Get expert insights and community support for your questions on IDNLearn.com. Ask any question and receive comprehensive, well-informed responses from our dedicated team of experts.
Sagot :
To calculate the arithmetic average (mean) of the given data, we should follow these steps:
1. Determine the Midpoints of Each Interval:
- For each interval of weekly wages, calculate the midpoint (average of the lower and upper limits of the interval).
2. Calculate the Total Sum of Weekly Wages:
- Multiply each midpoint by the number of workers in that interval to find the total wage contribution of each interval.
- Sum these contributions to get the total sum of weekly wages for all workers.
3. Find the Total Number of Workers:
- Sum the number of workers across all intervals.
4. Compute the Arithmetic Average:
- Divide the total sum of weekly wages by the total number of workers.
### Step-by-Step Solution
1. Calculate the Midpoints:
[tex]\[ \text{Midpoints} = \left\{ \frac{(100 + 105)}{2}, \frac{(105 + 110)}{2}, \frac{(110 + 115)}{2}, \frac{(115 + 120)}{2}, \frac{(120 + 125)}{2}, \frac{(125 + 130)}{2}, \frac{(130 + 135)}{2}, \frac{(135 + 140)}{2}, \frac{(140 + 145)}{2}, \frac{(145 + 150)}{2}, \frac{(150 + 155)}{2}, \frac{(155 + 160)}{2} \right\} \][/tex]
Calculating these midpoints:
[tex]\[ \text{Midpoints} = \left\{ 102.5, 107.5, 112.5, 117.5, 122.5, 127.5, 132.5, 137.5, 142.5, 147.5, 152.5, 157.5 \right\} \][/tex]
2. Calculate the Total Sum of Weekly Wages:
Multiply each midpoint by the corresponding number of workers and sum these products:
[tex]\[ \begin{align*} & 102.5 \times 200 \\ & + 107.5 \times 210 \\ & + 112.5 \times 230 \\ & + 117.5 \times 320 \\ & + 122.5 \times 350 \\ & + 127.5 \times 320 \\ & + 132.5 \times 410 \\ & + 137.5 \times 320 \\ & + 142.5 \times 280 \\ & + 147.5 \times 210 \\ & + 152.5 \times 160 \\ & + 157.5 \times 90 \\ \end{align*} \][/tex]
The resulting total sum of weekly wages is:
[tex]\[ \text{Total Sum of Weekly Wages} = 398000.0 \][/tex]
3. Find the Total Number of Workers:
Sum the number of workers across all intervals:
[tex]\[ \text{Total Number of Workers} = 200 + 210 + 230 + 320 + 350 + 320 + 410 + 320 + 280 + 210 + 160 + 90 \][/tex]
This sums to:
[tex]\[ \text{Total Number of Workers} = 3100 \][/tex]
4. Compute the Arithmetic Average:
Divide the total sum of weekly wages by the total number of workers:
[tex]\[ \text{Arithmetic Average} = \frac{\text{Total Sum of Weekly Wages}}{\text{Total Number of Workers}} = \frac{398000.0}{3100} \][/tex]
Simplifying this gives:
[tex]\[ \text{Arithmetic Average} \approx 128.38709677419354 \text{ Rs} \][/tex]
Therefore, the arithmetic average of the weekly wages of the workers is approximately Rs. 128.39.
1. Determine the Midpoints of Each Interval:
- For each interval of weekly wages, calculate the midpoint (average of the lower and upper limits of the interval).
2. Calculate the Total Sum of Weekly Wages:
- Multiply each midpoint by the number of workers in that interval to find the total wage contribution of each interval.
- Sum these contributions to get the total sum of weekly wages for all workers.
3. Find the Total Number of Workers:
- Sum the number of workers across all intervals.
4. Compute the Arithmetic Average:
- Divide the total sum of weekly wages by the total number of workers.
### Step-by-Step Solution
1. Calculate the Midpoints:
[tex]\[ \text{Midpoints} = \left\{ \frac{(100 + 105)}{2}, \frac{(105 + 110)}{2}, \frac{(110 + 115)}{2}, \frac{(115 + 120)}{2}, \frac{(120 + 125)}{2}, \frac{(125 + 130)}{2}, \frac{(130 + 135)}{2}, \frac{(135 + 140)}{2}, \frac{(140 + 145)}{2}, \frac{(145 + 150)}{2}, \frac{(150 + 155)}{2}, \frac{(155 + 160)}{2} \right\} \][/tex]
Calculating these midpoints:
[tex]\[ \text{Midpoints} = \left\{ 102.5, 107.5, 112.5, 117.5, 122.5, 127.5, 132.5, 137.5, 142.5, 147.5, 152.5, 157.5 \right\} \][/tex]
2. Calculate the Total Sum of Weekly Wages:
Multiply each midpoint by the corresponding number of workers and sum these products:
[tex]\[ \begin{align*} & 102.5 \times 200 \\ & + 107.5 \times 210 \\ & + 112.5 \times 230 \\ & + 117.5 \times 320 \\ & + 122.5 \times 350 \\ & + 127.5 \times 320 \\ & + 132.5 \times 410 \\ & + 137.5 \times 320 \\ & + 142.5 \times 280 \\ & + 147.5 \times 210 \\ & + 152.5 \times 160 \\ & + 157.5 \times 90 \\ \end{align*} \][/tex]
The resulting total sum of weekly wages is:
[tex]\[ \text{Total Sum of Weekly Wages} = 398000.0 \][/tex]
3. Find the Total Number of Workers:
Sum the number of workers across all intervals:
[tex]\[ \text{Total Number of Workers} = 200 + 210 + 230 + 320 + 350 + 320 + 410 + 320 + 280 + 210 + 160 + 90 \][/tex]
This sums to:
[tex]\[ \text{Total Number of Workers} = 3100 \][/tex]
4. Compute the Arithmetic Average:
Divide the total sum of weekly wages by the total number of workers:
[tex]\[ \text{Arithmetic Average} = \frac{\text{Total Sum of Weekly Wages}}{\text{Total Number of Workers}} = \frac{398000.0}{3100} \][/tex]
Simplifying this gives:
[tex]\[ \text{Arithmetic Average} \approx 128.38709677419354 \text{ Rs} \][/tex]
Therefore, the arithmetic average of the weekly wages of the workers is approximately Rs. 128.39.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.
