1. The Language of Statistical Decision Making
Lecture 3
Section 1.3
Mon, Jan 21, 2008

2. The Language of Statistical Decision Making - Part 2
Errors
Recall our conclusion that the distribution of M&M colors agreed with what the company said.
Could our conclusion have been wrong?
What would be the cause of our error?
Mon, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

3. The Language of Statistical Decision Making - Part 2
Errors
Had we concluded that the distribution was not what the company said it was, could we have been wrong?
What would be the cause of our error?
Mon, Jan 21, 2008
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, Jan 21, 2008
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, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

6. The Language of Statistical Decision Making - Part 2
Decisions and Errors
True distribution
It is what
company says
It is not what
company says
Correct
Type II
Error
It is what company says
Decision
Type I
Error
Correct
It is not what company says
Mon, Jan 21, 2008
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, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

8. 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, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

9. The Language of Statistical Decision Making - Part 2
Case Study 2
Hair May Help Reveal Eating Disorders
Mon, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

10. 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, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

11. 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, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

12. Type I Error vs. Type II Error
See Making Intelligent Errors, by Walter Williams.
Mon, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

13. 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, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

14. The Language of Statistical Decision Making - Part 2
Two Unusual Dice
Suppose that we have two 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, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

15. The Language of Statistical Decision Making - Part 2
Which Die Did We Pick?
We pick up one of the dice.
Suppose the null hypothesis is that we picked up die A and the alternative hypothesis is that we picked up die B.
We will roll the die one time and, based on the outcome, decide which die we think it is.
Mon, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

16. The Language of Statistical Decision Making - Part 2
The Decision
What should be our criterion (decision rule) for choosing between the two hypotheses?
That is, if the die turns up 1, which hypothesis do we choose? What if it turns up 6?
Describe a Type I error.
Describe a Type II error.
Mon, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

17. The Significance Level
What is the value of ?
What is the value of ?
Mon, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

18. Are Two Rolls Better Than One?
Suppose now that we roll the chosen 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, Jan 21, 2008
The Language of Statistical Decision Making - Part 2

19. Are Two Rolls Better Than One?
What would be a good criterion for deciding which die it is?
Based on this criterion,
What is ?
What is ?
Mon, Jan 21, 2008
The Language of Statistical Decision Making - Part 2