Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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]
[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]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
