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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

p−value =Pr(D≥dobs;H0),(5)

where the notation “;H0” means “assuming H0is true”.

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