1. Lecture 5 Section 1.4.3 Wed, Jan 23, 2008
What’s in the Bag?
Lecture 5
Section 1.4.3
Wed, Jan 23, 2008

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?
We use the p-value to measure this.

3. The p-value
In the Bag A/Bag B example, if the selected token is worth $50, what is the p-value?

4. Two Bags
-1000 10 20 30 40 60
1000 50
Bag A
Bag B
Observed
value

5. Two Bags Bag A Bag B At least as extreme as 50 -1000 10 20 30 40 50 60

6. Two Bags p-value = 2/20 = 0.10 Bag A Bag B At least as extreme as 50
-1000 10 20 30 40 50 60
1000
Bag B
-1000 10 20 30 40 50 60
1000

7. The p-value If the selected token is worth $30, what is the p-value?
Keep in mind, we may always compute the p-value regardless of our decision about which hypothesis to accept.

8. A Two-Sided Test 1 2 3 4 6 5
Bag E 8 10 9 7
Bag F 1 2 3 4 5 6 7 8 9 10

9. A Two-Sided Test
If the selected token is worth $8, what is the p-value?
First, what is the direction of extreme?
Which values are at least as extreme as 8?

10. A Two-Sided Test Bag E Bag F 1 2 3 4 6 5 8 10 9 7 Observed value 1 2 3

11. A Two-Sided Test Bag E Bag F 1 2 3 4 6 5 8 10 9 7 Equally extreme
Observed
value
Bag F 1 2 3 4 5 6 7 8 9 10

12. A Two-Sided Test Bag E Bag F At least as extreme as 81 2 3 4 5 6 7 8 9 10
Bag F 1 2 3 4 5 6 7 8 9 10

13. A Two-Sided Test p-value = 12/30 = 0.40 Bag E Bag F 1 2 3 4 5 6 7 8 910
Bag F 1 2 3 4 5 6 7 8 9 10

14. The p-value
If the selected token is worth $1, what is the p-value?

15. Shortcut
In a two-sided test, if the null distribution is symmetric, then you can compute the probability in one direction, and then double it to get the p-value.

16. Two Explanations of Unusual Observations
The null hypothesis leads us to a certain expectation of what the data will show.
If the data deviate from our expectation, then we need to explain that deviation.

17. Two Explanations of Unusual Observations
The null hypothesis is true; the deviation is due to chance.
The null hypothesis is false; we had the wrong expectation in the first place.

18. Two Explanations of Unusual Observations
Chance is a fine explanation if the deviation is small.
Chance is not a very good explanation if the deviation is large.
That’s where the p-value comes in.

19. Two Explanations of Unusual Observations
If the null hypothesis is true, then small deviations from the expected observation have high probabilities (large p-values).

20. Two Explanations of Unusual Observations
If the null hypothesis is true, then large deviations from the expected observation have low probabilities (small p-values).

21. Summary
Small p-values are significant; they lead to rejection of the null hypothesis.
Large p-values are not significant; they lead to acceptance of the null hypothesis.

22. Summary
Small p-values are significant; they lead to rejection of the null hypothesis.
Large p-values are not significant; they lead to acceptance of the null hypothesis.

23. Case Study 3
Incidence and risk of hypertension with sorafenib in patients with cancer: a systematic review and meta-analysis