1. Lecture 4 Section 1.4.3 Wed, Jan 19, 2005
What’s in the Bag?
Lecture 4
Section 1.4.3
Wed, Jan 19, 2005

2. How Strong is the Evidence?
Rather than give an accept/reject answer, we may ask a different question:
How strong is the evidence against H0?

3. The p-value
p-value – The likelihood of getting by chance, if H0 is true, a value at least as extreme as the one observed.

4. Two Bags
If the selected token is worth $50, what is the p-value?

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

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

7. Two Bags p-value = 2/20 = 0.10 Bag A Bag B -1000 10 20 30 40 50 60

8. Two More Bags
Bag E 1 2 3 4 5 6 7 8 9 10
Bag F 1 2 3 4 5 6 7 8 9 10

9. Two More Bags
If the selected token is worth $8, what is the p-value?

10. Two More Bags
Bag E 1 2 3 4 5 6 7 8 9 10
Bag F 1 2 3 4 5 6 7 8 9 10

11. Two More Bags p-value = 12/30 = 0.40 Bag E Bag F 1 2 3 4 5 6 7 8 9 10

12. The p-value If the null hypothesis is true, then Therefore,
The observed data is likely to be close to what the null hypothesis predicts.
Therefore,
Large discrepancies are less likely than small discrepancies.
The p-value measures the likelihood of a discrepancy at least as large as the one observed.

13. Interpretation of the p-value
Smaller p-values
Larger p-values 1
p-values

14. Interpretation of the p-value
More extreme values
Less extreme values 1
p-values

15. Interpretation of the p-value
Larger discrepancy
Smaller discrepancy 1
p-values

16. Interpretation of the p-value
Stronger evidence
against H0
Weaker evidence
against H0 1
p-values

17. Interpretation of the p-value
Statistically
more significant
Statistically
less significant 1
p-values

18. Two Explanations
For any discrepancy between the evidence and what is predicted by the null hypothesis, there are always two explanations:
The discrepancy occurred by chance (random sampling) even though the null hypothesis is true.
The discrepancy occurred because the null hypothesis is false.
Given the evidence, which explanation is more believable?

19. Two Explanations The null hypothesis gets the benefit of the doubt.
Therefore,
If the discrepancy is small (p-value is large), then we go with the first explanation and accept H0.
If the discrepancy is large (p-value is small), then we go with the second explanation and reject H0.

20. Let’s Do It!
Let’s do it! 1.9, p. 32 – p-value for a One-Sided Rejection Region to the Left.
Let’s do it! 1.10, p. 34 – p-value for a Two-Sided Rejection Region.
Example 1.9, p. 36 – Can You Picture the p-value?

21. Let’s Do It! Let’s do it! 1.7, p. 26 – Chromium Supplements.
Let’s do it! 1.8, p. 27 – Three Studies.
Let’s do it! 1.11, p. 39 – Machine A or Machine B?