1. Lecture 3 Section 1.4.1 – 1.4.2 Tue, Sep 5, 2006
What’s in the Bag?
Lecture 3
Section – 1.4.2
Tue, Sep 5, 2006

2. Two Bags
-1000 10 20 30 40 60
1000 50
Bag A
Bag B

3. The Hypotheses H0: The shown bag is Bag A. H1: The shown bag is Bag B.
-1000 10 20 30 40 60
1000 50
Bag A
Bag B

4. Decisions and Errors It is Bag A It is Bag B Win $1890 Lose $560
(Type II)
We decide it is Bag A
Lose $560
(Type I)
Win $1890
We decide it is Bag B

5. Decision Rules
Decision Rule

6. Decision Rules
Our contestant’s decision rule was to reject H0 if the bill was $1 or $5.
For that rule,  = 0.30 and = 0.30.
What if his rule was to reject H0 only if he drew a $1 bill?

7. Possible Decision Rules
Reject H0 if the voucher is worth either $60 or $1000, i.e.,  $60.
What is ?
What is ?
-1000 10 20 30 40 60
1000 50
Bag A
Bag B

8. Rejection and Accepance Regions
Rejection Region
Acceptance Region
Critical Value
The critical value itself will be included in the rejection region.

9. Decision Rule #1
$60

10. Decision Rule #1
$60
Critical Value

11. Decision Rule #1
$60
Rejection
Region
Critical Value

12. Decision Rule #1
$60
Rejection
Region
Acceptance Region
Critical Value

13. Compute 
Bag A
-1000 10 20 30 40 50 60
1000
Bag B
-1000 10 20 30 40 50 60
1000

14. Compute 
Bag A
-1000 10 20 30 40 50 60
1000
Bag B
-1000 10 20 30 40 50 60
1000

15. Compute 
Bag A
-1000 10 20 30 40 50 60
1000
Bag B
-1000 10 20 30 40 50 60
1000

16. Compute   = 1/20 = 0.05 Bag A Bag B -1000 10 20 30 40 50 60 1000

17. Compute 
Bag A
-1000 10 20 30 40 50 60
1000
Bag B
-1000 10 20 30 40 50 60
1000

18. Compute 
Bag A
-1000 10 20 30 40 50 60
1000
Bag B
-1000 10 20 30 40 50 60
1000

19. Compute 
Bag A
-1000 10 20 30 40 50 60
1000
Bag B
-1000 10 20 30 40 50 60
1000

20. Compute  Bag A  = 12/20 = 0.60 Bag B -1000 10 20 30 40 50 60 1000

21. Extreme Values Direction of Extreme Most Extreme Value
Next-most Extreme Value, etc.

22. Another Decision Rule Decision Rule #2
Reject H0 if the voucher is worth  $50.
What is ?
What is ?
-1000 10 20 30 40 60
1000 50
Bag A
Bag B

23.  vs.  If we decrease , we will increase , and
Is it possible to decrease both  and  at the same time?

24. Two-Sided Rejection Regions
See Example 1.6, p. 26.
Bag E
Bag F 1 2 3 4 5 6 7 8 9 10