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Lecture

# STAT231 Lecture Notes - Variance, Random Variable, Statistical Hypothesis Testing

13 pages30 viewsFall 2013

Department
Statistics
Course Code
STAT231
Professor
Matthias Schonlau

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Statistical Inference
We have looked at the sampling distribution of estimators and
conﬁdence intervals
Look at the idea of a hypothesis test and the p-value scale of
evidence
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Testing statistical hypotheses
There are often hypotheses that a statistician or scientist might
want to “test” in the light of observed data.
Two important types of hypotheses are
(1) that a parameter vector θhas some speciﬁed value θ0; we
denote this as H0:θ=θ0.
(2) that a random variable Yhas a speciﬁed probability distri-
bution, say with p.d.f. f0(y); we denote this as H0:Y
f0(y).
We shall concentrate on the ﬁrst of these.
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Statistical approach
Assume that the hypothesis H0will be tested using some ran-
dom data “Data”.
Deﬁne a test statistic (also called a discrepancy measure)D=
g(Data) that is constructed to measure the degree of “agree-
ment” between Data and the hypothesis H0.
It is conventional to deﬁne Dso that D=0represents the
best possible agreement between the data and H0, and so that
the larger Dis, the poorer the agreement.
Once speciﬁc observed “data” have been collected, let dobs =
g(data)be the corresponding observed value of D.
To test H, we now calculate the observed signiﬁcance level
(also called the p-value), deﬁned as
pvalue =Pr(Ddobs;H0),(5)
where the notation “;H0” means “assuming H0is true”.

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