To solve this problem, let's break it down step by step by analyzing the histogram with the given data and verifying each of the statements:
### Step-by-Step Solution:
1. List the Data:
The given dataset is:
```
runs = [1, 7, 1, 6, 1, 2, 8, 6, 8]
```
2. Create the Histogram:
We want to create a histogram to understand the distribution of the data. This histogram has bins ranging from 1 to 9.
The frequency counts in each bin `[1, 2)`, `[2, 3)`, `[3, 4)`, `[4, 5)`, `[5, 6)`, `[6, 7)`, `[7, 8)`, `[8, 9)` would be:
```
Bins Frequency
[1, 2) 3
[2, 3) 1
[3, 4) 0
[4, 5) 0
[5, 6) 0
[6, 7) 2
[7, 8) 1
[8, 9) 2
```
3. Clusters:
- Clusters can be defined as adjacent bins with non-zero values.
- We identify two clusters here:
- Cluster 1: Bins `[1, 2)` and `[2, 3)`
- Cluster 2: Bins `[6, 7)`, `[7, 8)`, and `[8, 9)`
Therefore, there are 2 clusters.
4. Peaks:
- Peaks are bins that have a higher frequency than their immediate neighbors.
- In our histogram:
- Bin `[1, 2)` is a peak since it holds the highest value in its vicinity with frequency 3.
- Bin `[6, 7)` cannot be a peak since the frequency 2 is not greater than that of next bin `[8, 9)`.
- Bin `[7, 8)` is a peak since the frequency decreases on both sides (from `[6, 7)` and `[8, 9)`).
- Bin `[8, 9)` is a peak as it has the highest value compared to surrounding bin `[7, 8)`.
Therefore, there are 3 peaks.
5. Specific Frequency:
- The frequency from `[4, 5)` (which corresponds to the bin `[4, 5)`) is 0.
Now, let's check the statements one-by-one:
- The histogram has 1 cluster: This is false.
- The histogram has 2 clusters: This is true.
- The histogram has 1 peak: This is false.
- The histogram has 4 peaks: This is false.
- The histogram has 0 frequency from 4-5: This is true.
### Conclusion:
- The statement "The histogram has 2 clusters" is correct.
- The statement "The histogram has 0 frequency from 4-5" is correct.
