To find the missing wage value, we need to ensure the arithmetic mean of the workers' wages matches the given value of 68.25. Let's lay out the steps in detail:
1. Known Values:
- Arithmetic Mean ([tex]\(\bar{x}\)[/tex]): 68.25
- Wages ([tex]\(w_i\)[/tex]): 50, 58, 60, 65, 70, [tex]\(?\)[/tex], 80, 100
- Number of Workers ([tex]\(n_i\)[/tex]): 2, 20, 5, 35, 8, 10, 16, 4
2. Sum of Workers:
Calculate the total number of workers:
[tex]\[
\text{Total Workers} = 2 + 20 + 5 + 35 + 8 + 10 + 16 + 4 = 100
\][/tex]
3. Calculate the Total Wages:
Let the missing wage value be [tex]\(x\)[/tex].
The total wages can be calculated using the formula:
[tex]\[
\text{Total Wages} = \sum (w_i \times n_i)
\][/tex]
Substitute known values:
[tex]\[
\text{Total Wages} = (50 \times 2) + (58 \times 20) + (60 \times 5) + (65 \times 35) + (70 \times 8) + (x \times 10) + (80 \times 16) + (100 \times 4)
\][/tex]
Calculate the product sums:
[tex]\[
\text{Total Wages} = 100 + 1160 + 300 + 2275 + 560 + 10x + 1280 + 400
\][/tex]
Simplify:
[tex]\[
\text{Total Wages} = 6075 + 10x
\][/tex]
4. Relationship with Arithmetic Mean:
The formula for the arithmetic mean is:
[tex]\[
\bar{x} = \frac{\text{Total Wages}}{\text{Total Workers}}
\][/tex]
Given the mean ([tex]\(\bar{x}\)[/tex]) is 68.25 and the total number of workers is 100, substitute these values:
[tex]\[
68.25 = \frac{6075 + 10x}{100}
\][/tex]
5. Solving for [tex]\(x\)[/tex]:
Multiply both sides by 100:
[tex]\[
6825 = 6075 + 10x
\][/tex]
Subtract 6075 from both sides:
[tex]\[
750 = 10x
\][/tex]
Divide both sides by 10:
[tex]\[
x = 75
\][/tex]
Therefore, the missing wage value is [tex]\( \boxed{75} \)[/tex].
