17. If the arithmetic mean is 68.25, find the missing value.

[tex]\[
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
Wages (₹) & 50 & 58 & 60 & 65 & 70 & $?$ & 80 & 100 \\
\hline
No. of Workers & 2 & 20 & 5 & 35 & 8 & 10 & 16 & 4 \\
\hline
\end{tabular}
\][/tex]

(Answer: Missing Value = 75)



Answer :

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].