Sure, let's work through each part of the question step-by-step.
## Part a: Constructing the frequency distribution
First, let's determine the class intervals. The task is to create five class intervals starting with a lower boundary of 5. We can use intervals of size 5:
1. 5 - 10
2. 10 - 15
3. 15 - 20
4. 20 - 25
5. 25 - 30
Now let's count the frequency of values falling into each interval:
### Step 1: Counting Frequencies
```
Class Interval Frequency
5 - 10 4
10 - 15 6
15 - 20 8
20 - 25 7
25 - 30 2
```
### Step 2: Calculating the Relative Frequency (%)
Relative frequency is calculated by dividing the frequency of each class by the total number of data points (27) and then multiplying by 100.
```
Class Interval Frequency Relative Frequency (%)
5 - 10 4 14.81
10 - 15 6 22.22
15 - 20 8 29.63
20 - 25 7 25.93
25 - 30 2 7.41
```
### Step 3: Calculating the Cumulative Frequency
Cumulative frequency is the sum of the frequencies of that class interval and all previous intervals:
```
Class Interval Frequency Relative Frequency (%) Cumulative Frequency
5 - 10 4 14.81 4
10 - 15 6 22.22 10
15 - 20 8 29.63 18
20 - 25 7 25.93 25
25 - 30 2 7.41 27
```
## Part b: Drawing the bar chart
To draw the bar chart for the frequency distribution:
1. Plot the class intervals on the x-axis.
2. Plot the frequency on the y-axis.
3. Draw bars for each class interval with heights corresponding to their frequencies.
Comment: The bar chart shows the distribution of returned items over the different ranges. It indicates that the most common range for the number of items returned is between 15 to 20, which might be an area of focus for the company.
## Part c: Checking if returned items are more than 19 per day in more than 35% of days
### Step 1: Count the days when returned items > 19
```
Items > 19: 22, 21, 25, 29, 24, 21, 20, 25 (8 days)
Total days: 27
```
### Step 2: Calculate the percentage of days with returned items > 19
[tex]\[
\text{Percentage} = \left(\frac{\text{Number of days with items > 19}}{\text{Total number of days}}\right) \times 100
\][/tex]
[tex]\[
\text{Percentage} = \left( \frac{8}{27} \right) \times 100 \approx 29.63\%
\][/tex]
Since 29.63% is less than 35%, the company's managers will not consider changing suppliers based on this sample.
In summary:
- Frequency distribution constructed as requested.
- Bar chart plotted and commentary provided.
- Calculated percentage of days with returned items > 19 and determined that the managers will not change the suppliers.
