1. H YP O TH E S IS TE S TIN G

2. In th is s e s s ion … .
 Wh at is h yp oth e s is te s ting?
 Inte rp re ting and s e le cting s ignificance le ve l
P R O BABILITY
 Typ e I and Typ e II e rrors
 O ne taile d and twoTIO NsS
D IS TR IBU taile d te ts
 H yp oth e s is te s ts for p op u lation m e an
 H yp oth e s is te s ts for p op u lation p rop ortion
 H yp oth e s is te s ts for p op u lation s tand ard d e viation

3. Wh at is H yp oth e s is Te s ting?
H yp oth e s is te s ting re fe rs to
• M aking an as s u m p tion, calle d h yp oth e s is , ab ou t a
- the B-school
p op u lation p aram e te r.
• C olle cting s am p le d ata.
• C alcu lating a s am p le s tatis tic.
• U s ing th e s am p le s tatis tic to e valu ate th e h yp oth e s is (h ow
like ly is it th at ou r h yp oth e s ize d p aram e te r is corre ct. To
te s t th e valid ity of ou r as s u m p tion we d e te rm ine th e
d iffe re nce b e twe e n th e h yp oth e s ize d p aram e te r valu e and
th e s am p le valu e .)

4. H YP O T H E
S IS
T E S T IN G
N u ll h yp oth e s is , H 0 Alte rnative
h yp oth e s is ,H A
S tate th e h yp oth e s ize d valu e of All p os s ib le alte rnative s oth e r
th e p aram e te r b e fore s am p ling. th an th e nu ll h yp oth e s is .
Th e as s u m p tion we wis h to te s t E .g µ ≠ 20
(or th e as s u m p tion we are trying
to re j ct)
e µ > 20
E .g p op u lation m e an µ = 20 µ < 20
 Th e re is no d iffe re nce b e twe e n Th e re is a d iffe re nce b e twe e n
coke and d ie t coke coke and d ie t coke

5. N u ll H yp oth e s is
Th e nu ll h yp oth e s is H 0 re p re s e nts a th e ory th at h as b e e n
- the B-school
p u t forward e ith e r b e cau s e it is b e lie ve d to b e tru e or
b e cau s e it is u s e d as a b as is for an argu m e nt and h as
not b e e n p rove n. F or e xam p le , in a clinical trial of a ne w
d ru g, th e nu ll h yp oth e s is m igh t b e th at th e ne w d ru g is no
b e tte r, on ave rage , th an th e cu rre nt d ru g. We wou ld write
H 0: th e re is no d iffe re nce b e twe e n th e two d ru gs on an
ave rage .

6. Alte rnative H yp oth e s is
Th e alte rnative h yp oth e s is , H A, is a s tate m e nt of wh at a
s tatis tical h yp oth e s is te s t is s e t u p to e s tab lis h . F or e xam p le ,
- the B-school
in th e clinical trial of a ne w d ru g, th e alte rnative h yp oth e s is
m igh t b e th at th e ne w d ru g h as a d iffe re nt e ffe ct, on ave rage ,
com p are d to th at of th e cu rre nt d ru g. We wou ld write
H A: th e two d ru gs h ave d iffe re nt e ffe cts , on ave rage .
or
H A: th e ne w d ru g is b e tte r th an th e cu rre nt d ru g, on ave rage .
Th e re s u lt of a h yp oth e s is te s t:
‘R e j ct H 0 in favou r of H A’ O R ‘D o not re j ct H 0’
e e

7. S e le cting and inte rp re ting s ignificance
le ve l
• D e cid ing on a crite rion for acce p ting or re j cting th e nu ll
e
h yp oth e s is .
- the B-school
• S ignificance le ve l re fe rs to th e p e rce ntage of s am p le m e ans
th at is ou ts id e ce rtain p re s crib e d lim its . E .g te s ting a
h yp oth e s is at 5% le ve l of s ignificance m e ans
 th at we re j ct th e nu ll h yp oth e s is if it falls in th e two re gions
e
of are a 0.025.
 D o not re j ct th e nu ll h yp oth e s is if it falls with in th e re gion of
e
are a 0.95.
5. Th e h igh e r th e le ve l of s ignificance , th e h igh e r is th e
p rob ab ility of re j cting th e nu ll h yp oth e s is wh e n it is tru e .
e
(acce p tance re gion narrows )

8. Typ e I and Typ e II E rrors
• Typ e I e rror re fe rs to th e s itu ation wh e n we re j ct th e nu ll
e
h yp oth e s is wh e n it is tru e (H 0 is wrongly re j cte d ).
e
- the B-school
e .g H 0: th e re is no d iffe re nce b e twe e n th e two d ru gs on
ave rage .
Typ e I e rror will occu r if we conclu d e th at th e two d ru gs
p rod u ce d iffe re nt e ffe cts wh e n actu ally th e re is n’t a d iffe re nce .
P rob (Typ e I e rror) = s ignificance le ve l = α
2. Typ e II e rror re fe rs to th e s itu ation wh e n we acce p t th e nu ll
h yp oth e s is wh e n it is fals e .
H 0: th e re is no d iffe re nce b e twe e n th e two d ru gs on ave rage .
Typ e II e rror will occu r if we conclu d e th at th e two d ru gs
p rod u ce th e s am e e ffe ct wh e n actu ally th e re is a d iffe re nce .
P rob (Typ e II e rror) = ß

9. Typ e I and Typ e II E rrors – E xam p le
You r nu ll h yp oth e s is is th at th e b atte ry for a h e art
p ace m ake r h as an ave rage life of 300 d ays , with th e
- the B-school
alte rnative h yp oth e s is th at th e ave rage life is m ore th an
300 d ays . You are th e qu ality control m anage r for th e
b atte ry m anu factu re r.
(b)Wou ld you rath e r m ake a Typ e I e rror or a Typ e II e rror?
(c)Bas e d on you r ans we r to p art (a), s h ou ld you u s e a h igh
or low s ignificance le ve l?

10. Typ e I and Typ e II E rrors – E xam p le
G ive n H 0 : ave rage life of p ace m ake r = 300 d ays , and
H A: Ave rage life of p ace m ake r > 300 d ays
- the eB-school is fals e i.e
(b)It is b e tte r to m ake a Typ e II rror (wh e re H 0
ave rage life is actu ally m ore th an 300 d ays b u t we acce p t
H 0 and as s u m e th at th e ave rage life is e qu al to 300 d ays )
(c)As we incre as e th e s ignificance le ve l (α) we incre as e th e
ch ance s of m aking a typ e I e rror. S ince h e re it is b e tte r to
m ake a typ e II e rror we s h all ch oos e a low α.

11. Two Tail Te s t
Two taile d te s t will re j ct th e nu ll h yp oth e s is if th e s am p le
e
m e an is s ignificantly h igh e r or lowe r th an th e h yp oth e s ize d
- the B-school
m e an. Ap p rop riate wh e n H 0 : µ = µ0 and H A: µ ≠ µ0
e .g Th e m anu factu re r of ligh t b u lb s wants to p rod u ce ligh t
b u lb s with a m e an life of 1 000 h ou rs . If th e life tim e is s h orte r
h e will los e cu s tom e rs to th e com p e tition and if it is longe r
th e n h e will incu r a h igh cos t of p rod u ction. H e d oe s not want
to d e viate s ignificantly from 1 000 h ou rs in e ith e r d ire ction.
Th u s h e s e le cts th e h yp oth e s e s as
H 0 : µ = 1 000 h ou rs and H A: µ ≠ 1 000 h ou rs
and u s e s a two tail te s t.

12. O ne Tail Te s t
A one -s id e d te s t is a s tatis tical h yp oth e s is te s t in wh ich th e
valu e s for wh ich we can re j ct th e nu ll h yp oth e s is , H 0 are
e
- the B-school
locate d e ntire ly in one tail of th e p rob ab ility d is trib u tion.
Lowe r taile d te s t will re j ct th e nu ll h yp oth e s is if th e s am p le
e
m e an is s ignificantly lowe r th an th e h yp oth e s ize d m e an.
Ap p rop riate wh e n H 0 : µ = µ0 and H A: µ < µ0
e .g A wh ole s ale r b u ys ligh t b u lb s from th e m anu factu re r in
large lots and d e cid e s not to acce p t a lot u nle s s th e m e an life
is at le as t 1 000 h ou rs .
H 0 : µ = 1 000 h ou rs and H A: µ < 1 000 h ou rs
and u s e s a lowe r tail te s t.
i.e h e re j cts H 0 only if th e m e an life of s am p le d b u lb s is
e
s ignificantly b e low 1 000 h ou rs . (h e acce p ts H A and re j cts th e
e
lot)

13. O ne Tail Te s t
U p p e r taile d te s t will re j ct th e nu ll h yp oth e s is if th e s am p le
e
m e an is s ignificantly h igh e r th an th e h yp oth e s ize d m e an.
- the B-school
Ap p rop riate wh e n H 0 : µ = µ0 and H A: µ > µ0
e .g A h igh way s afe ty e ngine e r d e cid e s to te s t th e load
b e aring cap acity of a 20 ye ar old b rid ge . Th e m inim u m load -
b e aring cap acity of th e b rid ge m u s t b e at le as t 1 0 tons .
H 0 : µ = 1 0 tons and H A: µ > 1 0 tons
and u s e s an u p p e r tail te s t.
i.e h e re j cts H 0 only if th e m e an load b e aring cap acity of th e
e
b rid ge is s ignificantly h igh e r th an 1 0 tons .

14. H yp oth e s is te s t for p op u lation m e an
n ( x − µ0 )
H 0 : µ = µ0 and Te s t s tatis tic =
∆
s
e - the B-school
F or H A: µ > µ0, re j ct H 0 if > t n −1,α
∆
F or H A: µ < µ0, re j ct H 0 if < −t n −1,α
e ∆
∆ > t n −1,α 2
F or H A: µ ≠ µ0, re j ct H 0 if
e
t n −1,α by zα
F or n ≥ 30, re p lace

15. H yp oth e s is te s t for p op u lation m e an
A we igh t re d u cing p rogram th at inclu d e s a s trict d ie t and
- the B-school
e xe rcis e claim s on its online ad ve rtis e m e nt th at it can h e lp an
ave rage ove rwe igh t p e rs on los e 1 0 p ou nd s in th re e m onth s .
F ollowing th e p rogram ’s m e th od a grou p of twe lve ove rwe igh t
p e rs ons h ave los t 8.1 5.7 1 1 .6 1 2.9 3.8 5.9 7.8 9.1
7.0 8.2 9.3 and 8.0 p ou nd s in th re e m onth s . Te s t at 5%
le ve l of s ignificance wh e th e r th e p rogram ’s ad ve rtis e m e nt is
ove rs tating th e re ality.

16. H yp oth e s is te s t for p op u lation m e an
S olu tion:
H 0: µ = 1 0 (µ0) H A: µ < 1 0 (µ0)
n = 1 2, x(b ar) = 8.027, s = 2.536, α = 0.05
12(8.075 − 10) 3.46 × −1.925
∆= = = −2.62
2.536 2.536
C ritical t-valu e = -tn-1 ,α= - t1 1 ,0.05 = -2. 201 (TIN V)
S ince ∆ < -tn-1 ,α we re j ct H 0 and conclu d e th at th e
e
p rogram is ove rs tating th e re ality.
(Wh at h ap p e ns if we take α = 0.01 ? Is th e p rogram
ove rs tating th e re ality at 1 % s ignificance le ve l?)

17. H yp oth e s is te s t for p op u lation p rop ortion
n ( p − p0 )
ˆ
H 0 : p = p 0 and Te s t s tatis tic =
∆
p0 (1 − p0 )
F or H A: p > p 0 re j ct H
e
-if∆ > z B-school
the α
0
F or H A: p < p 0 re j ct H 0 if∆ < −zα
e
∆ > zα 2
F or H A: p ≠ p 0 re j ct H 0 if
e

18. H yp oth e s is te s t for p op u lation p rop ortion
A ke tch u p m anu factu re r is in th e p roce s s of d e cid ing wh e th e r
- the B-school
to p rod u ce an e xtra s p icy b rand . Th e com p any’s m arke ting
re s e arch d e p artm e nt u s e d a national te le p h one s u rve y of
6000 h ou s e h old s and fou nd th e e xtra s p icy ke tch u p wou ld b e
p u rch as e d b y 335 of th e m . A m u ch m ore e xte ns ive s tu d y
m ad e two ye ars ago s h owe d th at 5% of th e h ou s e h old s
wou ld p u rch as e th e b rand th e n. At a 2% s ignificance le ve l,
s h ou ld th e com p any conclu d e th at th e re is an incre as e d
inte re s t in th e e xtra-s p icy flavor?

19. H yp oth e s is te s t for p op u lation p rop ortion
335
n = 6000, p=
ˆ = 0.05583
6000
- the B-school
H0 : p = 0.05( p0 ) H A : p > 0.05
n ( p − p0 )
ˆ 6000 × 0.00583
∆= =
p0 (1 − p0 ) 0.05 × 0.95
77.459 × 0.00583
= = 2.072
0.218
α = 0.02
Zα (the critical value of Z ) = 2.05 (N O R M S IN V
Q ∆ > Z we reject H i.e th e cu rre nt inte re s t is s ignificantly gre ate r
α 0
)
th an th e inte re s t of two ye ars ago.

20. H yp oth e s is te s t for p op u lation s tand ard
d e viation
(n − 1)s 2
H 0 : σ = σ 0 and Te s t s tatis tic ∆ =
σ 02
- the B-school
F or H A: σ > σ0 re j ct H 0 if
e ∆ > χ(2(−1),α
n
R)
F or H A: σ < σ0 re j ct H 0 if
e ∆ < χ(2(−1),1−α
n
R)
∆ < χ(2(−1),1−α 2
R)
∆ > χ(2(−1),α 2
R)
F or H A: σ ≠ σ0 re j ct H 0 if
e n
or n

21. H yp oth e s is te s t for com p aring two p op u lation
m e ans
C ons id e r two p op u lations with m e ans µ1 , µ2 and s tand ard d e viations σ1
and= σ12 .and µ x = µ2
µx µ
1 2
σx
am σ x
p op u lation2 re s p e ctive ly. - the B-school
are th e m e ans of th e sand p ling d is trib u tions of p op u lation1 and
1 2
d e note th e s tand ard e rrors of th e2
σ1 σ 22
µam xp2 ling d is trib u tions of th e m e ans .
sx− σ X −X = +
1 1 2
n1 n2
is th e m e an of th e d iffe re nce b e twe e n s am p le m e ans and
( X1 − X ) − ( µ − µ2 )H
is th e corre s p ond ing = tand ard 2e rror.1
∆ s 0
σ x −x
H 0 : µ1 = µ2 and te s t s tatis tic, 1 2
H e re ∆ d e note s th e s tand ard ize d
F or H A: µ1 > µ2 re j ct H 0 if ∆ > Z α
e d iffe re nce of s am p le m e ans
F or H A: µ1 < µ2 re j ct H 0 if∆ > - α 2
e ∆ < ZZ α
F or H A: µ1 ≠ µ2 re j ct H 0 if
e
(d e cis ion m ake rs m ay b e conce rne d with p aram e te rs of two p op u lations
e .g d o fe m ale e m p loye e s re ce ive lowe r s alary th an th e ir m ale

22. H yp oth e s is te s t for com p aring p op u lation
m e ans
A s am p le of 32 m one y m arke t m u tu al fu nd s was ch os e n on
- the B-school
Janu ary 1 , 1 996 and th e ave rage annu al rate of re tu rn ove r
th e p as t 30 d ays was fou nd to b e 3.23% and th e s am p le
s tand ard d e viation was 0.51 % . A ye ar e arlie r a s am p le of 38
m one y-m arke t fu nd s s h owe d an ave rage rate of re tu rn of
4.36% and th e s am p le s tand ard d e viation was 0.84% . Is it
re as onab le to conclu d e (at α = 0.05) th at m one y-m arke t
inte re s t rate s d e cline d d u ring 1 995?

23. H yp oth e s is te s t for com p aring p op u lation
m e ans
n1 = 32, x1 = 3.23, σ 1 = 0.51 n2 = 38, x2 = 4.36, σ 2 = 0.84
H0 : µ1 = µ2 H A : µ1 < µ2
σ X1 − X 2 =
σ 12 σ 22
+ =
0.26 0.71
+
- the B-school
= 0.026 = 0.163
n1 n2 32 38
( x1 − x2 ) − ( µ1 − µ2 )H0 −1.13 − 0
∆= = = −6.92
σ X1 − X 2 0.163
α = 0.05
Critical value of Z = −Zα = −1.64
Q ∆ < −Zα we reject H0 and conclude that there has
been a decline.

24. H yp oth e s is te s t for com p aring p op u lation
p rop ortions
p1
C ons id e r two s am p le s of s ize s n 1 and n 2 with p2
and as th e re s p e ctive
p rop ortions of s u cce s s e s . Th e n
p=
ˆ
n1p1 + n2 p2
n1 + n2
- the B-school
is th e e s tim ate d ove rall p rop ortion of s u cce s s e s in th e
two p op u lations .
ˆˆ ˆˆ
pq pq is th e e s tim ate d s tand ard e rror of th e d iffe re nce
σ p1 − p2 =
ˆ +
n1 n2 b e twe e n th e two p rop ortions .
( p1 − p2 ) − ( p1 − p2 )H0
H 0 : p 1 = p 2 and te s t s tatis tic,∆ = σ x1 −x2
ˆ
F or H A: p 1 > p 2 re j ct H 0 if ∆ > Z α
e A training d ire ctor m ay wis h to
F or H A: p 1 < p 2 re j ct H 0 if ∆ < - Z α
e d e te rm ine if th e p rop ortion of
∆ > Zα 2 p rom otab le e m p loye e s at one office
F or H A: p 1 ≠ p 2 re j ct H 0 if
e is d iffe re nt from th at of anoth e r.

25. H yp oth e s is te s t for com p aring p op u lation
p rop ortions
A large h ote l ch ain is trying to d e cid e wh e th e r to conve rt
- the B-school
m ore of its room s into non-s m oking room s . In a rand om
s am p le of 400 gu e s ts las t ye ar, 1 66 h ad re qu e s te d non-
s m oking room s . Th is ye ar 205 gu e s ts in a s am p le of 380
p re fe rre d th e non-s m oking room s . Wou ld you re com m e nd
th at th e h ote l ch ain conve rt m ore room s to non-s m oking?
S u p p ort you r re com m e nd ation b y te s ting th e ap p rop riate
h yp oth e s e s at a 0.01 le ve l of s ignificance .

26. H yp oth e s is te s t for com p aring p op u lation
p rop ortions
166 205
n1 = 400, p1 = = 0.415, n2 = 380, p2 = = 0.5395
400 380
H0 : p1 = p2 H A : p1 < p2
p=
ˆ =
- the B-school
n1p1 + n2 p2 400 × 0.415 + 380 × 0.5395 (P rop ortion of s u cce s s
= 0.4757 in th e two p op u lations )
n1 + n2 400 + 380
1 1  1 1 
σ p1 − p2 = pq  +  = 0.4757 × 0.5243 
ˆ ˆˆ +  = 0.0358
 n1 n2   400 380 
α = 0.01 Th e h ote l ch ain s h ou ld
Critical value of Z = −Zα = −2.32 conve rt m ore room s to
( p1 − p2 ) − ( p1 − p2 )H0 −0.1245 − 0 non-s m oking room s as
∆= = = −3.48 th e re h as b e e n a
σ p1 − p2
ˆˆ ˆ 0.0358 s ignificant incre as e in th e
Q ∆ < −Zα we reject H0 nu m b e r of gu e s ts s e e king
non-s m oking room s .