Given the distribution of student scores, we need to determine the pass marks when 78 out of 104 students passed the exam. Here's the step-by-step solution:
1. Tabulate the Given Data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
\text{Central Rank of Group} & 10 & 20 & 30 & 40 & 50 & 60 & 70 \\
\hline
\text{No. of Students} & 7 & 8 & 13 & 29 & 35 & 9 & 3 \\
\hline
\end{array}
\][/tex]
2. Calculate the Cumulative Distribution of Students:
We add the number of students in each group cumulatively to get the following:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
\text{Central Rank of Group} & 10 & 20 & 30 & 40 & 50 & 60 & 70 \\
\hline
\text{No. of Students} & 7 & 8 & 13 & 29 & 35 & 9 & 3 \\
\hline
\text{Cumulative No. of Students} & 7 & 15 & 28 & 57 & 92 & 101 & 104 \\
\hline
\end{array}
\][/tex]
3. Determine the Pass Mark:
We know that 78 students passed the examinations. To find the pass mark, we look for the cumulative count that first reaches or exceeds 78.
- For the central rank of 10: [tex]\(7\)[/tex] students (not enough)
- For the central rank of 20: [tex]\(7 + 8 = 15\)[/tex] students (not enough)
- For the central rank of 30: [tex]\(15 + 13 = 28\)[/tex] students (not enough)
- For the central rank of 40: [tex]\(28 + 29 = 57\)[/tex] students (not enough)
- For the central rank of 50: [tex]\(57 + 35 = 92\)[/tex] students (reaches and exceeds 78)
The cumulative number of students surpasses 78 at the central rank of 50. Thus, the pass mark is 50.
Therefore, the pass mark for the exam is 50.
