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Zorluk: ZorWeighted Averages in Applied Contexts

A manufacturing plant produces electronic components across three assembly lines: Line 1, Line 2, and Line 3. Line 1 accounts for 40%40\% of the plant's total production and has a defect rate of 2.0%2.0\%. Line 2 accounts for 35%35\% of total production and has a defect rate of 4.0%4.0\%. Line 3 produces the remaining components. If the overall defect rate for the entire manufacturing plant is 3.1%3.1\%, what is the defect rate of Line 3?

  1. A
    3.0%3.0\%
  2. B
    3.2%3.2\%
  3. C
    3.3%3.3\%
  4. 3.6%3.6\%Cevap
  5. E
    4.1%4.1\%

Cevap

3.6%3.6\%
The correct answer is 3.6%3.6\%. Since Line 1 accounts for 40%40\% of production at a 2.0%2.0\% defect rate (0.8%0.8\% overall contribution) and Line 2 accounts for 35%35\% at a 4.0%4.0\% defect rate (1.4%1.4\% overall contribution), their combined contribution is 2.2%2.2\%. Line 3 represents the remaining 25%25\% (0.250.25) of production. Setting up 2.2%+0.25×d=3.1%2.2\% + 0.25 \times d = 3.1\% gives 0.25d=0.9%0.25 d = 0.9\%, yielding d=3.6%d = 3.6\%.

Adım Adım Çözüm

1
Determine the proportion of total production for Line 3
The proportion for Line 3 is 100%−(40%+35%)=25%100\% - (40\% + 35\%) = 25\% (or 0.250.25).
The sum of the production proportions of all three lines must equal 100%100\% of total production.
2
Set up the weighted average equation for the overall defect rate
0.40(2.0%)+0.35(4.0%)+0.25(d%)=3.1%0.40(2.0\%) + 0.35(4.0\%) + 0.25(d\%) = 3.1\%, where d%d\% represents the defect rate of Line 3.
The overall weighted average defect rate is the sum of each component's defect rate multiplied by its relative production weight.
3
Calculate the defect contributions from Line 1 and Line 2
Contribution of Line 1 = 0.40×2.0%=0.8%0.40 \times 2.0\% = 0.8\%; Contribution of Line 2 = 0.35×4.0%=1.4%0.35 \times 4.0\% = 1.4\%. Combined contribution = 0.8%+1.4%=2.2%0.8\% + 1.4\% = 2.2\%.
Multiplying individual weights by their defect rates yields their weighted contribution to the overall plant defect rate.
4
Solve for the defect rate of Line 3 (dd)
2.2%+0.25d%=3.1%  ⟹  0.25d%=0.9%  ⟹  d=0.90.25=3.62.2\% + 0.25d\% = 3.1\% \implies 0.25d\% = 0.9\% \implies d = \frac{0.9}{0.25} = 3.6. Thus, the defect rate of Line 3 is 3.6%3.6\%.
Isolating the variable dd gives the exact required defect rate for Line 3.

Anahtar Kavram

Weighted Average in Combined Sets
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Weighted Averages in Applied Contexts

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