Let Y 1 , Y 2 , , Y n be n independent normal variables with common unknown variance 2 Let Y i have mean x i , i 1, 2, , n, where x 1 , x 2 , , x n are known but not all the same and is an unknown constant Find the likelihood ratio test for H 0 0 against all alternatives Show that this likelihood ratio test can be based on a statistic that has a well known distribution

Question: Let Y 1 , Y 2 , . . . , Y n be n independent normal variables with common unknown variance 2 .

Let Y₁, Y₂, . . . , Y_n be n independent normal variables with common unknown variance σ². Let Y_i have mean βx_i, i = 1, 2, . . ., n, where x₁, x₂, . . . , x_n are known but not all the same and β is an unknown constant. Find the likelihood ratio test for H₀ : β = 0 against all alternatives. Show that this likelihood ratio test can be based on a statistic that has a well-known distribution.

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