Answer :
Sure, let's go step by step and solve the problem.
### (a) Explain the formula used to compute the average score from this data.
To find the average (mean) score of the students, we use the formula for the weighted mean:
[tex]\[ \text{Mean} (\mu) = \frac{\sum (x_i \cdot f_i)}{N} \][/tex]
where:
- [tex]\( x_i \)[/tex] represents the marks.
- [tex]\( f_i \)[/tex] represents the number of students who received those marks.
- [tex]\( N \)[/tex] is the total number of students.
### (b) Find the total number of students.
To find the total number of students, we sum the frequencies (number of students) for each mark:
[tex]\[ N = \sum f_i \][/tex]
So, we have:
[tex]\[ N = 8 + 9 + 12 + 16 + 12 = 57 \][/tex]
The total number of students is [tex]\( 57 \)[/tex].
### (c) Find the mean of the given data.
To find the mean, we first compute the sum of the products of each mark and its corresponding number of students:
[tex]\[ \sum (x_i \cdot f_i) = (14 \cdot 8) + (15 \cdot 9) + (16 \cdot 12) + (17 \cdot 16) + (18 \cdot 12) \][/tex]
Thus:
[tex]\[ \sum (x_i \cdot f_i) = 112 + 135 + 192 + 272 + 216 = 927 \][/tex]
Now, we use the formula for the mean:
[tex]\[ \text{Mean} (\mu) = \frac{\sum (x_i \cdot f_i)}{N} = \frac{927}{57} \approx 16.263 \][/tex]
The mean of the given data is approximately [tex]\( 16.263 \)[/tex].
### (d) If the pass mark is 16, find the average mark of the failed students.
To find the average mark of the failed students, we consider only those students who scored below the pass mark (16). This includes students with marks of 14 and 15.
First, calculate the total number of failed students:
[tex]\[ \text{Total failed students} = 8 + 9 = 17 \][/tex]
Next, compute the sum of the products of the marks and the corresponding number of failed students:
[tex]\[ \sum (x_i \cdot f_i)_{\text{failed}} = (14 \cdot 8) + (15 \cdot 9) = 112 + 135 = 247 \][/tex]
Now, use the formula for the mean to find the average mark of the failed students:
[tex]\[ \text{Mean} (\mu)_{\text{failed}} = \frac{\sum (x_i \cdot f_i)_{\text{failed}}}{\text{Total failed students}} = \frac{247}{17} \approx 14.529 \][/tex]
The average mark of the failed students is approximately [tex]\( 14.529 \)[/tex].
### (a) Explain the formula used to compute the average score from this data.
To find the average (mean) score of the students, we use the formula for the weighted mean:
[tex]\[ \text{Mean} (\mu) = \frac{\sum (x_i \cdot f_i)}{N} \][/tex]
where:
- [tex]\( x_i \)[/tex] represents the marks.
- [tex]\( f_i \)[/tex] represents the number of students who received those marks.
- [tex]\( N \)[/tex] is the total number of students.
### (b) Find the total number of students.
To find the total number of students, we sum the frequencies (number of students) for each mark:
[tex]\[ N = \sum f_i \][/tex]
So, we have:
[tex]\[ N = 8 + 9 + 12 + 16 + 12 = 57 \][/tex]
The total number of students is [tex]\( 57 \)[/tex].
### (c) Find the mean of the given data.
To find the mean, we first compute the sum of the products of each mark and its corresponding number of students:
[tex]\[ \sum (x_i \cdot f_i) = (14 \cdot 8) + (15 \cdot 9) + (16 \cdot 12) + (17 \cdot 16) + (18 \cdot 12) \][/tex]
Thus:
[tex]\[ \sum (x_i \cdot f_i) = 112 + 135 + 192 + 272 + 216 = 927 \][/tex]
Now, we use the formula for the mean:
[tex]\[ \text{Mean} (\mu) = \frac{\sum (x_i \cdot f_i)}{N} = \frac{927}{57} \approx 16.263 \][/tex]
The mean of the given data is approximately [tex]\( 16.263 \)[/tex].
### (d) If the pass mark is 16, find the average mark of the failed students.
To find the average mark of the failed students, we consider only those students who scored below the pass mark (16). This includes students with marks of 14 and 15.
First, calculate the total number of failed students:
[tex]\[ \text{Total failed students} = 8 + 9 = 17 \][/tex]
Next, compute the sum of the products of the marks and the corresponding number of failed students:
[tex]\[ \sum (x_i \cdot f_i)_{\text{failed}} = (14 \cdot 8) + (15 \cdot 9) = 112 + 135 = 247 \][/tex]
Now, use the formula for the mean to find the average mark of the failed students:
[tex]\[ \text{Mean} (\mu)_{\text{failed}} = \frac{\sum (x_i \cdot f_i)_{\text{failed}}}{\text{Total failed students}} = \frac{247}{17} \approx 14.529 \][/tex]
The average mark of the failed students is approximately [tex]\( 14.529 \)[/tex].
