Transcript Document

Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 11.4
Using Inference to Make Decisions
Copyright © 2008 by W. H. Freeman & Company
Potential Wrong Decisions
• When we make decisions based on a
significance test we hope to make the right
decision - - but we could be wrong.
• Two types of errors:
– We reject H0 when H0 is in fact true.
– We fail the reject H0 when H0 is false.
• Each of these errors have consequences.
Type I and Type II Errors
Sampling Distribution Based on
the Truth of a Hypothesis
Reject H0
Fail to reject H0
Sampling
Distribution
when H0 is
false.
Sampling
Distribution
when H0 is
true.
α
β
Critical
α is the probability of a
type I error.
μ = 6.4 Value of μ = 6.7
(Ha)
(H0)
x
β is the probability
of a type II error.
α
1-β
1-α
β
Which type of error is more serious
• Depends – which error has the more
serious consequence. (Which error does
more harm.)
• Requires a realistic interpretation of what
constitutes a type I error and what
constitutes a type II.
Example 11.52
•
An outbreak of the mountain pine beetle has affected
several types of trees in British Columbia. The beetle
leaves behind a fungus that produces blue-colored
stains in the wood. Some customers might worry that
lumber obtained from the blue-stained trees is weaker
as the result of the fungus. A Canadian company
performed a test on the breaking strength of bluestained wood. The measured mean breaking strength
of a sample of 100 pieces of blue-stained pine. The
target breaking strength of lumber made from healthy
pine trees is 10,000 pounds per square inch.
1. State the appropriate null and alternative hypothesis for
the company test.
H 0 :   10,000 psi
H a :   10,000 psi
Example 11.52 Continued
2. Describe a Type I and a Type II error, and give the
consequences of each.
•
Type I error (Reject H0, when in fact H0 is true.) A type
I error is committed by telling the consumer that the
wood with blue stain is weaker when in fact it is not.
•
Type I consequence is that you must find more wood
without the stain.
•
Type II error (Fail to reject H0 when in fact H0 is False.)
A type II error is committed by not telling the consumer
that the blue-stained wood is weaker when in fact it is.
•
Type II consequence is that you may lose loyal
customers because they will lose their trust in the
company.
Example 11.52 Continued
3. Which type of error is more serious?
Why?
• While Type I error will cause the
company to spend more money to find
wood that does not contain blue stains,
the Type II error is more serious since
the company can not afford to lose a
portion of its customer base because of
perceived dishonesty.
Class Practice
Power of a Test
1-β
1 -α
Power
α
β
Definition of Power
• Power measures the sensitivity of the test to find a significant effect when
the truth of the alternative is far from the null hypothesis.
• High power is desirable.
Applet over Adjusting Power
• Look at Activity 11C on page 729.
• Power Applet.
Ways to Increase The Power
• Increase the sample size.
• Increase α.
• Consider an alternative hypothesis further
away from the null hypothesis.
• Decrease σ.
Advise: “…choose as high an α level (Type I error
probability) as your are willing to risk and as large a
sample size as you can afford.”
Standard
• Evolving standard:
– Sample size large enough for power (around
80%)
– 95% confidence interval
– Significance level for test, α = 0.05