Given the data set [tex]\(\{9, 1, 8, 7, 9, 9, 8, 9, 8, 7\}\)[/tex], we need to create a histogram and determine its properties regarding symmetry. Here is a step-by-step solution:
### Step 1: Calculate the Frequency Distribution
First, we count the occurrences of each unique number in the data set:
- The number 9 appears 4 times.
- The number 1 appears 1 time.
- The number 8 appears 3 times.
- The number 7 appears 2 times.
Thus, the frequency distribution is:
[tex]\[
\{9: 4, 1: 1, 8: 3, 7: 2\}
\][/tex]
### Step 2: Create the Histogram
The histogram can be visualized as follows:
- 9: #### (4 occurrences)
- 1: # (1 occurrence)
- 8: ### (3 occurrences)
- 7: ## (2 occurrences)
### Step 3: Determine the Symmetry of the Histogram
To check if the histogram is symmetrical:
1. Sort the unique values: [tex]\(\{1, 7, 8, 9\}\)[/tex].
2. Compare the frequency of elements from the start to the middle of the list with the frequency from the end to the middle of the list.
Let's compare the frequencies:
- Frequency of 1 (1 occurrence) compared with frequency of 9 (4 occurrences): Not equal
- Frequency of 7 (2 occurrences) compared with frequency of 8 (3 occurrences): Not equal
Since the frequencies are not mirrored, the histogram is not symmetrical.
### Conclusion
Given the analysis above, the correct answer is:
- The histogram is not symmetrical.
This confirms that the data are not evenly distributed and the left side of the graph is not the mirror image of the right side.
