1. The Language of Statistical Decision Making
Lecture 3
Section 1.3
Mon, Aug 27, 2007

2. The Language of Statistical Decision Making - Part 2
Errors
Recall our conclusion about the die being fair.
Could our conclusion have been wrong?
What would be the cause of our error?
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

3. The Language of Statistical Decision Making - Part 2
Errors
Had we concluded that the die was not fair, could we have been wrong?
What would be the cause of our error?
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

4. The Language of Statistical Decision Making - Part 2
Possible Errors
We might reject H0 when it is true.
This is a Type I error.
We might accept H0 when it is false.
This is a Type II error.
See Making Intelligent Errors, by Walter Williams.
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

5. The Language of Statistical Decision Making - Part 2
Decisions and Errors
State of Nature
H0 true
H0 false
Correct
Type II
Error
Accept H0
Decision
Type I
Error
Correct
Reject H0
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

6. The Language of Statistical Decision Making - Part 2
Decisions and Errors
State of Nature
H0 true
H0 false
Correct
Type II
Error
Accept H0
Decision
Type I
Error
Correct
Reject H0
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

7. The Language of Statistical Decision Making - Part 2
Decisions and Errors
State of Nature
H0 true
H0 false
Correct
Type II
Error
Accept H0
Decision
Type I
Error
Correct
Reject H0
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

8. The Language of Statistical Decision Making - Part 2
Example
Consider a study to determine the effectiveness of a new drug.
What are the two possible conclusions (hypotheses)?
Which should get the benefit of the doubt?
What are the two possible errors?
Which is more serious?
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

9. The Language of Statistical Decision Making - Part 2
Example
Now consider a study to determine the safety of a new drug.
What are the two possible conclusions (hypotheses)?
Which should get the benefit of the doubt?
What are the two possible errors?
Which is more serious?
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

10. Safe and Effective Criteria
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

11. The Language of Statistical Decision Making - Part 2
Significance Level
Significance Level – The likelihood of rejecting H0 when it is true, i.e., the likelihood of committing a Type I error.
 – The likelihood of a Type I error.
 – The likelihood of a Type II error.
That is,  is the significance level.
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

12. The Language of Statistical Decision Making - Part 2
Significance Level
Suppose that we have two very unusual dice.
Die A rolls a 1 80% of the time and a 6 only 20% of the time. (It never lands 2, 3, 4, or 5.)
Die B rolls a 1 only 10% of the time and a 6 90% of the time. (It never lands 2, 3, 4, or 5.)
Visually, the two dice are indistinguishable.
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

13. The Language of Statistical Decision Making - Part 2
Significance Level
We are given one of the dice and we roll it one time.
We get a 1.
Suppose the null hypothesis is that we rolled die A and the alternative hypothesis is that we rolled die B.
Which hypothesis do we choose?
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

14. The Language of Statistical Decision Making - Part 2
Significance Level
What is our criterion (decision rule) for choosing between the two hypotheses?
Describe a Type I error.
Describe a Type II error.
What is the value of ?
What is the value of ?
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

15. The Language of Statistical Decision Making - Part 2
Two Rolls
Suppose now that we roll the selected die twice and average the two rolls.
We must get either
A pair of 1s, with an average of 1.
A 1 and a 6, with an average of 3.5.
A pair of 6s, with an average of 6.
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

16. The Language of Statistical Decision Making - Part 2
Two Rolls
What would be a good criterion for deciding which die it is?
Based on this criterion,
What is ?
What is ?
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2

17. The Language of Statistical Decision Making - Part 2
Case Study 2
Hair May Help Reveal Eating Disorders
What were the hypotheses?
Describe a Type I error.
Describe a Type II error.
Mon, Aug 27, 2007
The Language of Statistical Decision Making - Part 2