statistics/statistical hypothesis

7/23/2019 Statistics/Statistical Hypothesis

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CHAPTER 10CHAPTER 10

SECTIONS 10.1 10.3:SECTIONS 10.1 10.3:

Statistical HypothesesStatistical

Hypotheses

Engineering StatisticsEn

gineering Statistics


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Quote

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Op

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Required Reading

R.E. Walpole, R.H. Myers, S.L.Myers, and K. Ye. (2012.!"ro#a#ili$y % S$a$is$i&s 'orEngineers and S&ien$is$s. )$* Ed."ren$i&e Hall+ e- ersey. /*ap$er10.110..


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Topics

Statistic Hypotheses: General

Concepts

Testing a Statistical Hypothesis

The Use of P-Values for ecision

!a"ing in Testing Hypotheses


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Hypothesis Testing Hypothesis state#ents $i%e% &ull Hypothesis

's% (lternati'e Hypothesis) either a*outpara#eters of one or #ore populations%

The &ull Hypothesis is +enote+ as H, .hichrefers to any hypothesis .e .ish to test%

The (lternati'e Hypothesis is +enote+ asH/%

The o*0ecti'e is to re0ect or fail to re0ect thenull hypothesis state#ent%

The re0ection of the null hypothesis lea+s tothe acceptance of an alternati'e hypothesis%


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Hypothesis Testing

( proce+ure lea+ing to a +ecision a*out aparticular hypothesis%

Hypothesis-testing proce+ures rely on using

the infor#ation in a ran+o# sa#ple fro# the

population of interest%

1f this infor#ation is consistent .ith the

hypothesis then .e .ill conclu+e that the

hypothesis is true% 1f this infor#ation is inconsistent .ith the

hypothesis .e .ill conclu+e that the

hypothesis is false%


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Figure 10.1 ecision criterion for

testing p ,%23 'ersus p4 ,%23

((


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Types of Error

Type 1 Error8 9ccurs .hen H,is re0ecte+ .hen it is true%

8 1ts pro*a*ility is +enote+ as .

Type 11 Error

8 9ccurs .hen H,is not re0ecte+ .hen it is false%

8 1ts pro*a*ility is +enote+ as .

Po.er of the Test

8 Po.er /- "(re3e&$ H04H0is 'alse


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So#eti#es the Type 1 Error is calle+ the significance le'el% The pro*a*ility of co##itting a type 1 error is +enote+ *y

the Gree" letter %

The pro*a*ility of co##itting a type 11 error is +enote+ *ythe Gree" letter .

The probability of committing both types of error can bereduced by increasing the sample size.

Table 10.1 Possi*le Situations for

Testing a Statistical Hypothesis


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The role of Sa#ple Sie

Consi+er the ne. 'accine *ut no. using

a sa#ple of /,, in+i'i+uals%

1f #ore than ;< surpass the 2-year

perio+ .e re0ect the null hypothesis thatp = an+ accept the alternati'e

hypothesis that p 4 =%

Critical 'alue is no. ; all outco#esa*o'e ;< constitute the critical region

an+ all possi*le scores less than or e?ual

to ;< fall in the acceptance region%


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Figure 10.2 Pro*a*ility of a

type 1 error


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Figure 10.3 Pro*a*ility of a type 11

error


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Hypothesis Testing

T.o-si+e+ Hypothesis

9ne-si+e+ Hypothesis

H0

:= 50 cm/s

H1

: 50 cm/s

H0

:= 50 cm/s

H1

:< 50 cm/s

H0

:= 50 cm/s

H1

: > 50 cm/s


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2-Si+e+ Hypothesis Testing

E6a#ple

Consi+er the null hypothesis that the

a'erage .eight of #ales stu+ents at

Te6as State Uni'ersity is


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Figure 10.4 Critical region $in *lue)

(ssu#e that the stan+ar+ +e'iation(ssu#e that the stan+ar+ +e'iation

of the population of .eights to *e of the population of .eights to *e

;%


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Fig!e 10." C!itical Regio#Fig!e 10." C!itical Regio#

$o!$o!testi#gtesti#g == %& 'e!ss%& 'e!ss

%&%&


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Figure 10.6 Pro*a*ility of type 11 error

for testing


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Figure 10.7 Type 11 error for testing


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Important Properties of a

Test of Hypothesis

1.1.ThTh(e)#itio# 10.*(e)#itio# 10.*


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The Use of P-Values for ecision

!a"ing in Testing Hyposthses

P-'alue

8 The s#allest le'el of significance that

.oul+ lea+ to re0ection of the null

hypothesis H0%8 Aor e6a#ple the p-'alue in the -test is

gi'en *y:


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Figure 10. ata that are li"ely

generate+ fro# populations ha'ing t.o

+ifferent #eans


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(pproach to Hypothesis Testing

.ith Ai6e+ Pro*a*ility of Type 1

Error/% Ta"e a ran+o# sa#ple%2% State H, an+H/%

;% Select so $*a$ $*e pro#a#ili$y o' 5ype

66 *as a sui$a#le s7all 8alue.

F% Specify the critical region that lea+s to the

re0ection of H,%

3% Co#pute the appropriate test statistic%


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Significance Testing

$P-Value (pproach)

State null an+ alternati'e hypotheses%

Choose an appropriate test statistic%

Co#pute the P-'alue *ase+ on theco#pute+ 'alue of the test statistic%

Use 0u+g#ent *ase+ on the P-'alue

an+ "no.le+ge of the scientificsyste#%

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