Let's solve the question step by step.
### Step 1: Finding the new mean after subtracting 5 from each number.
Given that the mean of five numbers is 20, we can calculate the total sum of these numbers.
1. **Calculate the total sum**:
\[
\text{Mean} = \frac{\text{Total Sum}}{\text{Number of Observations}}
\]
\[
20 = \frac{\text{Total Sum}}{5} \implies \text{Total Sum} = 20 \times 5 = 100
\]
2. **Subtract 5 from each number**:
If we subtract 5 from each of the 5 numbers, the total amount subtracted is:
\[
5 \times 5 = 25
\]
3. **Calculate the new total sum**:
\[
\text{New Total Sum} = 100 - 25 = 75
\]
4. **Calculate the new mean**:
\[
\text{New Mean} = \frac{\text{New Total Sum}}{\text{Number of Observations}} = \frac{75}{5} = 15
\]
So, the value of \( ulP \) is **15**.
### Step 2: Finding the median of the given data.
The data provided is: 32, 44, 53, 47, 37, 54, 34, 36, 40, 50.
1. **Arrange the data in ascending order**:
\[
32, 34, 36, 37, 40, 44, 47, 50, 53, 54
\]
2. **Count the number of observations**:
There are 10 observations (even number).
3. **Calculate the median**:
For an even number of observations, the median is the average of the two middle numbers. The two middle numbers are the 5th and 6th numbers:
\[
\text{5th number} = 40, \quad \text{6th number} = 44
\]
\[
\text{Median} = \frac{40 + 44}{2} = \frac{84}{2} = 42
\]
So, the value of \( ulQ \) is **42**.
### Step 3: Finding the mode of the given data.
The data provided is: 15, 17, 15, 19, 14, 18, 15, 14, 16, 15, 14, 20, 19, 14, 15.
1. **Count the frequency of each number**:
- 14 appears 4 times
- 15 appears 5 times
- 16 appears 1 time
- 17 appears 1 time
- 18 appears 1 time
- 19 appears 2 times
- 20 appears 1 time
2. **Identify the mode**:
The mode is the number that appears most frequently. Here, 15 appears 5 times, which is more than any other number.
So, the value of \( ulR \) is **15**.
### Final Answers:
- \( ulP = 15 \)
- \( ulQ = 42 \)
- \( ulR = 15 \)
