Let's solve each part of this problem step by step.
### a. Drawing a Bar Chart
To draw a bar chart, we will use the number of faulty pens as the categories (x-axis) and the number of boxes corresponding to each category as the values (y-axis).
Here is the data:
- Number of faulty pens: 0, 1, 2, 3, 4, 5, 6
- Number of boxes: 7, 10, 12, 20, 19, 18, 14
The bar chart will look something like this:
```
Number of Faulty Pens | Number of Boxes
-----------------------------------------
0 | 7
1 | 10
2 | 12
3 | 20
4 | 19
5 | 18
6 | 14
```
You can draw the bar chart by plotting this information manually or using statistical software.
### b. Stating the Modal Number of Faulty Pens
The mode is the value that appears most frequently in a data set.
From the data:
- Number of faulty pens: 0, 1, 2, 3, 4, 5, 6
- Number of boxes: 7, 10, 12, 20, 19, 18, 14
The number of boxes is highest when there are 3 faulty pens. Therefore, the modal number of faulty pens is 3.
### c. Finding the Median of the Distribution
The median is the middle value of the data set when it is arranged in ascending order. Since frequencies suggest that the number of faulty pens is already distributed, we need to find the cumulative frequency to pinpoint the median position.
1. List the data points as follows:
```
0 occurs 7 times
1 occurs 10 times
2 occurs 12 times
3 occurs 20 times
4 occurs 19 times
5 occurs 18 times
6 occurs 14 times
```
2. Cumulative frequencies:
```
0: 7
1: 7 + 10 = 17
2: 17 + 12 = 29
3: 29 + 20 = 49
4: 49 + 19 = 68
5: 68 + 18 = 86
6: 86 + 14 = 100
```
3. To find the median, locate the 50th percentile value.
The median is the value at:
```
(n + 1) / 2 = (100 + 1) / 2 = 50.5
```
So, it will be the value between the 50th and 51st observations.
From our cumulative frequencies, the 50th and 51st observations lie within the category where 3 faulty pens are found (cumulative frequency is 49 up to the value 2 and 49 + 20=69 up to the value 3).
Therefore, the median number of faulty pens is 3.
### Summary:
- Bar Chart: You can plot the bar chart using the provided data.
- Modal number: 3
- Median number: 3
