What’s in the Bag?
Lecture 4
Sections 1.4.1-1.4.2
Robb T. Koether
Hampden-Sydney College
Tue, Jan 19, 2010
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Outline
1
Homework Review
2
Bag A and Bag B
3
Possible Decision Rules
Decision Rule #1
Decision Rule #2
Decision Rule #3
4
Assignment
5
Answers to Even-Numbered Problems
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What’s in the Bag?
Tue, Jan 19, 2010
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Outline
1
Homework Review
2
Bag A and Bag B
3
Possible Decision Rules
Decision Rule #1
Decision Rule #2
Decision Rule #3
4
Assignment
5
Answers to Even-Numbered Problems
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What’s in the Bag?
Tue, Jan 19, 2010
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Homework Review
Exercise 1.11, p. 67.
A consumer protection agency received many complaints that the
sodium content in a six-ounce can of Star Kist tuna is greater then the
250 mg that is stated on the label of each can. In response to these
complaints, the protection agency tested the following hypotheses:
H0 : The average sodium content of all six-ounce cans is 250 mg.
H1 : The average sodium content of all six-ounce cans is greater than
250 mg.
The data from the study were not statistically significant.
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What’s in the Bag?
Tue, Jan 19, 2010
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Homework Review
Exercise 1.7, p. 68.
(a) What hypothesis was supported?
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Homework Review
Exercise 1.7, p. 68.
(a) What hypothesis was supported?
The null hypothesis was supported.
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What’s in the Bag?
Tue, Jan 19, 2010
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Homework Review
Exercise 1.7, p. 68.
(a) What hypothesis was supported?
The null hypothesis was supported.
(b) Was a complaint registered with the provider, the Star Kist Co.?
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What’s in the Bag?
Tue, Jan 19, 2010
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Homework Review
Exercise 1.7, p. 68.
(a) What hypothesis was supported?
The null hypothesis was supported.
(b) Was a complaint registered with the provider, the Star Kist Co.?
No.
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What’s in the Bag?
Tue, Jan 19, 2010
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Homework Review
Exercise 1.7, p. 68.
(a) What hypothesis was supported?
The null hypothesis was supported.
(b) Was a complaint registered with the provider, the Star Kist Co.?
No.
(c) Could a mistake have been made? If so, describe the mistake.
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Homework Review
Exercise 1.7, p. 68.
(a) What hypothesis was supported?
The null hypothesis was supported.
(b) Was a complaint registered with the provider, the Star Kist Co.?
No.
(c) Could a mistake have been made? If so, describe the mistake.
Yes, a Type II error could have been made. A Type II error would
be to conclude that the average sodium content of all six-ounce
cans is greater than 250 mg when, in fact, it is equal to 250 mg.
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What’s in the Bag?
Tue, Jan 19, 2010
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Outline
1
Homework Review
2
Bag A and Bag B
3
Possible Decision Rules
Decision Rule #1
Decision Rule #2
Decision Rule #3
4
Assignment
5
Answers to Even-Numbered Problems
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What’s in the Bag?
Tue, Jan 19, 2010
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Two Bags
Two bags contain vouchers with various dollar values.
Bag A
10
20
30
40
50
60
10
20
30
40
50
60
Bag B
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What’s in the Bag?
Tue, Jan 19, 2010
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Two Bags
The vouchers in Bag A are worth $470.
The vouchers in Bag B are worth $930.
You get to choose whichever bag you want, cash in the vouchers,
and keep the money.
Which bag should you choose?
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What’s in the Bag?
Tue, Jan 19, 2010
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Two Bags
The vouchers in Bag A are worth $470.
The vouchers in Bag B are worth $930.
You get to choose whichever bag you want, cash in the vouchers,
and keep the money.
Which bag should you choose?
Oh, one more thing...
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Two Bags
The vouchers in Bag A are worth $470.
The vouchers in Bag B are worth $930.
You get to choose whichever bag you want, cash in the vouchers,
and keep the money.
Which bag should you choose?
Oh, one more thing...
The bags are indistinguishable from the outside and you are not
allowed to look inside.
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
8 / 41
Two Bags
The vouchers in Bag A are worth $470.
The vouchers in Bag B are worth $930.
You get to choose whichever bag you want, cash in the vouchers,
and keep the money.
Which bag should you choose?
Oh, one more thing...
The bags are indistinguishable from the outside and you are not
allowed to look inside.
However, you may reach inside, choose one voucher, and read its
value.
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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The Hypotheses
We are shown one of the bags.
The hypotheses are:
I
I
H0 : The bag shown is Bag A.
H1 : The bag shown is Bag B.
We will choose one voucher at random from the shown bag and
read its value.
From that information, we will decide which bag we believe we are
holding.
Then we will choose the one that we believe is Bag B, take the
money and run.
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Decisions and Their Consequences
We decide it is Bag A
We decide it is Bag B
Robb T. Koether (Hampden-Sydney College)
It is Bag A
Win $930
Win $470
What’s in the Bag?
It is Bag B
Win $470
Win $930
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Outline
1
Homework Review
2
Bag A and Bag B
3
Possible Decision Rules
Decision Rule #1
Decision Rule #2
Decision Rule #3
4
Assignment
5
Answers to Even-Numbered Problems
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Outline
1
Homework Review
2
Bag A and Bag B
3
Possible Decision Rules
Decision Rule #1
Decision Rule #2
Decision Rule #3
4
Assignment
5
Answers to Even-Numbered Problems
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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A Decision Rule
Decision Rule #1
Reject H0 if the voucher is worth $60.
What is α?
What is β?
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What’s in the Bag?
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Rejection and Acceptance Regions
Definition (Rejection Region)
The rejection region is the set of possible observed values that will
lead to rejection of the null hypothesis.
Definition (Acceptance Region)
The acceptance region is the set of possible observed values that will
lead to acceptance of the null hypothesis.
Definition (Critical Value(s))
The critical value(s) is the value that separates the the two regions.
The critical value itself will be included in the rejection region.
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What’s in the Bag?
Tue, Jan 19, 2010
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Decision Rule #1
10
20
30
40
50
60
Critical value
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Tue, Jan 19, 2010
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Decision Rule #1
Rejection region
10
20
30
40
50
60
Critical value
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What’s in the Bag?
Tue, Jan 19, 2010
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Decision Rule #1
Acceptance region
10
20
30
40
50
Rejection region
60
Critical value
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Compute α
Bag A
10
20
30
40
50
60
10
20
30
40
50
60
Bag B
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Compute α
Bag A
10
20
30
40
50
60
10
20
30
40
50
60
Bag B
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Compute α
(reject H0)
Bag A
10
20
30
40
50
60
10
20
30
40
50
60
Bag B
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What’s in the Bag?
Tue, Jan 19, 2010
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Compute α
α = 1/20
Bag A
10
20
30
40
50
60
10
20
30
40
50
60
Bag B
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What’s in the Bag?
Tue, Jan 19, 2010
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Compute β
Bag A
10
20
30
40
50
60
10
20
30
40
50
60
Bag B
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Compute β
Bag A
10
20
30
40
50
60
10
20
30
40
50
60
Bag B
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What’s in the Bag?
Tue, Jan 19, 2010
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Compute β
Bag A
10
20
40
50
60
50
60
(accept H0)
Bag B
10
Robb T. Koether (Hampden-Sydney College)
30
20
30
40
What’s in the Bag?
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Compute β
Bag A
10
30
40
50
60
40
50
60
β = 13/20
Bag B
10
Robb T. Koether (Hampden-Sydney College)
20
20
30
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Extreme Values
Definition (Direction of Extreme)
The direction of extreme is the direction, left or right or both, in which
we are more likely to reject H0 .
Normally, either
I
I
The larger the observed value, the more likely we are to reject H0 ,
or
The smaller the observed value, the more likely we are to reject H0 .
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What’s in the Bag?
Tue, Jan 19, 2010
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Outline
1
Homework Review
2
Bag A and Bag B
3
Possible Decision Rules
Decision Rule #1
Decision Rule #2
Decision Rule #3
4
Assignment
5
Answers to Even-Numbered Problems
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Another Decision Rule
Decision Rule #2
Reject H0 if the voucher is worth at least $50.
What is α?
What is β?
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Compute α and β
Bag A
10
20
30
40
50
60
10
20
30
40
50
60
Bag B
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Outline
1
Homework Review
2
Bag A and Bag B
3
Possible Decision Rules
Decision Rule #1
Decision Rule #2
Decision Rule #3
4
Assignment
5
Answers to Even-Numbered Problems
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Yet Another Decision Rule
Decision Rule #3
Reject H0 if the voucher is worth at least $40.
What is α?
What is β?
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Compute α and β
Bag A
10
20
30
40
50
60
10
20
30
40
50
60
Bag B
Robb T. Koether (Hampden-Sydney College)
What’s in the Bag?
Tue, Jan 19, 2010
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Die A and Die B
Recall Die A and Die B.
Outcome
1
4
Robb T. Koether (Hampden-Sydney College)
Die A
0.80
0.20
What’s in the Bag?
Die B
0.10
0.90
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Die A and Die B
If we base our decision on the average of four rolls, the
probabilities are
Outcome
Average Die A Die B
Three 1s
1
0.512 0.001
Two 1s, one 4
2
0.384 0.027
One 1, two 4s
3
0.096 0.243
Three 4s
4
0.008 0.729
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What’s in the Bag?
Tue, Jan 19, 2010
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Die A and Die B
0.5
Die A
0.4
0.3
0.512
0.2
0.384
0.1
0.096
0.008
3
4
Average
0.0
1
2
0.7
0.6
0.5
Die B
0.4
0.729
0.3
0.2
0.243
0.1
0.001
0.027
1
2
Average
0.0
3
4
What is a good decision rule?
What are the values of α and β?
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What’s in the Bag?
Tue, Jan 19, 2010
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α vs. β
If we modify the decision rule to decrease α, we will increase β,
and
If we modify the decision rule to decrease β, we will increase α.
Is it possible to decrease both α and β at the same time?
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What’s in the Bag?
Tue, Jan 19, 2010
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Outline
1
Homework Review
2
Bag A and Bag B
3
Possible Decision Rules
Decision Rule #1
Decision Rule #2
Decision Rule #3
4
Assignment
5
Answers to Even-Numbered Problems
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What’s in the Bag?
Tue, Jan 19, 2010
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Assignment
Homework
Read Section 1.4 - 1.4.2, pages 16 - 28.
Let’s Do It! 1.7, 1.8.
Page 67, exercises 13 - 15, 17 - 23, 25.
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What’s in the Bag?
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Outline
1
Homework Review
2
Bag A and Bag B
3
Possible Decision Rules
Decision Rule #1
Decision Rule #2
Decision Rule #3
4
Assignment
5
Answers to Even-Numbered Problems
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Answers to Even-Numbered Problems
Page 67
1.14 The answer is (e).
1.18 (a) “The proportion of newborns that are girls is not 0.50.”
(b) Two-sided.
1.20 The answer is (b)
1.22 (a) The frequency plots:
2
4
6
8 10 12 14 16 18
2
Bag A
4
6
8 10 12 14 16 18
Bag B
The direction of extreme is two-sided.
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What’s in the Bag?
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Answers to Even-Numbered Problems
Page 67
1.22 (b) H0 : The bag is Bag A.
H1 : The bag is Bag B.
(c) Reject H0 if the value of the voucher is more than 14 or less than 6.
(or 12 and 8, or 16 and 4)
2
6
= 0.12. (or 14
(d) α = 50
50 = 0.28, or 50 = 0.04.)
6
30
(e) β = 16
50 = 0.32. (or 50 = 0.12, or 50 = 0.60.)
(f) (i) Accept H0 . (or Accept H0 , or Reject H0 .)
(g)
(ii) Type II error. (or Type II error, or Type I error.)
(i) Reject H0 . (or Reject H0 , or Accept H0 .)
(ii) Type I error. (or Type I error, or Type II error.)
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