Let's tackle the given problem step by step. We have a set of marks secured by 25 students, and we are asked to arrange this data into a frequency distribution using the class intervals [tex]\(20-29\)[/tex], [tex]\(30-39\)[/tex], [tex]\(40-49\)[/tex], [tex]\(50-59\)[/tex], and [tex]\(60-69\)[/tex]. We also need to provide the cumulative frequencies for each interval.
### Step 1: List of Marks
Here are the marks provided:
[tex]\[ 83, 28, 90, 82, 35, 35, 36, 40, 41, 43, 44, 45, 48, 49, 52, 58, 54, 56, 56, 58, 61, 62, 65, 64 \][/tex]
### Step 2: Defining Class Intervals and Frequency Distribution
The class intervals are:
- [tex]\(20-29\)[/tex]
- [tex]\(30-39\)[/tex]
- [tex]\(40-49\)[/tex]
- [tex]\(50-59\)[/tex]
- [tex]\(60-69\)[/tex]
We will now arrange the marks into these intervals and count the number of marks within each interval (frequency distribution).
### Frequency Distribution:
- [tex]\(20-29\)[/tex]: 1 mark
- Marks: [28]
- [tex]\(30-39\)[/tex]: 3 marks
- Marks: [35, 35, 36]
- [tex]\(40-49\)[/tex]: 7 marks
- Marks: [40, 41, 43, 44, 45, 48, 49]
- [tex]\(50-59\)[/tex]: 6 marks
- Marks: [52, 54, 56, 56, 58, 58]
- [tex]\(60-69\)[/tex]: 4 marks
- Marks: [61, 62, 64, 65]
Summarizing the frequency distribution in a table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
Class Interval & Frequency \\
\hline
20-29 & 1 \\
\hline
30-39 & 3 \\
\hline
40-49 & 7 \\
\hline
50-59 & 6 \\
\hline
60-69 & 4 \\
\hline
\end{tabular}
\][/tex]
### Step 3: Calculating Cumulative Frequencies
To find the cumulative frequency for each interval, we add the frequencies of all preceding intervals.
- Cumulative frequency for [tex]\(20-29\)[/tex]: 1 (only the first interval)
- Cumulative frequency for [tex]\(30-39\)[/tex]: [tex]\(1 + 3 = 4\)[/tex]
- Cumulative frequency for [tex]\(40-49\)[/tex]: [tex]\(4 + 7 = 11\)[/tex]
- Cumulative frequency for [tex]\(50-59\)[/tex]: [tex]\(11 + 6 = 17\)[/tex]
- Cumulative frequency for [tex]\(60-69\)[/tex]: [tex]\(17 + 4 = 21\)[/tex]
### Cumulative Frequency Table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
Class Interval & Cumulative Frequency \\
\hline
20-29 & 1 \\
\hline
30-39 & 4 \\
\hline
40-49 & 11 \\
\hline
50-59 & 17 \\
\hline
60-69 & 21 \\
\hline
\end{tabular}
\][/tex]
### Final Summary:
In summary, the frequency distribution and cumulative frequency table for the given marks are as follows:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
Class Interval & Frequency & Cumulative Frequency \\
\hline
20-29 & 1 & 1 \\
\hline
30-39 & 3 & 4 \\
\hline
40-49 & 7 & 11 \\
\hline
50-59 & 6 & 17 \\
\hline
60-69 & 4 & 21 \\
\hline
\end{tabular}
\][/tex]
This detailed table provides a clear arrangement of the given marks into class intervals with their corresponding frequencies and cumulative frequencies.
