Answer :
Let's tackle the question step by step.
### (a) Construct a frequency distribution table
First, we need to create bins/classes for the marks and count how many marks fall into each class. The classes are as follows:
- 30 - 39
- 40 - 49
- 50 - 59
- 60 - 69
- 70 - 79
- 80 - 89
Then, we count the number of students whose marks fall into each class.
Frequency distribution table:
| Class Interval | Frequency |
|----------------|-----------|
| 30 - 39 | 2 |
| 40 - 49 | 8 |
| 50 - 59 | 11 |
| 60 - 69 | 11 |
| 70 - 79 | 5 |
| 80 - 89 | 3 |
### (b) Draw a histogram for the above data
In a histogram, the x-axis represents the classes (30-39, 40-49, etc.), and the y-axis represents the frequency of marks in those classes.
The histogram bars will look something like this:
```
Frequency
^
11| ________
10| | |
9| | |
8| ________ | |
7| | | | |
6| | | | |
5| | | |______|
4| | | | |
3| | |______ | | _____
2| | | || |_____
1| |______| || | | |
|________________________________________________ Class Intervals
30-39 40-49 50-59 60-69 70-79 80-89
```
### (c) Construct a "less than" cumulative frequency distribution
To construct a "less than" cumulative frequency distribution, we add the frequencies cumulatively from the lowest class to the highest class.
"Less than" cumulative frequency distribution table:
| Class Interval | Cumulative Frequency |
|--------------------|----------------------|
| Less than 40 | 2 |
| Less than 50 | 10 |
| Less than 60 | 21 |
| Less than 70 | 32 |
| Less than 80 | 37 |
| Less than 90 | 40 |
### (d) Draw a "less than" cumulative frequency polygon
A cumulative frequency polygon can be drawn by plotting the cumulative frequencies against the class boundaries. Then, we connect the points with straight lines.
The cumulative frequency polygon will look something like this:
```
Frequency
^
40|
35| _______
30| ______|
25| _________|
20| ________|
15| ___________|
10| ____|
5|______|
0|___________________________________________________ Class Intervals
< 40 < 50 < 60 < 70 < 80 < 90
```
Each point on the polygon represents the cumulative frequency at the upper boundary of each class. For example, at "Less than 50", the cumulative frequency is 10, and so on.
### (a) Construct a frequency distribution table
First, we need to create bins/classes for the marks and count how many marks fall into each class. The classes are as follows:
- 30 - 39
- 40 - 49
- 50 - 59
- 60 - 69
- 70 - 79
- 80 - 89
Then, we count the number of students whose marks fall into each class.
Frequency distribution table:
| Class Interval | Frequency |
|----------------|-----------|
| 30 - 39 | 2 |
| 40 - 49 | 8 |
| 50 - 59 | 11 |
| 60 - 69 | 11 |
| 70 - 79 | 5 |
| 80 - 89 | 3 |
### (b) Draw a histogram for the above data
In a histogram, the x-axis represents the classes (30-39, 40-49, etc.), and the y-axis represents the frequency of marks in those classes.
The histogram bars will look something like this:
```
Frequency
^
11| ________
10| | |
9| | |
8| ________ | |
7| | | | |
6| | | | |
5| | | |______|
4| | | | |
3| | |______ | | _____
2| | | || |_____
1| |______| || | | |
|________________________________________________ Class Intervals
30-39 40-49 50-59 60-69 70-79 80-89
```
### (c) Construct a "less than" cumulative frequency distribution
To construct a "less than" cumulative frequency distribution, we add the frequencies cumulatively from the lowest class to the highest class.
"Less than" cumulative frequency distribution table:
| Class Interval | Cumulative Frequency |
|--------------------|----------------------|
| Less than 40 | 2 |
| Less than 50 | 10 |
| Less than 60 | 21 |
| Less than 70 | 32 |
| Less than 80 | 37 |
| Less than 90 | 40 |
### (d) Draw a "less than" cumulative frequency polygon
A cumulative frequency polygon can be drawn by plotting the cumulative frequencies against the class boundaries. Then, we connect the points with straight lines.
The cumulative frequency polygon will look something like this:
```
Frequency
^
40|
35| _______
30| ______|
25| _________|
20| ________|
15| ___________|
10| ____|
5|______|
0|___________________________________________________ Class Intervals
< 40 < 50 < 60 < 70 < 80 < 90
```
Each point on the polygon represents the cumulative frequency at the upper boundary of each class. For example, at "Less than 50", the cumulative frequency is 10, and so on.
