Suppose that X₁,..., Xₙ are i.i.d. N(μ, σ2). To test the null hypothesis H₀: μ = μ₀, the t test is often used:
t = X−μ₀/sX
Under H₀, t follows a t² distribution with n − 1 df.
Show that the likelihood ratio test of this H₀ is equivalent to the t² test
(a) You might want to use results you obtain from Problem 2 (b)
(b) The following equation is also helpful. Prove it before you use it (it is not very difficult).
ᵢ₌₁Σⁿ (Xi - μ₀)² = ᵢ₌₁Σⁿ (X1 - X)² + ᵢ₌₁Σⁿ (X - μ₀)²