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.