This problem shows you how to make a better blend of almost anything.

Let x₁, x₂, … x_n be independent random variables with respective variances σ²₁, σ²₂ , … , σ²_n.

Let c₁, c₂, ………., c_n be constant weights such that 0 ≤ c_i ≤ 1 and c₁ + c₂ + … + c_n = 1. The linear combination w = c₁x₁ + c₂ x₂ + … + c_nx_n is a random variable with variance

 (a) Two types of epoxy resin are used to make a new blend of superglue. Both resins have about the same mean breaking strength and act independently. The question is how to blend the resins (with the hardener) to get the most consistent breaking strength. Why is this important, and why would this require minimal σ²_W?

We don’t want some bonds to be really strong while others are very weak, resulting in inconsistent bonding.

Let x₁ and x₂ be random variables representing breaking strength (lb) of each resin under uniform testing conditions. If σ₁ = 8 lb σ₂ = 12 lb, show why a blend of about 69% resin 1 and 31% resin 2 will result in a superglue with smallest σ_W² and most consistent bond strength.

(b) Use c₁ = 0.69 and c₂ = 0.31 to compute σ_w and show that σ_w is less than both s₁ and s₂. The dictionary meaning of the word synergetic is “working together or cooperating for a better overall effect.” Write a brief explanation of how the blend w = c₁x₁ + c₂ x₂ has a synergetic effect for the purpose of reducing variance?

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