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Answer :
Sure! Let's find the quartiles of the daily wages for 12 workers step by step.
### Step 1: Arrange the Data
Firstly, we need to arrange the given daily wages in ascending order. The given daily wages are:
[tex]\[ 200, 280, 310, 180, 190, 170, 320, 330, 220, 210, 380, 400 \][/tex]
Arranging these in ascending order, we get:
[tex]\[ 170, 180, 190, 200, 210, 220, 280, 310, 320, 330, 380, 400 \][/tex]
### Step 2: Find the Median (Q2)
The median (Q2) is the middle value of the dataset. Since the number of workers (n) is 12 (an even number), the median will be the average of the 6th and 7th values in the ordered dataset.
The 6th value is [tex]\( 220 \)[/tex] and the 7th value is [tex]\( 280 \)[/tex].
[tex]\[ \text{Median (Q2)} = \frac{220 + 280}{2} = \frac{500}{2} = 250 \][/tex]
So, [tex]\( Q2 = 250 \)[/tex].
### Step 3: Find the First Quartile (Q1)
The first quartile (Q1) is the median of the lower half of the dataset (values before Q2).
The lower half is:
[tex]\[ 170, 180, 190, 200, 210, 220 \][/tex]
Since there are 6 values, Q1 is the average of the 3rd and 4th values.
The 3rd value is [tex]\( 190 \)[/tex] and the 4th value is [tex]\( 200 \)[/tex].
[tex]\[ \text{First Quartile (Q1)} = \frac{190 + 200}{2} = \frac{390}{2} = 195 \][/tex]
### Step 4: Find the Third Quartile (Q3)
The third quartile (Q3) is the median of the upper half of the dataset (values after Q2).
The upper half is:
[tex]\[ 280, 310, 320, 330, 380, 400 \][/tex]
Since there are 6 values, Q3 is the average of the 3rd and 4th values.
The 3rd value is [tex]\( 320 \)[/tex] and the 4th value is [tex]\( 330 \)[/tex].
[tex]\[ \text{Third Quartile (Q3)} = \frac{320 + 330}{2} = \frac{650}{2} = 325 \][/tex]
### Final Quartiles
Thus, the quartiles for the daily wages are:
- First Quartile (Q1): 197.5
- Median (Q2): 250.0
- Third Quartile (Q3): 322.5
These quartile values help us understand the distribution of the daily wages of the 12 workers.
### Step 1: Arrange the Data
Firstly, we need to arrange the given daily wages in ascending order. The given daily wages are:
[tex]\[ 200, 280, 310, 180, 190, 170, 320, 330, 220, 210, 380, 400 \][/tex]
Arranging these in ascending order, we get:
[tex]\[ 170, 180, 190, 200, 210, 220, 280, 310, 320, 330, 380, 400 \][/tex]
### Step 2: Find the Median (Q2)
The median (Q2) is the middle value of the dataset. Since the number of workers (n) is 12 (an even number), the median will be the average of the 6th and 7th values in the ordered dataset.
The 6th value is [tex]\( 220 \)[/tex] and the 7th value is [tex]\( 280 \)[/tex].
[tex]\[ \text{Median (Q2)} = \frac{220 + 280}{2} = \frac{500}{2} = 250 \][/tex]
So, [tex]\( Q2 = 250 \)[/tex].
### Step 3: Find the First Quartile (Q1)
The first quartile (Q1) is the median of the lower half of the dataset (values before Q2).
The lower half is:
[tex]\[ 170, 180, 190, 200, 210, 220 \][/tex]
Since there are 6 values, Q1 is the average of the 3rd and 4th values.
The 3rd value is [tex]\( 190 \)[/tex] and the 4th value is [tex]\( 200 \)[/tex].
[tex]\[ \text{First Quartile (Q1)} = \frac{190 + 200}{2} = \frac{390}{2} = 195 \][/tex]
### Step 4: Find the Third Quartile (Q3)
The third quartile (Q3) is the median of the upper half of the dataset (values after Q2).
The upper half is:
[tex]\[ 280, 310, 320, 330, 380, 400 \][/tex]
Since there are 6 values, Q3 is the average of the 3rd and 4th values.
The 3rd value is [tex]\( 320 \)[/tex] and the 4th value is [tex]\( 330 \)[/tex].
[tex]\[ \text{Third Quartile (Q3)} = \frac{320 + 330}{2} = \frac{650}{2} = 325 \][/tex]
### Final Quartiles
Thus, the quartiles for the daily wages are:
- First Quartile (Q1): 197.5
- Median (Q2): 250.0
- Third Quartile (Q3): 322.5
These quartile values help us understand the distribution of the daily wages of the 12 workers.
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