Sure, let's create a histogram step by step from the given data:
### Step-by-step Solution:
#### Step 1: Gather the Data
The data collected by Manuel is as follows:
3, 22, 6, 4, 8, 7, 11, 8, 12, 12, 8, 15, 6, 10, 3, 7, 10, 9, 5, 6, 14, 13, 9, 3, 10, 8, 8, 10
#### Step 2: Determine the Range and Bins
We need to create 8 bins for this histogram. First, we find the minimum and maximum values in the data.
- Minimum Value: 3
- Maximum Value: 22
The range of the data is:
[tex]\[ \text{Range} = \text{Maximum Value} - \text{Minimum Value} = 22 - 3 = 19 \][/tex]
Since we need 8 bins, we divide the range by the number of bins to determine the bin width:
[tex]\[ \text{Bin Width} = \frac{\text{Range}}{\text{Number of Bins}} = \frac{19}{8} \approx 2.375 \][/tex]
To make it easier, we can round this to around 2 or a little more for uniformity. Let’s assume a bin width of 2.5 for simplicity. We define the bins as follows:
- Bin 1: [tex]\[ 3 - 5.5 \][/tex]
- Bin 2: [tex]\[ 5.5 - 8 \][/tex]
- Bin 3: [tex]\[ 8 - 10.5 \][/tex]
- Bin 4: [tex]\[ 10.5 - 13 \][/tex]
- Bin 5: [tex]\[ 13 - 15.5 \][/tex]
- Bin 6: [tex]\[ 15.5 - 18 \][/tex]
- Bin 7: [tex]\[ 18 - 20.5 \][/tex]
- Bin 8: [tex]\[ 20.5 - 23 \][/tex]
#### Step 3: Tally the Data into Bins
Sort the data into these bins:
- Bin 1: [tex]\( 3, 3, 3, 4, 5 \)[/tex] → 5 values
- Bin 2: [tex]\( 6, 6, 6, 7, 7, 8, 8, 8 \)[/tex] → 8 values
- Bin 3: [tex]\( 8, 8, 9, 9, 10, 10, 10, 10 \)[/tex] → 8 values
- Bin 4: [tex]\( 11, 12, 12 \)[/tex] → 3 values
- Bin 5: [tex]\( 13, 14, 15 \)[/tex] → 3 values
- Bin 6: [tex]\( 15 \)[/tex] → 1 value
- Bin 7: -- 0 values
- Bin 8: [tex]\( 22 \)[/tex] → 1 value
#### Step 4: Plot the Histogram
| Bin Range | Count |
|:------------- |------:|
| 3 - 5.5 | 5 |
| 5.5 - 8 | 8 |
| 8 - 10.5 | 8 |
| 10.5 - 13 | 3 |
| 13 - 15.5 | 3 |
| 15.5 - 18 | 1 |
| 18 - 20.5 | 0 |
| 20.5 - 23 | 1 |
To visualize this as a histogram, we draw bars where the height of each bar represents the frequency of data in each bin:
```
Frequency
10 |
9 |
8 | #######
7 | #######
6 | #######
5 | ###
4 | ###
3 | ### ### ###
2 | ### ### ###
1 | ### ### ### #
-----------------------------------
Bin Ranges (3-5.5, 5.5-8, etc...)
```
Feel free to use any graphing tool, including spreadsheets or graphing calculators, to accurately plot these bins and their frequencies to create a visual representation of the histogram.
