11.Hypothesis Test for a Single Mean

H YPOTHESIS TESTS

FOR A SINGLE MEAN

11.1 TESTING A HYPOTHESI HYPOTHESIS S CONCERNIN CONCERNING G THE

#es# s#a#is#i* is (rea#er #er #han

 z  DISTRIBUTION

MEAN BY USE OF THE

+ z α /

 H o : μ = μo

!ess

Test Statistic: z – test ((if if

is known, or

√ 

No#e.

√ 

 μ0=¿

%e&ia#ion

 z

¿

 is #he *omu#e% #es# s#a#is#i*

MEAN BY USE OF THE  The

ou ou!a# !a#ion ion

t 

%is#ri+ %is #ri+u#io u#ion n is #he aro aroria# ria#e e +asis +asis for

s#an% s#an%ar ar% %

%is#ri+u#e% +u#

s =¿  sam!e s#an%ar% %e&ia#ion

 The #es# re'uires re'uires #ha# #he sam!e si$e

Test Statistic: t – test (if ( if  x´ − μ 0 t = s/ n

n ≥ 30

  where:

 μ0=¿

C!itica Rei&"

 h"o#hesi$e% mean

n =¿  sam!e si$e

% – #a)e

 H 0

if #he *omu#e% #es# s#a#is#i* is (rea#er #er #han

Re-e*# Ho if #he

 p – value = P ( z  z > z

¿

Ate!"a ti#e H$% &t'e sis

Re-e*#

− z α  .

Re-e*#

 H  A : μ >

 H 0

if #he *omu#e% #es# s#a#is#i* is !ess #han

 H 0

if #he *omu#e%

Re-e*# Ho if #he

 p – value = P ( z  z < z

C!itica Rei&"

Re-e*#

is !ess #han α

+ z α .

 H  A : μ ≠

 x ´ =¿  sam!e mean

s =¿  sam!e s#an%ar% %e&ia#ion

Re-e*#

 H  A : μ <

is unknown)

√ 

is ea* ea*#! #!" " norm norma! a!!" !" %is# %is#ri ri+u +u#e #e% % If #he #he sam sam!i !in( n( %is#ri+u#ion is norma!, #he #es# is aroria#e for an" sam!e si$e

 H  A : μ >

σ   is no# known

 H o : μ = μo

is unknown, un!ess #he sam!in( ou!a#ion

Ate!"a ti#e H$% &t'e sis

t   DISTRIBUTION

%e#ermi %e#erminin nin( ( #he s#an%ar s#an%ar%i$e% %i$e% #es# s#a#is#i s#a#is#i* * when #he sam sam!i !in( n( %is# %is#ri ri+u +u#i #ion on of #he #he mean mean is nor norma!! ma!!" "

n =¿  sam!e si$e

when

#han

11.* TESTING A HYPOTHESI HYPOTHESIS S CONCERNIN CONCERNING G THE

 h"o#hesi$e% mean

σ =¿

is !ess #han α

2

 x´ =¿  sam!e mean

  where:

or

− z α  / / .

n ≥ 30  if

is unknown)  x ´ − μ0  x´ − μ 0  z = ∨ z = σ / n s/ n

σ 

2

¿

if 

#he *omu#e% #es# s#a#is#i* s#a#is#i* is (rea#er #han

Re-e*# Ho if #he

¿

 p – value = P ( t > t  ) is !ess #han α

+ t α .

)  H  A : μ <

is !ess #han α

Re-e*# Ho if #he

 p – value =2 P ( z  z >| z

 H 0

% – #a)e

¿

Re-e*#

 H 0

if 

#he *omu#e% #es# s#a#is#i* s#a#is#i* is !ess !ess #han #han

Re-e*# Ho if #he

 p – value = P ( t < t  ) ¿

is !ess #han α

Pa(e 1 of *

H YPOTHESIS TESTS

FOR A SINGLE MEAN

−t α  . Re-e*#

 H  A : μ ≠

 H 0

if 

#he *omu#e% #es# s#a#is#i* is (rea#er #han + t α /2 or !ess

Re-e*# Ho if #he

 p – value =2 P ( t >|t  | ¿

is !ess #han α

#han

−t α  / .

E+a,%e -: A manufa*#urer *on#em!a#in( #he ur*hase of new #oo! makin( e'uimen# has se*i9e% #ha#, on a&era(e, #he e'uimen# shou!% no# re'uire more #han /7min of se#u #ime er hour of oera#ion  The ur*hasin( a(en# &isi#s a *oman" where #he e'uimen# +ein( *onsi%ere% is ins#a!!e%@ from re*or%s #here #he a(en# no#es #ha# 06 ran%om!" se!e*#e% hours of oera#ion in*!u%e% a #o#a! of

---

2

¿

No#e. /)

t   is #he *omu#e% #es# s#a#is#i* 0)

(n – 1 )

t α /2

an% #he

 p1&a!ues

 %e(rees of free%om If

#he sam!e si$e is !ar(e 2 use% in !a*e of #he

are +ase% on

σ   is unknown +u#

n ≥ 30 ¿ , #he

 z 1#es# is

t  1#es#

E+a,%e -1:  34N5IN %onu#s *!aim #ha# #he wai#in( #ime of *us#omers for ser&i*e is norma!!" %is#ri+u#e% wi#h a mean of #hree minu#es an% a s#an%ar% %e&ia#ion of one minu#e The 'ua!i#" assuran*e %ear#men# foun% in a sam!e of 67 *us#omers #ha# #he mean wai#in( #ime is 086 minu#es A# a 776 !e&e! of si(ni9*an*e, *an we *on*!u%e #ha# #he mean wai#in( #ime is !ess #han #hree minu#es: E+a,%e -*: Home ;i%eos In* sur&e"s <67 househo!%s an% 9n%s #ha# #he mean amoun# sen# for ren#in( or +u"in( &i%eos is P/=6 a mon#h an% #he s#an%ar% %e&ia#ion of #he sam!e is P>606 Is #his e&i%en*e su?*ien# #o *on*!u%e #ha# #he mean amoun# sen# is (rea#er #han P/0>67 er mon#h a# a 7706 !e&e! of si(ni9*an*e:

E+a,%e -/: A #ea*hersB union wou!% !ike #o es#a+!ish #ha# #he a&era(e sa!ar" for hi(h s*hoo! #ea*hers in a ar#i*u!ar s#a#e is !ess #han C=0,677 A ran%om sam!e of /77 u+!i* hi(h s*hoo! #ea*hers in #he ar#i*u!ar s#a#e has a mean sa!ar" of C=/,6>8 I# is known from as# his#or" #ha# #he s#an%ar% %e&ia#ion of #he sa!aries for #he #ea*hers in #he s#a#e is C<, *oies er minu#e 2as a%&er#ise% in na#iona! +usiness ma(a$ines an% e!sewhere) Suose #ha# #he *oman" wan#s #o #es# whe#her #he new #wo1*o!or *oier has #he same a&era(e see% as i#s s#an%ar% *oma*# *oier an% i# *on%u*#s a #es# of 0< runs of #he new ma*hines, (i&in(  x ´ 0< an% sam!e s#an%ar% a sam!e mean of %e&ia#ion

s =7.4

si(ni9*an*e !e&e!

2*oies er minu#e) 4sin( #he

α =0.05 , is #here e&i%en*e #o

*on*!u%e #ha# #he a&era(e see% of #he new ma*hine is %ieren# from #he s#an%ar% ma*hine:

Pa(e * of *