What

1
Whats in the Bag?

• Lecture 5
• Section 1.4.3
• Wed, Aug 29, 2007

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
Observed value
5
Two Bags
Bag A
50
-1000
10
20
30
40
60
1000
Bag B
50
-1000
10
20
30
40
60
1000
6
Two Bags
p-value 2/20 0.10
Bag A
50
-1000
10
20
30
40
60
1000
Bag B
50
-1000
10
20
30
40
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
Bag F
5
1
2
3
4
6
8
10
9
7
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
Observed value
Bag F
5
1
2
3
4
6
8
10
9
7
11
A Two-Sided Test
Equally extreme
Observed value
Bag F
5
1
2
3
4
6
8
10
9
7
12
A Two-Sided Test
At least as extreme as 8
At least as extreme as 8
Bag E
5
1
2
3
4
6
8
10
9
7
Bag F
5
1
2
3
4
6
8
10
9
7
13
A Two-Sided Test
p-value 12/30 0.40
Bag E
5
1
2
3
4
6
8
10
9
7
Bag F
5
1
2
3
4
6
8
10
9
7
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
The p-value

• A small p-value is strong evidence against the
  null hypothesis.
• Why?

17
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.

18
Two Explanations of Unusual Observations

• Two explanations
• 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.

19
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.
• Thats where the p-value comes in.

20
Two Explanations of Unusual Observations

• Small deviations are consistent with the null
  hypothesis.
• They have high probability (large p-values) if
  the null hypothesis is true.

21
Two Explanations of Unusual Observations

• Large deviations are inconsistent with the null
  hypothesis.
• They have low probability (small p-values) if the
  null hypothesis is true.

22
Case Study 3

• Long-term Proton Pump Inhibitor Therapy and Risk
  of Hip Fracture