Let X₁, , X_m be a random sample from N(₁, ² ), and Y₁, , Y_n be a random sample from N(₂, ² ), where , ₁ and ₂ are unknown. Assume the two random samples are independent. Let X_m and y_n be the sample means of X_i 's and Y_j 's, respectively.

Also, S²_X = i=1m(XiX^m)2 and S²_Y = i=1n(YiY^n)2 .

Consider test H0 : ₁ = ₂ vs _{1 2}.

(i) What distribution does X_m - y_n follow under H0?

(ii) What distribution does S²_X + S²_Y follow?

(iii) Show that X_m - y_n is independent of S²_X + S²_Y

(iiii) set

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