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STATISTICAL INFERENCES FOR
MEANS AND VARIANCES
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Tests on Means with Variance Known
Hypothesis Test StatisticCriteria for
RejectionH0: µ µ0H!: µ ≠ µ0
"#0" $ #α%&
H0: µ µ0H!: µ ' µ0
#0 ' (#α
H0: µ µ0
H!: µ $ µ0
#0 $ #α
n y Z σ
00
−=
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E)a*+,eA -en.or s/*its ,ots o1 1aric to a te)ti,e
*an/1act/rer2 The *an/1act/rer wants to 3now i1the ,ot a-era4e
rea3in4 stren4th e)cee.s &00 +si2 i1so5 she wants to acce+t the
,ot2 6ast e)+eriencein.icates that a reasona,e -a,/e 1or the
-ariance o1
rea3in4 stren4th is !00 7+si8&2 The h9+otheses toe teste.
are
H0: µ &00
H!: µ $ &00Fo/r s+eci*ens are ran.o*,9 se,ecte.5 an.
thea-era4e rea3in4 stren4th oser-e. is &! +si2
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E)a*+,e &02!An asse*,9 +rocess has a stan.ar. ti*e o1
02;& *in2 A
trainee asse*,e. hat test is a++ro+riate to .eter*ine i1 the
a-era4e
2 6er1or* the tests 1or the two cases e,ow2
– H0: µ µ0 -ers/s H!: µ ≠ µ0 – H0: µ µ0
-ers/s H!: µ ' µ0
c2 Deter*ine the re?/ire. n -a,/e to ens/re that β 02&0 1or
the one(si.e. test2 >hat is β 1or the one(si.e. test ao-e@
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E)a*+,e &02
Two ?/a,it9 contro, en4ineers want to co*+are the
a-era4e stren4th o1 a +,astic *a.e 9 si*i,ar +rocessesat two
+,ants2 The +rocesses are we,, esta,ishe.5 an.
the σ -a,/es are 3nown2 The .ata is:
Are the *eans e?/a,@
Process n (lb/in2) σσσσ (lb/in2)
! = =
& !B ; ;
X
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Tests on Means o1 Nor*a,
Distri/tions with Variance n3nown
Hypothesis Test StatisticCriteria for
RejectionH0: µ µ0H!: µ ≠ µ0
"t0" $ tα%&5 n(!
H0: µ µ0H!: µ ' µ0
t0 ' (tα5 n(!
H0: µ µ0
H!: µ $ µ0
t0 $ tα5 n(!
nS yt 0
0 −=
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E)a*+,e !=2
A *eteoro,o4ist is st/.9in4 the ann/a, rain1a,, in
a .esert re4ion2 >ith a,, a-ai,a,e recor.s thea-era4e has
een
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Tests on Means with Variance
n3nown
Hypothesis Test StatisticCriteria for
Rejection
H0: µ! µ&H!: µ! ≠ µ& "t0" $ tα%&5 ν
11
21
21
0
+
−=
nnS
y yt
p
H0: µ! µ&H!: µ! ' µ& t0 ' (tα5 ν
H0: µ! µ&H!: µ! $ µ& t0 $ tα5 ν( ) ( )
11
2
2
2
2
2
2
1
2
1
2
1
2
2
2
2
1
2
1
2
2
2
1
2
1
21
0
21
−+
−
+
=
+
−=
−+=
n
nS
n
nS
n
S
n
S
n
S
n
S
y yt
nn
ν
ν
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Tests on Variance o1 Nor*a,
Distri/tion
Hypothesis Test Statistic Criteria for Rejection
H0: σ!& σ&
&
H!: σ!& ≠ σ&
& 1,1,20 21or−−> nnF F α
H0: σ!& σ&
&
H!: σ!& ' σ&
&
H0: σ!& σ&
&
H!: σ!& $ σ&
&
2
2
2
1
0S
S F =
1,1,21021
−−− nnα
1,1,0 12 −−> nnF F α
1,1,0 21 −−> nnF F α
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Co*+arin4 a sin4,e -ariance to a s+eci1ie.
-a,/e@@@
→ se c (s?/are str /t on
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6aire. Co*+arison
d nS
d t 0 =
iii
i
i
y yd
d n
d
21
1
−=
= ∑=
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E)a*+,e
Consi.er a har.ness testin4 *achine that
+resses a ro. with a +ointe. ti+ into a *eta,s+eci*en with a
3nown 1orce2 9 *eas/rin4 the.e+th o1 the .e+ression ca/se. 9 ti+5
the
har.ness o1 the s+eci*en is .eter*ine.2 Two.i11erent ti+s are
a-ai,a,e 1or this *achine5 an.a,tho/4h the +recision o1 the
*eas/re*ents
*a.e 9 the two ti+s see*s to e the sa*e5 it iss/s+ecte. that one
ti+ +ro./ces .i11erentrea.in4s than the other2
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E)a*+,eSpecimen Tip Tip 2
!&
<
B
;
;
B
<
=
!0
&=
;
&
=
;
