Let X 1 , , X m be a random sample from N( 1 , 2 ), and Y 1 , , 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, S 2 X i 1 m ( X i X m ) 2 and S 2 Y i 1 n ( Y i Y 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 S 2 X S 2 Y follow (iii) Show that X m y n is independent of S 2 X S 2 Y (iiii) set m n 2 1 U 1)1 2 (83 $3 ) 1 2

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Question: Let X 1 , , X m be a random sample from N( 1 , 2 ), and Y 1 , , Y n be

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_mand 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_Yfollow?

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

(iiii) set

Let X1, , Xm be a random sample from N(1, 2 ),

m + n - 2 1/ U : 1)1/2 (83 + $3 ) 1/2

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