Certainly! Let's analyze the given data set in a detailed step-by-step manner to find the mode.
### Step-by-Step Solution
1. Understand the Table:
The data is presented in a stem-and-leaf format. Each row consists of a "stem" to the left of the vertical line and "leaves" to the right.
- Stem 2: Leaves 8
- Stem 3: Leaves 1, 1, 6
- Stem 4: Leaves 0, 2, 4
- Stem 5: Leaves 3, 5
2. Construct the Complete List of Data Points:
From the stem-and-leaf plot, we construct the complete list of values. Each leaf is concatenated with its corresponding stem to produce the final numbers.
- Stem 2 with leaf 8 gives us 28.
- Stem 3 with leaves 1, 1, 6 gives us 31, 31, 36.
- Stem 4 with leaves 0, 2, 4 gives us 40, 42, 44.
- Stem 5 with leaves 3, 5 gives us 53, 55.
3. Compile the Data Set:
The data set compiled from the above steps is:
[tex]\[
\{28, 31, 31, 36, 40, 42, 44, 53, 55\}
\][/tex]
4. Identify the Frequency of Each Number:
We need to count the occurrences of each number in the data set:
- 28 occurs 1 time.
- 31 occurs 2 times.
- 36 occurs 1 time.
- 40 occurs 1 time.
- 42 occurs 1 time.
- 44 occurs 1 time.
- 53 occurs 1 time.
- 55 occurs 1 time.
5. Determine the Mode:
The mode of a data set is the number that appears most frequently. From our frequency count, we observe:
- The number 31 appears 2 times, which is more frequent than any other number in the data set.
6. Conclusion:
Therefore, the mode of the given data set is:
[tex]\[
31
\][/tex]
This detailed analysis leads us to conclude that the mode of the data set derived from the given stem-and-leaf plot is indeed [tex]\(31\)[/tex].
