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Sagot :
Let's solve each part of the question step-by-step.
### (i) What is the mid-value of the class (50 - 60)?
The mid-value of a class interval is calculated as the average of the lower and upper class boundaries.
For the class interval \( 50 - 60 \):
[tex]\[ \text{Mid-value} = \frac{50 + 60}{2} = \frac{110}{2} = 55.0 \][/tex]
### (ii) Compute the average wages per hour.
To compute the average wages per hour, we need to use the following formula for the weighted average:
[tex]\[ \text{Average wage} = \frac{\sum (\text{mid-value of each class} \times \text{number of workers in that class})}{\sum (\text{number of workers})} \][/tex]
Let's calculate this step-by-step.
1. Calculate Mid-values for each class:
- \(50 - 60: \text{Mid-value} = 55.0\)
- \(60 - 70: \text{Mid-value} = \frac{60 + 70}{2} = 65.0\)
- \(70 - 80: \text{Mid-value} = \frac{70 + 80}{2} = 75.0\)
- \(80 - 90: \text{Mid-value} = \frac{80 + 90}{2} = 85.0\)
- \(90 - 100: \text{Mid-value} = \frac{90 + 100}{2} = 95.0\)
2. Total wages and total workers:
- For the class \(50-60\): \(55.0 \times 6 = 330\)
- For the class \(60-70\): \(65.0 \times 8 = 520\)
- For the class \(70-80\): \(75.0 \times 12 = 900\)
- For the class \(80-90\): \(85.0 \times 8 = 680\)
- For the class \(90-100\): \(95.0 \times 6 = 570\)
- Total Wages: \(330 + 520 + 900 + 680 + 570 = 3000\)
- Total Workers: \(6 + 8 + 12 + 8 + 6 = 40\)
3. Calculate the Average wage:
[tex]\[ \text{Average wage} = \frac{3000}{40} = 75.0 \][/tex]
Thus, the average wage per hour is \(75.0\) Rs.
### (iii) Justify Purnima's statement that for any number of workers having wages Rs (70 - 80) per hour, the average wage of all workers remains unchanged.
Purnima's statement can be justified through the concept of a weighted average. The overall average wage is calculated based on the total wages and the total number of workers.
When we are dealing with classes of wages and their respective number of workers, any increase or decrease in the number of workers within a single class (here \(70-80\) with mid-value \(75.0\)) doesn't alter the weighted average if the changes are proportionate within that class. This is because the mid-value for the \(70-80\) class is equal to the computed average wage (75.0 Rs).
In simpler terms, adding or removing workers within this wage class keeps the ratio of total wages to total workers the same, provided the mid-value of the wages in the adjusted section is equivalent to the overall average wage. This results in no change to the overall average wage as calculated.
Therefore, Purnima's assertion is correct.
### (i) What is the mid-value of the class (50 - 60)?
The mid-value of a class interval is calculated as the average of the lower and upper class boundaries.
For the class interval \( 50 - 60 \):
[tex]\[ \text{Mid-value} = \frac{50 + 60}{2} = \frac{110}{2} = 55.0 \][/tex]
### (ii) Compute the average wages per hour.
To compute the average wages per hour, we need to use the following formula for the weighted average:
[tex]\[ \text{Average wage} = \frac{\sum (\text{mid-value of each class} \times \text{number of workers in that class})}{\sum (\text{number of workers})} \][/tex]
Let's calculate this step-by-step.
1. Calculate Mid-values for each class:
- \(50 - 60: \text{Mid-value} = 55.0\)
- \(60 - 70: \text{Mid-value} = \frac{60 + 70}{2} = 65.0\)
- \(70 - 80: \text{Mid-value} = \frac{70 + 80}{2} = 75.0\)
- \(80 - 90: \text{Mid-value} = \frac{80 + 90}{2} = 85.0\)
- \(90 - 100: \text{Mid-value} = \frac{90 + 100}{2} = 95.0\)
2. Total wages and total workers:
- For the class \(50-60\): \(55.0 \times 6 = 330\)
- For the class \(60-70\): \(65.0 \times 8 = 520\)
- For the class \(70-80\): \(75.0 \times 12 = 900\)
- For the class \(80-90\): \(85.0 \times 8 = 680\)
- For the class \(90-100\): \(95.0 \times 6 = 570\)
- Total Wages: \(330 + 520 + 900 + 680 + 570 = 3000\)
- Total Workers: \(6 + 8 + 12 + 8 + 6 = 40\)
3. Calculate the Average wage:
[tex]\[ \text{Average wage} = \frac{3000}{40} = 75.0 \][/tex]
Thus, the average wage per hour is \(75.0\) Rs.
### (iii) Justify Purnima's statement that for any number of workers having wages Rs (70 - 80) per hour, the average wage of all workers remains unchanged.
Purnima's statement can be justified through the concept of a weighted average. The overall average wage is calculated based on the total wages and the total number of workers.
When we are dealing with classes of wages and their respective number of workers, any increase or decrease in the number of workers within a single class (here \(70-80\) with mid-value \(75.0\)) doesn't alter the weighted average if the changes are proportionate within that class. This is because the mid-value for the \(70-80\) class is equal to the computed average wage (75.0 Rs).
In simpler terms, adding or removing workers within this wage class keeps the ratio of total wages to total workers the same, provided the mid-value of the wages in the adjusted section is equivalent to the overall average wage. This results in no change to the overall average wage as calculated.
Therefore, Purnima's assertion is correct.
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