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Unformatted text preview: h Carolina; Slide 37 One Last Time!
Airplanes approaching a runway for a
landing are required to stay within the
localizer (certain distance left and right of
the runway). When an airplane deviates
from the localizer, it is called an
exceedence. Historically a given airline has
experienced 12% exceedence at a certain
airport. After major renovations to the
runway a random sample of 250 landings
had 35 exceedences. Is there reason to
think that the exceedence rate has
changed?
changed? G. baker, Department of Statistics
G.
University of South Carolina; Slide 38 H0:
Ha: α = 0.05 0.025 0.025 1.96 Z= pvalue = Ha +1.96 H0 Ha G. baker, Department of Statistics
G.
University of South Carolina; Slide 39 Given population parameter p and test value p0:
For H0: p = p0
Ha: p ≠ p0 α/2 α/2 Ha H0 Ha: p > p0 α
H0 Ha: p < p0 Ha Ha α Ha H0 G. baker, Department of Statistics
G.
University of South Carolina; Slide 40 There are Two Mistakes We Can Make
We can reject H0, finding for Ha, when H0 iis
s
true.
true.
Or
We can fail to reject H0, continuing to
continuing
assume H0 is true, when Ha is true.
assume
G. baker, Department of Statistics
G.
University of South Carolina; Slide 41 Controlling Risk
How do we control the probability of
rejecting H0 when H0 is true?
rejecting
when
This is called a type I error and the
probability of it happening is alpha.
probability G. baker, Department of Statistics
G.
University of South Carolina; Slide 42 H0: p < 0.07 Ha: p > 0.07
n = 250 30 p = 0.07 25 20 15 10 5 0
0 0.05 0.1 0.15 0.2 0.25 phat 0.0965 H0 Ha G. baker, Department of Statistics
G.
University of South Carolina; Slide 43 Suppose p = 0.14
30 p = 0.07 25 p = 0.14 20 15 10 β 5 α 0
0 0.05 0.1 0.15 0.2 0.25 phat 0.0965 H0 Ha G. baker, Department of Statistics
G.
University of South Carolina; Slide 44 Probability of Type II error = β
Probability
P (fail to reject H0  Ha iis true at p = 0.14)
s
ˆ
P ( p < 0.0965  p = 0.14)
0.0965
0.0965 − 0.14
Z=
= −1.982
0.14(0.86)
250 P (Z < 1.982) = 0.0237 = β
(Z
1.982)
G. baker, Department of Statistics
G.
University of South Carolina; Slide 45 Power of a Hypothesis Test
Power = P (reject H0  Ha is true at p = 0.14)
Power = 1 – β
Power
Power of this test at p = 0.14:
1 – 0.0237 = 0.9763
0.0237 G. baker, Department of Statistics
G.
University of South Carolina; Slide 46 How can We Control β?
How
30 p = 0.07 25 p = 0.14 20 15 10 β 5 α 0
0 0.05 0.1 0.15 0.2 0.25 phat 0.0965 H0 Ha G. baker, Department of Statistics
G.
University of South Carolina; Slide 47 Summary
α = P (committing Type I error)
α = P (reject H0  H0 is true)
Control α by setting it prior to test
Control
β = P (committing Type II error)
β = P (fail to reject H0  Ha is true at some value)
Control β by choosing appropriate sample size.
Control
Power = P (reject H0  Ha is true at some value)
Power = 1  β
Power
G. baker, Department of Statistics
G.
University of South Carolina; Slide 48 You Try It
H0: p > 0.12
Ha: p < 0.12
α = 0.05 n = 250 What is the power of this test at p = 0.07? G. baker, Department of Statistics
G.
University of South Carolina; Slide 49...
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 Fall '13
 Wang
 Statistics

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