Let X 1 ,X 2 , , X n be a random sample from the distribution N( 1 , 2 ) Show that the likelihood ratio principle for testing H 0 2 ' 2 specified, and 1 unspecified against H 1 2 ' 2 , 1 unspecified, leads to a test that rejects when n 1 (xi x) 2 c 1 or n 1 (x i x) 2 c 2 , where c 1 c 2 are selected appropriately

Question: Let X 1 ,X 2 , . . . , X n be a random sample from the distribution N( 1 , 2 ).

Let X₁,X₂, . . . , X_n be a random sample from the distribution N(θ₁, θ₂). Show that the likelihood ratio principle for testing H₀ : θ₂ = θ'₂ specified, and θ₁ unspecified against H₁ : θ₂ ≠ θ'₂, θ₁ unspecified, leads to a test that rejects when Σⁿ₁ (xi − ‾x)² ≤ c₁ or Σⁿ₁ (x_i − x)² ≥ c₂, where c₁ < c₂ are selected appropriately.

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