Let's go through each part of the problem step by step.
### (a) Explanation of the Formula for Computing the Average Score
The average score, or mean, in the context of grouped data, can be calculated using the formula:
[tex]\[ \text{Mean} = \frac{\sum (f_i \cdot x_i)}{\sum f_i} \][/tex]
- [tex]\( f_i \)[/tex] represents the frequency of the [tex]\( i \)[/tex]-th observation (number of students who obtained a particular mark).
- [tex]\( x_i \)[/tex] represents the [tex]\( i \)[/tex]-th value of the marks.
- [tex]\( \sum (f_i \cdot x_i) \)[/tex] is the sum of the products of each mark and its frequency.
- [tex]\( \sum f_i \)[/tex] is the sum of all frequencies (total number of students).
### (b) Total Number of Students
The total number of students is the sum of all the frequencies provided in the data.
Given frequencies: 8, 9, 12, 16, 12
[tex]\[ \text{Total Students} = 8 + 9 + 12 + 16 + 12 = 57 \][/tex]
### (c) Mean of the Given Data
To find the mean, we need to compute the sum of the products of each mark and its frequency, then divide by the total number of students.
Step-by-step calculation for the products of marks and frequencies:
- [tex]\( 14 \times 8 = 112 \)[/tex]
- [tex]\( 15 \times 9 = 135 \)[/tex]
- [tex]\( 16 \times 12 = 192 \)[/tex]
- [tex]\( 17 \times 16 = 272 \)[/tex]
- [tex]\( 18 \times 12 = 216 \)[/tex]
Sum of the products:
[tex]\[ 112 + 135 + 192 + 272 + 216 = 927 \][/tex]
Now, the mean is given by:
[tex]\[ \text{Mean} = \frac{\sum (f_i \cdot x_i)}{\sum f_i} = \frac{927}{57} \approx 16.263 \][/tex]
### (d) Average Mark of the Failed Students (Marks < 16)
To find the average mark of the students who failed (students with marks less than 16), we focus on the students with marks 14 and 15.
Marks and corresponding frequencies of failed students:
- Marks = 14, Frequency = 8
- Marks = 15, Frequency = 9
Total number of failed students:
[tex]\[ \text{Total Failed Students} = 8 + 9 = 17 \][/tex]
Sum of the products of failed students' marks and their frequencies:
- [tex]\( 14 \times 8 = 112 \)[/tex]
- [tex]\( 15 \times 9 = 135 \)[/tex]
Sum of these products:
[tex]\[ 112 + 135 = 247 \][/tex]
Now, the average mark of the failed students is given by:
[tex]\[ \text{Failed Mean} = \frac{\sum (f_i \cdot x_i)}{\sum f_i} = \frac{247}{17} \approx 14.529 \][/tex]
### Summary
- Total number of students: 57
- Mean score of the students: 16.263
- Number of students who failed: 17
- Average mark of the failed students: 14.529
