Upload
pately5315
View
235
Download
2
Embed Size (px)
Citation preview
7/23/2019 Statistics/Statistical Hypothesis
1/23
CHAPTER 10CHAPTER 10
SECTIONS 10.1 10.3:SECTIONS 10.1 10.3:
Statistical HypothesesStatistical
Hypotheses
Engineering StatisticsEn
gineering Statistics
7/23/2019 Statistics/Statistical Hypothesis
2/23
Quote
Op
Op
ThTh
7/23/2019 Statistics/Statistical Hypothesis
3/23
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..
7/23/2019 Statistics/Statistical Hypothesis
4/23
Topics
Statistic Hypotheses: General
Concepts
Testing a Statistical Hypothesis
The Use of P-Values for ecision
!a"ing in Testing Hypotheses
7/23/2019 Statistics/Statistical Hypothesis
5/23
Hypothesis Testing Hypothesis state#ents $i%e% &ull Hypothesis
's% (lternati'e Hypothesis) either a*outpara#eters of one or #ore populations%
The &ull Hypothesis is +enote+ as H, .hichrefers to any hypothesis .e .ish to test%
The (lternati'e Hypothesis is +enote+ asH/%
The o*0ecti'e is to re0ect or fail to re0ect thenull hypothesis state#ent%
The re0ection of the null hypothesis lea+s tothe acceptance of an alternati'e hypothesis%
7/23/2019 Statistics/Statistical Hypothesis
6/23
Hypothesis Testing
( proce+ure lea+ing to a +ecision a*out aparticular hypothesis%
Hypothesis-testing proce+ures rely on using
the infor#ation in a ran+o# sa#ple fro# the
population of interest%
1f this infor#ation is consistent .ith the
hypothesis then .e .ill conclu+e that the
hypothesis is true% 1f this infor#ation is inconsistent .ith the
hypothesis .e .ill conclu+e that the
hypothesis is false%
7/23/2019 Statistics/Statistical Hypothesis
7/23
Figure 10.1 ecision criterion for
testing p ,%23 'ersus p4 ,%23
((
7/23/2019 Statistics/Statistical Hypothesis
8/23
Types of Error
Type 1 Error8 9ccurs .hen H,is re0ecte+ .hen it is true%
8 1ts pro*a*ility is +enote+ as .
Type 11 Error
8 9ccurs .hen H,is not re0ecte+ .hen it is false%
8 1ts pro*a*ility is +enote+ as .
Po.er of the Test
8 Po.er /- "(re3e&$ H04H0is 'alse
7/23/2019 Statistics/Statistical Hypothesis
9/23
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
7/23/2019 Statistics/Statistical Hypothesis
10/23
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%
7/23/2019 Statistics/Statistical Hypothesis
11/23
Figure 10.2 Pro*a*ility of a
type 1 error
7/23/2019 Statistics/Statistical Hypothesis
12/23
Figure 10.3 Pro*a*ility of a type 11
error
7/23/2019 Statistics/Statistical Hypothesis
13/23
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
7/23/2019 Statistics/Statistical Hypothesis
14/23
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
7/23/2019 Statistics/Statistical Hypothesis
15/23
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
;%
7/23/2019 Statistics/Statistical Hypothesis
16/23
Fig!e 10." C!itical Regio#Fig!e 10." C!itical Regio#
$o!$o!testi#gtesti#g == %& 'e!ss%& 'e!ss
%&%&
7/23/2019 Statistics/Statistical Hypothesis
17/23
Figure 10.6 Pro*a*ility of type 11 error
for testing
7/23/2019 Statistics/Statistical Hypothesis
18/23
Figure 10.7 Type 11 error for testing
7/23/2019 Statistics/Statistical Hypothesis
19/23
Important Properties of a
Test of Hypothesis
1.1.ThTh(e)#itio# 10.*(e)#itio# 10.*
7/23/2019 Statistics/Statistical Hypothesis
20/23
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:
7/23/2019 Statistics/Statistical Hypothesis
21/23
Figure 10. ata that are li"ely
generate+ fro# populations ha'ing t.o
+ifferent #eans
7/23/2019 Statistics/Statistical Hypothesis
22/23
(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%
7/23/2019 Statistics/Statistical Hypothesis
23/23
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#%