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Homework 8 solutions
Exercise 2
X � Y ⇠ N(µ1 � µ2,
q�219 +
�22
12 ). Therefore we can write:
Z =X � Y � (µ1 � µ2)q
�219 +
�22
12
.
And since �21 = 3�2
2 we get:
Z =X � Y � (µ1 � µ2)p
�22(
39 + 1
12 ).
Now we need to define a �2 random variable. Because X and Y are independent we have:
(9 � 1)S21
�21
+(12 � 1)S2
2
�22
⇠ �212+9�2 ⇠ �2
19.
Using again �21 = 3�2
2 we get:
13 8S
21 + 11S2
2
�22
⇠ �219.
Now we can define a variable that has a t distribution as follows:
t =
X�Y �(µ1�µ2)p�22( 39+ 1
12)
r138S2
1+11S2
2�22
19
⇠ t19
Finally we get:
t =X � Y � (µ1 � µ2)p
13 8S
21 + 11S2
2
p19p
39 + 1
12
=X � Y � (µ1 � µ2)p
13 8S
21 + 11S2
2
q228
5.
We can use the above t19 random variable to construct a 95% confidence interval for µ1 � µ2. We want:
P (�t↵2
;19 X � Y � (µ1 � µ2)p
13 8S
21 + 11S2
2
q228
5 t↵
2;19) = 1 � ↵.
After some manipulation we find that µ1 � µ2 will fall in the following interval with 95% confidence:
X � Y ± t↵2
;19
q8
3S21 + 11S2
2
q5
228.
Exercise 6Exercise 2
Exercise 3
Exercise 4
Exercise 1
Exercise 5