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Lecture

Week 7 lecture Slides.pdf

by OneClass235004 , Fall 2013
13 Pages
59 Views

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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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 Go Back Full Screen Close Quit •First• Prev •Next •Last • Go Back •Full Screen •Close •Quit                                     Home Page 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. Go Back Full Screen Close Quit     •Firs•Prev •Next•Last•Go Back •Full Scre•Close•Quit                   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”. •Fir•Prev•NextLast•Go Bac•Full Scr•Clos•Quit                                 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 Go Back (b) H 0 is false. Full Screen Close Quit       • Firs•Prev•Next •Last•Go Back•Full Scree•Close•Quit                               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. Go Back Full Screen Close Quit       •First•Prev •Next •Last •Go Back •Full Screen•Close •Quit                               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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