = 31 \][/tex]
To find Q1, use the formula to find its position:
[tex]\[ \text{Position of Q1} = \frac{N + 1}{4} = \frac{31 + 1}{4} = \frac{32}{4} = 8 \][/tex]
Since we're looking for the value of the 8th learner in the cumulative frequency:
[tex]\[ \begin{tabular}{|c|c|c|} \hline \text{Mark} & \text{Frequency} & \text{Cumulative Frequency} \\ \hline 16 & 1 & 1 \\ \hline 19 & 3 & 4 \\ \hline 21 & 2 & 6 \\ \hline 22 & 5 & 11 \\ \hline 23 & 2 & 13 \\ \hline 25 & 2 & 15 \\ \hline 37 & 2 & 17 \\ \hline 38 & 3 & 20 \\ \hline 39 & 2 & 22 \\ \hline 40 & 2 & 24 \\ \hline 43 & 1 & 25 \\ \hline 44 & 2 & 27 \\ \hline A & 1 & 28 \\ \hline \end{tabular} \][/tex]
The 8th rank occurs between the first 6 plus 2 more, which lands in mark 22. Therefore, the lower quartile (Q1) is 22.
Next, we need to determine the number of learners who scored higher than this value.
- Those scoring higher rated as (position of 8th + 1 to the total learners):
Number of people above Q1 = 31 - 8 = 23.
- Percentage:
[tex]\[ \frac{23}{31} \times 100 \approx 74.19\% \][/tex]
Given that 74.19% > 64.5%,
Ms. Sibeko's claim is correct.
### 1.1.3 Determine the value of [tex]\(A\)[/tex]
To find the value of [tex]\(A\)[/tex], consider the given average mark is 45:
[tex]\[ \text{Total sum of marks} = \text{Average mark} \times \text{Total frequency} = 45 \times 31 \][/tex]
Count the current sum without [tex]\(A's\)[/tex] influence,
[tex]\[ \text{Sum of marks excluding A} = 16 \times 1 + 19 \times 3 + 21 \times 2 + 22 \times 5 + 23 \times 2 + 25 \times 2 + 37 \times 2 + 38 \times 3 + 39 \times 2 + 40 \times 2 + 43 \times 1 + 44 \times 2 \][/tex]
Sum with contributions:
Sum underneath including A:
Given average total (1457)
Calculate [tex]\(A\)[/tex] value
[tex]\[ \text{A value} = 31 \times 45 - \text{sum individual} = 1395 ... therefore A = 507 \][/tex]
### Answers Summary:
1.1.1 The modal mark is 22
1.1.2 Ms Sibeko's claim that only [tex]\(64.5\%\)[/tex] of the learners obtained marks higher than the lower quartile is accurate.
1.1.3 The value of A is 507 in the table.
