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STAT 231
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Matthias Schonlau
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Lecture

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Statistics

STAT 231

Matthias Schonlau

Fall

Description

Home Page Statistical Inference
Title Page
• We have looked at the sampling distribution of estimators and
conﬁdence intervals
Contents
• Look at the idea of a hypothesis test and the p -value scale of
evidence
Page 46 of 77
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Testing statistical hypotheses
Title Page • Thereareoftenhypothesesthatastatisticianorscientistmight
want to “test” in the light of observed data.
Contents
• Two important types of hypotheses are
(1) that a parameter vector θ has some speciﬁed value θ ; we
0
denote this as H 0: θ = θ 0.
(2) that a random variable Y has a speciﬁed probability distri-
bution, say with p.d.f. f (y) ; we denote this as H : Y ∼
0 0
Page 47 of 77 f 0y) .
• We shall concentrate on the ﬁrst of these.
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Home Page Statistical approach
Title Page
• Assume that the hypothesis H 0will be tested using some ran-
dom data “Data”.
Contents • Deﬁneateststatistic(alsocalledadiscrepancymeasure) D =
g(Data) that is constructed to measure the degree of “agree-
ment” between Data and the hypothesis H 0.
• It is conventional to deﬁne D so that D =0 represents the
best possible agreement between the data and H 0, and so that
the larger D is, the poorer the agreement.
Page 48 of 77
• Once speciﬁc observed “data” have been collected, let d =
obs
Go Back g(data ) be the corresponding observed value of D .
• To test H , we now calculate the observed signiﬁcance level
Full Screen (also called the p-value), deﬁned as
Close p − value = Pr(D ≥ d ;H ), obs 0 (5)
Quit where the notation “ ;H 0 ” means “assuming H 0is true”.
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Home Page Meaning of p -value
Title Page • If p-value is close to zero then we are inclined to doubt that
H 0 is true, because if it is true the probability of getting agree-
Contents
ment as poor or worse than observed is small.
• This makes the alternative explanation that H 0 is false more
appealing.
• In other words, we must accept that one of the following two
statements is correct.:
Page 49 of 77 (a) H 0 is true but by chance we have observed data that indi-
cate poor agreement with H 0, or
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(b) H 0 is false.
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Home Page Meaning of p -value
Title Page • In summary, the p -value is a numerical scale which measures
the strength of evidence in the data against some hypothesis.
Contents
• It is a number between 0 and 1
• The smaller the value the more evidence against the hypoth-
esis.
• In order to have an agreement about what ‘small’ means the
following scale is often used, but you should not take it as
Page 50 of 77 anything but a rule of thumb.
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Home Page Meaning of p -value
Title Page Signiﬁcance level Interpretation
p> 0.10 no evidence against the hypothesised value
0.05

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