02 Bài Giảng Xác Suất Thống Kê

Xc sut &Thng k

Nguyn c Phng

TP. HCM, Ngy 2 thng 9 nm 2014

Bi

gin

g

H v tn:

Mssv:

Mc lc

Mc lc i

1 Bin c, xc sut ca bin c 1

1.1 Php th, bin c . . . . . . . . . . . . . . . . . . . . . . . 11.2 Quan h gia cc bin c . . . . . . . . . . . . . . . . . . . 21.3 nh ngha xc sut . . . . . . . . . . . . . . . . . . . . . 41.4 Xc sut c iu kin, s c lp . . . . . . . . . . . . . . 5

1.4.1 Xc sut c iu kin . . . . . . . . . . . . . . . . . 51.4.2 S c lp ca hai bin c . . . . . . . . . . . . . . 8

1.5 Cc cng thc tnh xc sut . . . . . . . . . . . . . . . . . 91.5.1 Cng thc cng . . . . . . . . . . . . . . . . . . . . 91.5.2 Cng thc nhn . . . . . . . . . . . . . . . . . . . . 101.5.3 Cng thc xc sut y . . . . . . . . . . . . . . 141.5.4 Cng thc xc sut Bayes . . . . . . . . . . . . . . 15

1.6 Bi tp chng 1 . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Bin ngu nhin 26

2.1 Khi nim bin ngu nhin . . . . . . . . . . . . . . . . . 262.2 Phn phi xc sut ca bin ngu nhin . . . . . . . . . . 27

2.2.1 X l bin ngu nhin ri rc . . . . . . . . . . . . . 272.2.2 X l bin ngu nhin lin tc . . . . . . . . . . . . 302.2.3 Hm phn phi xc sut . . . . . . . . . . . . . . . 31

2.3 Cc c trng s ca bin ngu nhin . . . . . . . . . . . 352.3.1 K vng - EX . . . . . . . . . . . . . . . . . . . . . . 35

Trang ii Mc lc

2.3.2 Phng sai - VarX . . . . . . . . . . . . . . . . . . 372.3.3 ModX . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 Bi tp chng 2 . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Mt s phn phi xc sut thng dng 49

3.1 Phn phi Bernoulli . . . . . . . . . . . . . . . . . . . . . . 493.2 Phn phi nh thc . . . . . . . . . . . . . . . . . . . . . . 503.3 Phn phi siu bi . . . . . . . . . . . . . . . . . . . . . . . 523.4 Phn phi Poisson . . . . . . . . . . . . . . . . . . . . . . . 543.5 Phn phi chun . . . . . . . . . . . . . . . . . . . . . . . . 563.6 Bi tp chng 3 . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Lut s ln v cc nh l gii hn 69

4.1 Hi t theo xc sut v phn phi . . . . . . . . . . . . . . 694.2 Bt ng thc Markov, Chebyshev . . . . . . . . . . . . . 70

4.2.1 Bt ng thc Markov . . . . . . . . . . . . . . . . 704.2.2 Bt ng thc Chebyshev . . . . . . . . . . . . . . 70

4.3 Lut s ln . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.4 nh l gii hn trung tm . . . . . . . . . . . . . . . . . . 724.5 Lin h gia cc phn phi xc sut . . . . . . . . . . . . 73

4.5.1 Lin h gia phn phi nh thc v chun . . . . . 734.5.2 Lin h gia nh thc v Poisson . . . . . . . . . . 744.5.3 Lin h gia siu bi v nh thc . . . . . . . . . . 75

5 Vct ngu nhin 77

5.1 Khi nim vct ngu nhin . . . . . . . . . . . . . . . . . 775.2 Phn phi xc sut ca .X; Y / . . . . . . . . . . . . . . . . 77

5.2.1 .X; Y / l vct ngu nhin ri rc . . . . . . . . . . 775.2.2 .X; Y / l vct ngu nhin lin tc . . . . . . . . . 81

5.3 Bi tp chng 5 . . . . . . . . . . . . . . . . . . . . . . . . 86

6 L thuyt mu 92

Mc lc Trang iii

6.1 Tng th, mu . . . . . . . . . . . . . . . . . . . . . . . . . 926.2 M t d liu . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2.1 Phn loi mu ngu nhin . . . . . . . . . . . . . . 936.2.2 Sp xp s liu . . . . . . . . . . . . . . . . . . . . . 93

6.3 Cc c trng ca mu . . . . . . . . . . . . . . . . . . . . 956.3.1 Trung bnh mu . . . . . . . . . . . . . . . . . . . . 956.3.2 Phng sai mu . . . . . . . . . . . . . . . . . . . . 966.3.3 Phng sai mu c hiu chnh . . . . . . . . . . . . 96

7 c lng tham s 101

7.1 Khi nim chung . . . . . . . . . . . . . . . . . . . . . . . 1017.2 c lng im . . . . . . . . . . . . . . . . . . . . . . . . 1017.3 Khong tin cy . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.1 M t phng php. . . . . . . . . . . . . . . . . . . 1037.3.2 Khong tin cy cho trung bnh . . . . . . . . . . . 1037.3.3 Khong tin cy cho t l . . . . . . . . . . . . . . . 106

7.4 Bi tp chng 7 . . . . . . . . . . . . . . . . . . . . . . . . 108

8 Kim nh gi thit 111

8.1 Bi ton kim nh gi thit . . . . . . . . . . . . . . . . . 1118.1.1 Gi thit khng, i thit . . . . . . . . . . . . . . . 1118.1.2 Min ti hn . . . . . . . . . . . . . . . . . . . . . . 1128.1.3 Hai loi sai lm . . . . . . . . . . . . . . . . . . . . 1128.1.4 Phng php chn min ti hn . . . . . . . . . . . 113

8.2 Kim nh gi thit v trung bnh . . . . . . . . . . . . . . 1138.3 Kim nh gi thit v t l . . . . . . . . . . . . . . . . . 1158.4 So snh hai gi tr trung bnh . . . . . . . . . . . . . . . . 1168.5 So snh hai t l . . . . . . . . . . . . . . . . . . . . . . . . 1188.6 Bi tp chng 8 . . . . . . . . . . . . . . . . . . . . . . . . 120

A Cc bng gi tr xc sut 135

Trang iv Mc lc

A.1 Bng gi tr f .z/ . . . . . . . . . . . . . . . . . . . . . . . . 136A.2 Bng gi tr '.x/ . . . . . . . . . . . . . . . . . . . . . . . 138A.3 Bng gi tr tn . . . . . . . . . . . . . . . . . . . . . . . . . 140

Ti liu tham kho 142

Chng 1

Bin c, xc sut ca bin c

Mc lc chng 11.1 Php th, bin c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Quan h gia cc bin c . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 nh ngha xc sut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Xc sut c iu kin, s c lp . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Cc cng thc tnh xc sut . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Bi tp chng 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.1 Php th, bin c

- Php th l vic thc hin mt th nghim hoc quan st mt hintng no . Php th c gi l ngu nhin nu ta khng th d botrc chnh xc kt qu no s xy ra.

- Mi kt qu ca php th, ! c gi l mt bin c s cp.

V d 1.1. Thc hin php th tung mt ng xu. C hai kt qu c thxy ra khi tung ng xu l xut hin mt sp-S hoc mt nga-N:

Kt qu ! D S l mt bin c s cp. Kt qu ! D N l mt bin c s cp.

- Tp hp tt c cc kt qu, ! c th xy ra khi thc hin php thgi l khng gian cc bin c s cp, k hiu l .

V d 1.2. Tung ngu nhin mt con xc sc. Quan st s chm trnmt xut hin ca xc sc, ta c 6 kt qu c th xy ra l:1, 2, 3, 4,

Chng 1. Bin c, xc sut ca bin c

5, 6. Khng gian cc bin c s cp, D f1; 2; 3; 4; 5; 6g. S phn t ca, jj D 6:

- Mi tp con ca khng gian cc bin c s cp gi l bin c.

V d 1.3. Thc hin php th tung mt xc sc. Ta bit Df1; 2; 3; 4; 5; 6g

t A D f2; 4; 6g , A gi l bin c S chm trn mt xut hinl s chn. Thay v lit k cc phn t ca A, ta t tn cho A

A: S chm trn mt xut hin l s chn

Ngc li, nu ta gi bin c:

B: S chm trn mt xut hin ln hn 4

th khi B D f5; 6g

- Xt bin c A, khi thc hin php th ta c kt qu !.

Nu trong ln th ny kt qu ! 2 A ta ni bin c A xy ra. Ngc li nu trong ln th ny kt qu ! A ta ni bin c Akhng xy ra.

V d 1.4. Mt sinh vin thi kt thc mn xc sut thng k.A : Sinh vin ny thi t A D f4I : : : I 10g

Gi s sinh vin ny i thi c kt qu ! D 6 2 A lc ny ta nibin c A xy ra (Sinh vin ny thi t).

Ngc li nu sinh vin ny thi c kt qu ! D 2 A th ta nibin c A khng xy ra (Sinh vin ny thi khng t).

1.2 Quan h gia cc bin c

a) Quan h ko theo .A B/ W Nu bin c A xy ra th ko theo bin cB xy ra.

1.2 Quan h gia cc bin c Trang 3

V d 1.5. Theo di 3 bnh nhn phng ang c iu tr. Gi cc binc:

Ai : C i bnh nhn t vong, i D 0; 1; 2; 3B : C nhiu hn mt bnh nhn t vong

Ta c A2 B, A3 B, A1 6 B

b) Hai bin c A v B c gi l bng nhau nu A B v B A, khiu A D B.

c) Bin c tng AC B .A [ B/ xy ra khi v ch khi A xy ra hoc Bxy ra trong mt php th. (t nht mt trong hai bin c xy ra)

V d 1.6. Hai x th cng bn vo mt mc tiu, mi ngi bn mtpht. Gi cc bin c:

A : Ngi th nht bn trung mc tiuB : Ngi th hai bn trng mc tiu

Bin c AC B: C it nht mt ngi bn trng mc tiu

d) Bin c tch AB .A\B/ xy ra khi v ch khi c hai bin c A v Bcng xy ra trong mt php th.

V d 1.7. Mt sinh vin thi kt thc 2 mn hoc. Gi cc bin c:A : Sinh vin thi t mn th nhtB : Sinh vin thi t mn th hai

Bin c AB: Sinh vin thi t c hai mn

e) Hai bin c A v B gi l xung khc nu chng khng cng xy ratrong mt php th .AB D ;/.

f) Bin c khng th: l bin c khng bao gi xy ra khi thc hinphp th, k hiu ;.

g) Bin c chc chn: l bin c lun xy ra khi thc hin php th,k hiu .

h) Bin c NA c gi l bin c b ca bin c A hay ngc li khi vch khi (

A \ NA D ;A [ NA D


1.3 nh ngha xc sut

nh ngha 1.1 (nh ngha c in). Xt mt php th ng kh nng,c khng gian cc bin c s cp

D f!1; !2; : : : ; !ng ; jj D n

1.4 Xc sut c iu kin, s c lp Trang 5

a) Hai bi trng.

b) t nht mt bi trng.

Gii.

Ch : Trong cu b), chng ta tnh xc sut ca bin c b s ngin hn. Ta c

NB W Ly c khng bi trng

P .B/ D 1 P NB D 1 C 04C 36C 310

1.4 Xc sut c iu kin, s c lp

1.4.1 Xc sut c iu kin

nh ngha 1.3 (Xc sut c iu kin). P .AjB/ l xc sut xy ra binc A bit rng bin c B xy ra (P .B/ > 0).

V d 1.11. Mt l c 4 vin bi trng v 6 vin bi en. T l ny ly lnlt ra 2 vin bi, mi ln ly mt bi (ly khng hon li). Tm xc sut ln ly th hai c vin bi trng bit ln ly th nht ly cvin bi trng.

Gii. 4 bi trng6 bi en

B xy ra! ly 1 bi trng

3 bi trng6 bi en


V d 1.12. T mt b bi ty (4 cht, 52 l), rt ngu nhin ra 2 l.Tnh xc sut:

a) Rt c hai l bi c.

b) Rt c 2 l bi c bit rng 2 l bi ny mu .

Gii.

V d 1.13. Mt nhm 100 ngi c:

+ 20 ngi ht thuc.

+ 30 n, trong c 5 ngi ht thuc.

1.4 Xc sut c iu kin, s c lp Trang 7

Chn ngu nhin mt ngi trong nhm 100 ngi ny. Tnh xcsut:

a. Ngi ny ht thuc bit rng ngi ny l n.b. Ngi ny l n bit rng ngi ny ht thuc.

30 n 20 ngi ht thuc

5 n ht thuc

Gii.

Cng thc xc sut iu kin

P .AjB/ D P .AB/P .B/

; P .B/ > 0

Tnh cht 1.4. Xc sut c iu kin c cc tnh cht:

i. 0 P .AjB/ 1 vi mi bin c A.ii. Nu A A0 th P .AjB/ P .A0jB/.iii. P .AjB/ D 1 P NAjB :

V d 1.14. Mt cng ty cn tuyn 4 nhn vin. C 10 ngi np nd tuyn, trong c 4 n (kh nng trng tuyn ca cc ng c vinl nh nhau). Tnh xc sut:


a) C 4 n trng tuyn.

b) C t nht mt n trng tuyn.

c) C 4 n trng tuyn, bit rng c t nht mt n trng tuyn.

Gii.

1.4.2 S c lp ca hai bin c

nh ngha 1.5 (S c lp). A v B l hai bin c c lp nu B c xyra hay khng cng khng nh hng n kh nng xy ra A v ngcli, ngha l:

P .AjB/ D P Aj NBTnh cht 1.6. Nu A v B c lp th

i. P .AjB/ D P .A/ v P .BjA/ D P .B/ii. A v NB; NA v B; NA v NB c lp.

V d 1.15. Tung mt xc sc 2 ln. Gi cc bin c:A : Ln 1 xut hin mt 6 chmB : Ln 2 xut hin mt 6 chm

Hai bin c A v B c c lp?

1.5 Cc cng thc tnh xc sut Trang 9

Gii.

V d 1.16. Mt l ng 4 bi trng v 6 bi en, thc hin hai ln ly bi.Mi ln ly 1 bi (ly khng hon li). t cc bin c:

A : Ln 1 ly c bi enB : Ln 2 ly c bi trng

Hai bin c A v B c c lp?

Gii.

1.5 Cc cng thc tnh xc sut

1.5.1 Cng thc cng

P .AC B/ D P .A/C P .B/ P .AB/Ch : Nu A v B xung khc .AB D ;/ th

P .AC B/ D P .A/C P .B/


V d 1.17. Mt lp hc c 20 hc sinh trong c 10 hc sinh gii ton,8 hc sinh gii vn v 6 hc sinh gii c ton v vn. Chn ngu nhinmt hc sinh, tnh xc sut hc sinh ny gii t nht mt mn.

Gii.

Cng thc cng 3 bin c:

P .AC B C C/ DP .A/C P .B/C P .C / P .AB/ P .AC/ P .BC/C P .ABC/

Ch : Nu A;B;C xung khc tng i mt th

P .AC B C C/ D P .A/C P .B/C P .C /

1.5.2 Cng thc nhn

P .AB/ D P .A/P .BjA/ D P .B/P .AjB/Ch : Nu A v B c lp th P .AB/ D P .A/P .B/

M rng cng thc nhn: Cho n bin c A1; A2; : : : ; An

P .A1A2 : : : An/ D P .A1/P .A2jA1/ : : :P .AnjA1A2 : : : An1/Ch : Nu Ai ; i D 1; : : : ; n c lp ton b th

P .A1 : : : An/ D P .A1/ : : :P .An/V d 1.18. Mt ngi c 4 con g mi, 6 con g trng nht trong mtlng. Hai ngi n mua (ngi th nht mua xong ri n lt ngi


th hai mua, mi ngi mua 2 con) v ngi bn bt ngu nhin tlng. Tnh xc sut ngi th nht mua c mt g trng v ngi thhai mua hai g trng.

Gii.

V d 1.19. Trong mt k thi, mi sinh vin phi thi 2 mn. Mt sinhvin A c lng rng: xc sut t mn th nht l 0,8. Nu t mnth nht th xc sut t mn th hai l 0,6; nu khng t mn thnht th xc sut t mn th hai l 0,3. Tnh xc sut sinh vin A:

a. t mn th hai.b. t i mn, i D 0; 1; 2:c. t t nht mt mn.d. t mn th hai bit rng sinh vin ny t mt mn.e. t mn th hai bit rng sinh vin ny t t nht mt mn.

Gii.


V d 1.20. Mt ngi c 3 con g mi, xc sut trng trong ngyca con g I, II, III ln lt l 0,4; 0,7; 0,8. Tnh xc sut:

a) C i con g trng trong ngy, i D 0; 1; 2; 3:

b) C t nht 1 con g trng trong ngy.

c) C nhiu nht 2 con g trng trong ngy.


d) Con g th I trng trong ngy bit rng trong ngy c 1 con trng.

e) Con g th I trng trong ngy bit rng trong ngy c t nht1 con trng.

f) Con g th I trng trong ngy bit rng trong ngy c nhiunht 2 con trng.

Gii.


1.5.3 Cng thc xc sut y

nh ngha 1.7 (H y ). n bin c A1; A2; : : : ; An c gi l h y nu chng xung khc tng i mt v lun c t nht mt bin c xyra trong mt php th. Ngha l(

Ai \ Aj D ;; 8i jA1 C A2 C C An D

V d 1.21. T mt l c 4 bi trng v 6 bi en ly ra 2 bi.A0 : Ly c 0 bi enA1 : Ly c 1 bi enA2 : Ly c 2 bi en

Khi A0IA1IA2 l h y .

Cng thc xc sut y : Cho A1IA2I : : : IAn (P .Ai/ > 0 ) l hy cc bin c v B l mt bin c bt k. Xc sut xy ra bin c B

P .B/ D P .A1/P .BjA1/C P .A2/P .BjA2/C C P .An/P .BjAn/V d 1.22. Mt m ng c s n ng bng na s n b. Xc sut n ng b bnh tim l 0,06 v n b l 0,036. Chn ngu nhin 1ngi t m ng, tnh xc sut ngi ny b bnh tim.

Gii.


1.5.4 Cng thc xc sut Bayes

Gi thit ging cng thc xc sut y . Xc sut:

P .Ai jB/ D P .AiB/P .B/

D P .Ai/P .BjAi /P .B/

; i D 1; 2; : : : ; n

V d 1.23. Mt lp c s hc sinh nam bng 3 ln s hc sinh n. Tl hc sinh n gii ton l 30% v t l hc sinh nam gii ton l 40%.Chn ngu nhin mt hc sinh trong lp ny. Tnh xc sut:

a. Hc sinh ny gii ton.b. Hc sinh ny l nam bit rng hc sinh ny gii ton.

Gii.


V d 1.24. C hai chung g: Chung I c 10 g trng v 8 g mi;Chung II c 12 trng v 10 mi. C hai con g chy t chung I sangchung II. Sau c hai con g chy ra t chung II. Tnh xc sut:

a. Hai con g chy t chung I sang chung II l 2 con trng v haicon g chy ra t chung II cng l hai con trng.

b. Hai con g chy ra t chung II l hai con trng.c. Bit rng hai con g chy ra t chung II l hai con trng, tnh xc

sut hai con g chy t chung I sang chung II l 2 con g trng.

Gii.

1.6 Bi tp chng 1 Trang 17

1.6 Bi tp chng 1

Bi tp 1.1. Mt nhm kho st s thch tit l thng tin l trong nmqua:

45% ngi xem Tivi thch xem phim tnh cm Hn quc. 25% ngi xem Tivi thch xem phim hnh ng M. 10% thch xem c hai th loi trn.

Tnh t l nhm ngi thch xem t nht mt trong hai th loi trn.(60%)

Gii.

Bi tp 1.2. C ba l hng mi l c 20 sn phm, s sn phm loi Ac trong l I, II, III ln lt l: 12; 14; 16. Bn mua chn ngu nhin tmi l hng 3 sn phm, nu l no c 3 sn phm u loi A th bnmua nhn mua l hng . Tnh xc sut:

a. L th i c mua, i D 1; 2; 3: (0,193; 0,3193; 0,4912)b. C i l c mua, i D 0; 1; 2; 3: (0,2795; 0,4678; 0,2225; 0,0303)c. C nhiu nht hai l c mua. (0,9697)d. C t nht mt l c mua.(0,7205)e. Gi s c t nht mt l c mua. Tnh xc sut trong l II c

mua. (0,4432)f. Gi s c t nht mt l c mua. Tnh xc sut trong l I v IIc mua.(0,0855)


g. Gi s c mt l c mua. Tnh xc sut l II c mua. (0,2803)

Gii.

Bi tp 1.3. Mt hp bng bn c 15 bng mi v 8 bng c. Ln th I


ly ra 2 bng s dng sau cho vo li hp; ln th II ly ra 3 bng.Tnh xc sut

a. Ln th I ly c i bng c, i D 0; 1; 2. (0,4150; 0,4743; 0,1107)b. Ln I ly 1 bng c v ln II l 3 bng mi. (0,0975)

c. Ln th II ly c 3 bng mi. (0,1929)

d. Bit ln th II ly c 3 bng mi, tnh xc sut ln th I ly c1 bng c. (0,5054)

Gii.


Bi tp 1.4. C 3 bnh ng bi: bnh I c 4 bi trng v 6 bi en; bnh IIc 7 bi trng v 3 bi en; bnh III c 6 bi trng v 8 bi en. T bnh I vbnh II, mi bnh ly 1 bi v b sang bnh III. Tip theo, t bnh III lyra tip 3 bi. Tnh xc sut:

a. Hai bi ly ra t bnh I v II c i bi trng, i D 0; 1; 2: (0,18; 0,54;0,28)

b. Ba bi ly ra t bnh III c hai bi trng. (0,3424)c. Gi s ba bi ly t bnh III c hai bi trng, tnh xc sut hai bi ly

t bnh I v II l hai bi en. (0,1408)

Gii.


Bi tp 1.5. Mt thng kn ng 2 loi thuc: S lng l thuc loi Abng 2/3 thuc s lng l thuc loi B. T l l thuc A, B ht hn sdng ln lt l 10% v 8%. T thng ly ngu nhin mt l thuc.

a. Tnh xc sut ly c l thuc A ht hn s dng. (0,04)b. Tnh xc sut l thuc ly ra t thng ht hn s dng. (0,088)c. Gi s ly c l thuc cn hn s dng, tnh xc sut l ny l l

thuc B. (0,6053)

Gii.

Bi tp 1.6. 1 Mt ngi bn 3 pht n vo mt mc tiu mt cch clp. Xc sut trng mc tiu mi pht ln lt l 0,55; 0,6; 0,7. Xcsut mc tiu b h khi bi trng 1, 2, 3 pht n ln lt l 0,2; 0,4; 0,8.Tnh xc sut:

1Sinh vin h cao ng khng phi lm cc cu c, e, f.


a. C i pht trng mc tiu, i D 0; 1; 2; 3: (0,054; 0,273; 0,442; 0,231)b. C nhiu nht 2 pht trng mc tiu. (0,769)c. Tnh xc sut mc tiu b h. (0,4162)d. Gi s c 2 pht trng mc tiu, tnh xc sut pht th I trng

mc tiu. (0,5724)e. Gi s mc tiu b h. Tnh xc sut phat th nht trng mc tiu.

(0,7189)

f. Bit rng c nhiu nht 2 pht trng mc tiu, tnh xc sut mctiu b h. (0,3009)

Gii.


Bi tp 1.7. Nh my c hai phn xng, sn lng ca phn xng Igp 3 ln sn lng ca phn xng II. T l ph phm ca phn xngI, II ln lt l 7% v 12%. Chn ngu nhin mt sn phm ca nhmy, tnh:

a. Xc sut chn c sn phm tt do phn xng I sn xut.(0,6975)

b. Xc sut chn c ph phm. (0,0825)c. Gi s chn c sn phm tt, tnh xc sut sn phm ny do

phn xng I sn xut. (0,7602)

Gii.


Bi tp 1.8. Mt ngi bun bn bt ng sn ang c gng bn mtmnh t ln. ng ta tin rng nu nn kinh t tip tc pht trin, khnng mnh t c mua l 80%; ngc li nu nn kinh t ngng phttrin, ng ta ch c th bn c mnh t vi xc sut 40%. Theod bo ca mt chuyn gia kinh t, xc sut nn kinh t tip tc tngtrng l 65%. Tnh xc sut bn c mnh t. (0,66)

Gii.

Bi tp 1.9. 2 C hai hp ng bi: hp I c 5 bi trng v 7 bi en; hpII c 6 bi trng v 4 bi en. Ly 1 bi t hp I b sang hp II, ri t hpII ly ra 1 bi. Tnh xc sut

a. Bi ly t hp II l bi trng. (7/12)b. Gi s bi ly t hp II l bi trng, tnh xc sut bi ly t hp I l bi

trng. (5/11)c. Gi s bi ly ra t hp II l bi trng, tnh xc sut bi ny ca hp

I. 512

111=

712

d. Gi s bi ly ra t hp II l bi trng, tnh xc sut bi ny ca hp

II. 611=

712

Gii.

2Sinh vin h cao ng khng phi lm cc cu c, d.

Chng 2

Bin ngu nhin

Mc lc chng 22.1 Khi nim bin ngu nhin . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Phn phi xc sut ca bin ngu nhin . . . . . . . . . . . . . . . . . . 27

2.3 Cc c trng s ca bin ngu nhin . . . . . . . . . . . . . . . . . . . 35

2.4 Bi tp chng 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1 Khi nim bin ngu nhin

- Xt mt php th c khng gian cc bin c s cp . t

X W ! R! 7! X.!/ D x

X c gi l bin ngu nhin, x gi l gi tr ca bin ngu nhin X .

fX 2 I g

X

I R

fX 2 I g D f! W X.!/ 2 I g D A

Hnh 2.1: Bin ngu nhin X

V d 2.1. Thc hin php th gieo ng thi 2 ng xu cn i, chngta c khng gian cc bin c s cp

D fN1N2IN1S2IS1N2IS1S2g

2.2 Phn phi xc sut ca bin ngu nhin Trang 27

t X.!/ l s ng xu sp khi kt qu php th l !. Ta c:

X.N1N2/ D 0I X.N1S2/ D 1I X.S1N2/ D 1I X.S1S2/ D 2Khi ta gi X l bin ngu nhin s ng xu sp khi tung 2 ngxu.

- C hai loi bin ngu nhin:

Bin ngu nhin ri rc l bin ngu nhin m gi tr c th ca nl mt tp hu hn hoc v hn m c.

Bin ngu nhin lin tc l bin ngu nhin m gi tr c th can lp y mt khong trn trc s.

V d 2.2.

S chm trn mt xut hin khi tung mt xc sc l bin ngunhin ri rc (gi tr ca X l tp hu hn).

S cuc gi n tng i in thoi trong 1 gi l bin ngu nhinri rc (gi tr ca X l tp v hn m c).

Thi gian hon thnh 1 sn phn ca mt cng nhn l bin ngunhin lin tc.

2.2 Phn phi xc sut ca bin ngu nhin

2.2.1 X l bin ngu nhin ri rc

m t phn phi xc sut ca bin ngu nhin ri rc ngi ta sdng bng phn phi xc sut:

X x1 x2 xn P f .x1/ f .x2/ f .xn/

Trong :

Dng 1 lit k gi tr c th ca X. f .xi/ D P .X D xi / ; i D 1; 2; : : : gi l xc sut X nhn gi tr xi : Nu x0 fx1; : : : ; xn; : : :g th f .x0/ D 0:

Chng 2. Bin ngu nhin

V d 2.3. Thc hin php th tung mt xc sc. Gi X l s chm trnmt xut hin ca xc sc. X c bng phn phi nh sau:

X 1 2 3 4 5 6

P 1=6 1=6 1=6 1=6 1=6 1=6

Nhn xt:

f .x1/C f .x2/C C f .xn/C D 1: P .a < X < b/ D P

a

2.2 Phn phi xc sut ca bin ngu nhin Trang 29

V d 2.6. Mt x th c 6 vin n, bn ln lt tng vin vo mtmc tiu mt cch c lp. Xc sut trng mc tiu mi ln bn l0,7. Nu c 3 vin trng mc tiu hoc ht n th dng. Gi X l svin n bn, lp bng phn phi xc sut ca X:

Gii.

V d 2.7. Mt l c 3 bi trng v 7 bi en. T l ny ly ra ngu nhin4 bi. Gi X l s bi en ln trong 4 bi ly ra, lp bng phn phi xc

Chng 2. Bin ngu nhin

sut ca X:

Gii.

2.2.2 X l bin ngu nhin lin tc

nh ngha 2.1 (Hm mt ). Hm s f .x/ 0;8x 2 R c gi lhm mt ca bin ngu nhin lin tc X nu

P .X 2 A/ DZA

f .x/dx;8A R

Nhn xt. Vi nh ngha hm mt ta c

i. Nu X l bin ngu nhin lin tc th xc sut X thuc mt tpA R c tnh bng tch phn ca hm mt f .x/ trn tp A:

ii. Mi hmmt phi tha hai iu kin f .x/ 0 vC1R1

f .x/dx D1

V d 2.8. Cho hm s

f .x/ D8 1/ :

V d 3.9. im Toeic ca sinh vin sp tt nghip trng i hcc phn phi chun vi gi tr trung bnh 560 v lch chun 78.Tnh:

a. T l sinh vin c im nm gia 600 v 700.b. T l sinh vin c im Toeic trn 500.c. Gi s nh trng mun xc nh im Toeic ti thiu sinh vin

c th ra trng vi t l 80%. Tnh im Toeic ti thiu (ly phnnguyn).

Gii.

3.5 Phn phi chun Trang 59

V d 3.10. Tui th ca my ct c l bin ngu nhin c phn phichun vi trung bnh l 82 thng. Nh sn xut bo hnh sn phmkhi bn ra l 33 thng. Gi s 2,5% sn phm b tr li (hng) trongthi gian bo hnh. Tnh:

a. lch chun ca tui th sn phm ny.

b. Xc sut mt my loi ny c tui th trn 50 thng.c. Mt ca hng bn 10 my ct c loi ny. Tnh:

i) Xc sut c 2 my hng trong thi gian bo hnh.

ii) S my trung bnh hng trong thi gian bo hnh.

Chng 3. Mt s phn phi xc sut thng dng

Gii.

3.6 Bi tp chng 3

Bi tp 3.1. Mt nh vn trng 121 cy mai vi xc sut n hoa cami cy trong dp tt nm nay l 0,75. Gi bn 1 cy mai n hoa l 0,5triu ng.


a. Tnh s cy trung bnh n hoa trong dp tt. (90,75 cy)b. Gi s nh vn bn ht nhng cy mai n hoa, tnh s tin trong

dp tt nm nay nh vn thu c chc chn nht. (45,5 triung)

Gii.

Bi tp 3.2. Ch vn lan nhm 20 chu lan c hoa mu vi100 chu lan c hoa mu tm (lan cha n hoa). Mt khch hng chnngu nhin 15 chu t 120 chu lan (chn 1 ln).

a. Tnh xc sut c t 5 n 6 chu lan c hoa mu . (0,0723)b. Gi X l s chu lan c hoa mu khch chn c. Tnh gi tr

ca EX v VarX . (5/2; 125/68)

Gii.


Bi tp 3.3. Ti bnh vin A trung bnh 3 gi c 8 ca m. Tnh

a. S ca m chc chn nht s xy ra ti bnh vin A trong 25 gi. (66ca)

b. Tnh xc sut trong 5 gi c t 10 n 12 ca m. (0,2821)

Gii.

Bi tp 3.4. Mt l hng cha 20 sn phm trong c 4 ph phm.Chn lin tip 3 ln (c hon li) t l hng, mi ln chn ra 4 snphm. Tnh xc sut trong 3 ln chn c:

a. ng 1 ln chn c khng qu 1 ph phm. (0,066)b. Trung bnh s ln chn c khng qu 1 ph phm. (2,514)

Gii.


Bi tp 3.5. Gi c ph trn th trng l bin ngu nhin c phn phichun vi trung bnh l 26000 ng/kg v lch chun 2000 ng. kl gi tr ti c ph c gi ln hn k vi xc sut 90%. Tnh gi tr k:(23420 ng)

Gii.

Bi tp 3.6. Thi gian mang thai ca sn ph l bin ngu nhin cphn phi chun vi trung bnh 280 ngy. Cho bit t l mt sn phmang thai trn 290 ngy l 25,14%, tnh lch chun ca thi gianmang thai. (15 ngy)

Gii.


Bi tp 3.7. Chiu di ca loi linh kin in t A ti ca hng B lbin ngu nhin X (mm) c phn phi chun N.12I 2; 5/. Mt cng ty cnmua loi linh kin ny vi chiu di t 11,98mm n 13mm v h chnln lt 7 chic t ca hng B. Tnh xc sut trong 7 chic c chnc:

a. T 5 n 6 chic s dng c. (1,06%)

b. t nht mt chic s dng c. (0,8531)

Gii.

Bi tp 3.8. Thi gian chi th thao trong mt ngy ca mt thanh nin


l bin ngu nhin X (gi/ngy) c hm mt

f .x/ D8 00 ni khc

; D 0; 013

a. Tnh tui th trung bnh ca ngi dn a phng. (76,9231tui)

b. Tnh t l ngi dn th trn 60 tui. (0,4584)c. Trung bnh c bao nhiu ngi th trn 60 tui tui trong 1000

dn. (458,4)

Gii.


Bi tp 3.10. Thi gian hc rnh ngh sa ti vi ca mt ngi l mtbin ngu nhin - X (nm) c hm mt .

f .x/ D8 0,

P

Snn

"

! 0

khi n!C1:

Chng 4. Lut s ln v cc nh l gii hn

Chng minh. Bi v X1; : : : ; Xn l c lp v cng phn phi, ta c

Var

Sn

n

D

2

nv E

Sn

n

D : p dng bt ng thc Chebyshev, vi

mi " > 0,

P

Snn

"

2

n"2

C nh " v khi n!C1

P

Snn

"

! 0

Sn=n l trung bnh ca cc bin ngu nhin Xi , .i D 1; : : : ; n/, do ngi ta thng gi lut s ln l lut trung bnh.

4.4 nh l gii hn trung tm

nh l 4.5. Cho X1; : : : ; Xn l cc bin ngu nhin c lp v cngphn phi vi k vng v phng sai 2 hu hn. Ta t

Sn D X1 C C XnKhi n!1 th bin ngu nhin

SnF! X; vi X N .E .Sn/ IVar .Sn//

Nhn xt: nh l trn cho ta kt qu l khi n ln phn phi ca binngu nhin Sn c xp x bng phn phi chun N .E .Sn/ IVar .Sn//. n gin ta vit Sn

: N .E .Sn/ IVar .Sn//, du : ngha l xp xphn phi.

V d 4.2. Tung 1000 ln 1 xc sc, tnh xc sut tng s chm trong1000 ln tung ln hn 3600.

Gii.

4.5 Lin h gia cc phn phi xc sut Trang 73

4.5 Lin h gia cc phn phi xc sut

4.5.1 Lin h gia phn phi nh thc v chun

Cho X1; : : : ; Xn l cc bin ngu nhin c lp v Xi B.p/: Ta c

X D X1 C C Xn B.nIp/

Khi n ln (np > 5 v nq > 5) th SnN.npInpq/. Khi :

P .a X < b/ 'b nppnpq

'

a nppnpq

(4.1)

v

P .X D k/ D C kn pkqnk 1pnpq

f

k nppnpq

(4.2)

trong f .x/ c tra bng A.2.

V d 4.3. Trong mt kho la ging c t l ht la lai l 20%. Tnh xcsut sao cho khi chn ln lt 1000 ht la ging trong kho th c:

a. ng 192 ht la lai.b. C t 185 n 195 ht la lai.


Gii.

4.5.2 Lin h gia nh thc v Poisson

Cho X B.n; p/; khi n ln (n 20) v p nh (p 0; 05) th Sn P./ v

P .Sn D k/ D C kn pkqnk ke

k(4.3)

V d 4.4. Mt l hng tht ng lnh ng gi nhp khu c cha 0,6%b nhim khun. Tm xc sut khi chn ngu nhin 1.000 gi tht tl hng ny c:

a. Khng qu 2 gi b nhim khun.b. ng 40 gi b nhim khun.

Gii.

4.5 Lin h gia cc phn phi xc sut Trang 75

4.5.3 Lin h gia siu bi v nh thc

Cho X H.N;NA; n/; khi n N (n=N 0; 01) th X B.nINA=N/ v

P .X D k/ D CkNAC nkNNAC kn

C kn pkqnk (4.4)

trong p D NA=n:V d 4.5. Mt ao c c 10.000 c da trn, trong c 1.000 con ctra.

a. Tnh xc sut khi bt ngu nhin 20 con t ao th c 5 con ctra.

b. Tnh xc sut khi bt ngu nhin 50 con t ao th c 10 conc tra.

Gii.

Chng 5

Vct ngu nhin

Mc lc chng 55.1 Khi nim vct ngu nhin . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Phn phi xc sut ca .X; Y / . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3 Bi tp chng 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.1 Khi nim vct ngu nhin

Mt b c th t n bin ngu nhin .X1; : : : ; Xn/ gi l mt vctngu nhin n chiu.

Vct ngu nhin n chiu l lin tc hay ri rc nu, cc bin ngunhin thnh phn l lin tc hay ri rc.

V d 5.1. Nng xut la mt tha rung a phng A l bin ngunhin X , nu xt n lng phn Y th ta c vct ngu nhin hai chiu.X; Y /, cn nu xt thm lng nc Z th ta c vct ngu nhin 3 chiu.X; Y;Z/:

Trong gii hn ca chng trnh ta ch xt vct ngu nhin haichiu, k hiu .X; Y /:

5.2 Phn phi xc sut ca .X; Y /

5.2.1 .X; Y / l vct ngu nhin ri rc

a) Phn phi xc sut ng thi: Vct ngu nhin ri rc .X; Y /c biu din bng bng phn phi xc sut ng thi:

Chng 5. Vct ngu nhin

XY

y1 y2 yj yn Tng dngx1 f .x1; y1/ f .x1; y2/ f .x1; yj / f .x1; yn/ f .x1; /x2 f .x2; y1/ f .x2; y2/ f .x2; yj / f .x2; yn/ f .x2; /:::

:::::: ::: ::: :::

xi f .xi ; y1/ f .xi ; y2/ f .xi ; yj / f .xi ; yn/ f .xi ; /:::

:::::: ::: ::: :::

xm f .xm; y1/ f .xm; y2/ f .xm; yj / f .xm; yn/ f .xm; /Tng ct f .; y1/ f .; y2/ f .; yj / f .; yn/ 1

Trong :

f .xi ; yj / D PX D xi IY D yj

mPiD1

nPjD1

f .xi Iyj / D 1

V d 5.2. Bin ngu nhin ri rc X nhn cc gi tr 6, 7 v 8. Binngu nhin Y nhn cc gi tr 1, 2, 3. Phn phi ng thi ca vctngu nhin .X; Y / cho bi bng

XY 1 2 3

6 0,1 0,15 0,057 0,1 0,2 0,18 0,05 0,2 0,05

Tnh:

a. P .X D 6IY D 2/ IP .X D 4IY D 6/ :b. P .X 7IY 2/ :

Gii.

5.2 Phn phi xc sut ca .X; Y / Trang 79

b) Phn phi xc sut thnh phn (l)

Bng phn phi xc sut ca X

X x1 x2 xmP.X D x/ f .x1; / f .x3; / f .xm; /

Trong f .xi ; / l tng dng i: Bng phn phi xc sut ca Y

Y y1 y2 ynP.Y D y/ f .; y1/ f .; y2/ f .; yn/

Trong f .; yj / l tng ct j:V d 5.3. Cho bng phn phi xc sut ng thi nh sau:

XY 1 2 3

6 0,1 0,15 0,057 0,1 0,2 0,18 0,05 0,2 0,05

a. Lp bng phn phi xc sut ca X:b. Tnh P .X > 6/ :c. Lp bng phn phi xc sut ca Y:d. Tnh P .Y < 3/ :

Gii.


c) Phn phi xc sut c iu kin

Bng phn phi xc sut ca X vi iu kin Y D yj

X x1 x2 xmP.X D xjY D yj /

f .x1; yj /

f .; yj /f .x2; yj /

f .:yj / f .xm; yj /

f .; yj /

Bng phn phi xc sut ca Y vi iu kin X D xiY y1 y2 yn

P.Y D yjX D xi/f .xi ; y1/

f .xi ; /f .xi ; y2/

f .xi ; / f .xi ; yn/

f .xi ; /V d 5.4. Cho bng phn phi xc sut ng thi nh sau:

XY 1 2 3

6 0,1 0,15 0,057 0,1 0,2 0,18 0,05 0,2 0,05

a. Lp bng phn phi xc sut ca X bit Y D 2:b. Tnh xc sut P .X > 6jY D 2/ :c. Lp bng phn phi xc sut ca Y bit X D 6:d. Tnh xc sut P .Y > 1jX D 6/ :

Gii.


5.2.2 .X; Y / l vct ngu nhin lin tc

a) Hm mt ng thi

nh ngha 5.1 (Hm mt ng thi). Hm s f .x; y/ 0;8.x; y/ 2R2 c gi l hm mt ng thi ca .X; Y / nu

P ..X; Y / 2 A/ DA

f .x; y/dxdy; A R2

Nhn xt. Vi nh ngha hm mt ng thi ca vct ngu nhinta c

i. Nu .X; Y / l vct ngu nhin lin tc th xc sut .X; Y / thucmt tp A R2 c tnh bng tch phn ca hm mt f .x; y/trn tp A:

ii. Mi hm mt ng thi ca vct ngu nhin (X,Y) phi tha


hai iu kin f .x; y/ 0 v

P.X; Y / 2 R2 D

R2

f .x; y/dxdy D 1

V d 5.5. Cho hm s

f .x; y/ D10x2y khi 0 < y < x < 10 ni khc

a. Chng t f .x; y/ l hm mt .X; Y /:b. Tnh P .2Y > X/ :

Gii.

0

1

0 1x

y D xD W

0 < x < 1

0 < y < xhoc

0 < y < 1

y < x < 1


0

1

0 1x

y D x

y D x=2D0 W

0 < x < 1

x=2 < y < x

b) Hm mt thnh phn (l)

Hm mt ca X:

fX .x/ DC1Z1

f .x; y/dy

Hm mt ca Y:

fY .y/ DC1Z1

f .x; y/dx

V d 5.6. Cho vct ngu nhin .X; Y / c hm mt


a. Tm hm mt ca X:b. Tm hm mt ca Y:c. Tnh P .X > 1=2/ v EX:d. Tnh P .Y < 1=2/ v EX:


Gii.

0

1

0 1

y D xD W

0 < x < 1

0 < y < xhoc

0 < y < 1

y < x < 1

x


c) Hm mt c iu kin

Hm mt ca X vi iu kin Y D y

fX .xjY D y/ Df .x; y/

fY .y/

Hm mt ca Y vi iu kin X D x

fY .yjX D x/ Df .x; y/

fX .x/

V d 5.7. Cho vct ngu nhin .X; Y / c hm mt


a. Tm hm mt ca X vi iu kin Y D 1=2:b. Tm hm mt ca Y vi iu kin X D 1=3:c. Tnh P .X > 2=3jY D 1=2/ v E.X jY D 1=2/:d. Tnh P .Y < 1=4jX D 1=3/ v E.Y jX D 1=3/:


5.3 Bi tp chng 5

Bi tp 5.1. Chi ph qung co (X : triu ng) v doanh thu (Y : triung) ca mt ca hng c bng phn phi ng thi cho nh sau:

X

Y 500 700 900(400-600) (600-800) (800-1000)

30 0,10 0,05 050 0,15 0,20 0,0580 0,05 0,05 0,35

a. Lp bng phn phi xc sut chi ph chi cho qung co.b. Cho doanh thu l 500 triu, lp bng phn phi xc sut chi ph

qung co.c. Lp bng phn phi xc sut doanh thu ca ca hng.d. Cho bit chi ph qung co l 30 triu, lp bng phn phi xc sut

ca doanh thu.e. Tnh chi ph chi cho qung co trung bnh.f. Cho doanh thu l 500 triu, tnh chi ph qung co trung bnh.


g. Tnh doanh thu trung bnh ca ca hng.

h. Cho chi ph qung co l 30 triu, tnh doanh thu trung bnh.

Gii.


Bi tp 5.2. Nng sut la X(tn/ha) v lng phn Ur Y(x 100 kg) chm mt ng thi

f .x; y/ D8 30 n 30;X N.I 2/

Bit 2 " Dpnt1

2" D p

nt1

2

(t12tra bng A.2) (t1

2tra bng A.2)

Khng bit 2 " Dspnt1

2" D sp

ntn1

(t12tra bng A.2) (tn1 tra bng A.3).

Chng 7. c lng tham s

V d 7.2. Kho st v thi gian t hc X (gi/tun) trong tun ca mts sinh vin h chnh quy trng i hc A trong thi gian gn y,ngi ta thu c bng s liu

X 5 6 7 8 9 10S SV 10 35 45 36 10 8

c lng thi gian t hc trung bnh ca mt sinh vin vi tin cy95% cho hai trng hp:

a. Bit D 2b. Cha bit

Gii. T mu ta tnh c n D 144I Nx D 7; 1736I s D 1; 2366.Gi l thi gian t hc trung bnh ca sinh vin. khong tin cy

cho vi tin cy 95% c dng

.1I2/ D . Nx "I Nx C "/Tip theo ta tnh " cho tng trng hp:

a. Bit D 2" D p

nt1

2D 2p

1441; 96 D 0; 3267

Vy khong tin cy

.1I2/ D .7; 1736 0; 3267I 7; 1736C 0; 3267/ D .6; 8469I 7; 5003/

Ch . Cho trc tin cy l 1 D 0; 95 cho nn ta c 12D 0; 475.

Tra bng A.2 ta c t0;475 D 1; 96.b. Khng bit

" D spnt1

2D 1; 2366p

1441; 96 D 0; 202

Vy khong tin cy

.1I2/ D .7; 1736 0; 202I 7; 1736C 0; 202/ D .6; 9716I 7; 3756/

Ch . Vi t0;475 D 1; 96 c tnh nh cu a.V d 7.3. Kho st cn nng (kg) ca g khi xut chung, ngi ta cnmt s con v kt qu cho nh sau:

7.3 Khong tin cy Trang 105

2,1; 1,8; 2,0; 2,3; 1,7; 1,5; 2,0; 2,2; 1,8

Gi s cn nng ca g l bin ngu nhin c phn phi chun. Vi tin cy 95% c lng cn nng trung bnh ca g khi xut chung:

a. Bit D 0; 3.b. Khng bit .

Gii. T mu ta tnh c n D 9I Nx D 1; 9333I s D 0; 2549.Gi l cn nng trung bnh ca g khi xut chung.

a. Cho bit D 0; 3

" D pnt1

2D 0; 3p

91; 96 D 0; 196

Vy khong tin cy

.1I2/ D .1; 9333 0; 196I 1; 9333C 0; 196/ D .1; 7373I 2; 1293/b. Khng bit

" D spntn1 D

0; 2549p9

2; 306 D 0; 1959

Vy khong tin cy

.1I2/ D .1; 9333 0; 1959I 1; 9333C 0; 1959/ D .1; 7374I 2; 1292/

Ch . Cho trc tin cy l 1 D 0; 95 cho nn ta c D 0; 05. Trabng A.3 ta c t80;05 D 2; 306.

Ch . Cc ch tiu c lng trung bnh. Ta nhn thy trong c lngtrung bnh c 3 ch tiu chnh "; 1 ; n: Nu bit hai ch tiu th s xcnh c ch tiu th 3.

a. Xc nh c mu n nh nht sao cho chnh xc khng ln hn "v tin cy l 1 ( y ta lun gi s c mu ln). Ta c

n "t1

2

2hoc n

s"t1

2

2

n nh nht tha iu kin trn l

n D" t12

2 C 1hoc n D

s"t12

2 C 1


b. Xc nh tin cy ca c lng khi bit chnh xc ca c

lng. Trc ht xc nh gi tr t12D "

pn

s. V t y d dng

tnh c 1 :V d 7.4. Cn th 121 sn phm (n v tnh bng kg) ta tnh cs2 D 5; 76.

a. Xc nh chnh xc nu mun c lng trng lng trung bnhvi tin cy 95%.

b. Xc nh c mu nh nht lng trng lng trung bnh vi tin cy 95% v chnh xc nh hn 0,4.

c. Xc nh tin cy nu mun c lng trung bnh vi chnhxc l " D 0; 5.

Gii.

a. Xc nh chnh xc:

" D spnt1

2D 2; 4p

1211; 96 D 0; 4276

b. Xc nh c mu n.

n Ds"t12

2C 1 D2; 40; 4

1; 96

2 C 1 D 139c. Xc nh tin cy, trc ht ta tnh

t12D "

pn

sD 0; 5

p121

2; 4D 2; 29

Tra bng A.2 ta tnh c 12D 0; 489. T suy ra 1 D 0; 978

7.3.3 Khong tin cy cho t l

Gi p l t l phn t A cha bit ta tm khong .p1Ip2/ cha p sao choP .p1 < p < p2/ D 1 . Khong tin cy

.p1Ip2/ D .f "If C "/trong

7.3 Khong tin cy Trang 107

f l t l phn t A tnh trn mu. " gi l chnh xc ca c lng c tnh nh sau:

" Drf .1 f /

nt1

2

V d 7.5. Kho st t l ph phm do mt nh my sn xut ra, ngita quan st 800 sn phm thy c 8 ph phm. Vi tin cy 95% hyc lng t l ph phn ca nh my.

Gii. Gi

f l t l ph phm trn mu.f D 8

800D 0; 01

:

p l t l ph phm ca nh my.

chnh xc ca c lng t l

" Drf .1 f /

nt1

2Dr0; 01.1 0; 01/

8001; 96 D 0; 0069

Vy khong tin cy cho p vi tin cy 95% l

.p1Ip2/ D .0; 01 0; 0069I 0; 01C 0; 0069/ D .0; 0031I 0; 0169/

Ch . Xc nh cc ch tiu c lng

a Xc nh c mu n nh nht sao cho chnh xc khng ln hn "

v tin cy l 1 Ta c n f .1 f /"2

t1

2

2. n nh nht tha

iu kin trn l

n Df .1 f /"2

t1

2

2C 1b Xc nh tin cy ca c lng khi bit chnh xc ca clng. Trc ht xc nh gi tr

t12D "

rn

f .1 f /:

V t y d dng tnh c 1 bng bng A.2.


V d 7.6. Quan st 800 sn phm do mt x nghip sn xut ra thyc 128 mu loi A.

a. Xc nh chnh xc nu mun c lng t l sn phm loi Avi tin cy 95%.

b. Xc nh c mu nh nht c lng t l sn phm loi A vi chnh xc nh hn 0,023 v tin cy 95%.

c. Xc nh tin cy nu mun c lng t l sn phm A vi chnh xc l 0,022.

Gii. Gi:

f l t l sn phm loi A tnh trn muf D 128

800D 0; 16

.

p l t l sn phm loi A do x nghip sn xut ra.

a. chnh xc ca c lng

" Drf .1 f /

nt1

2Dr0; 16.1 0; 16/

8001; 96 D 0; 0254

b. Xc nh n

n Df .1 f /"2

t1

2

2 C 1 D0; 16.1 0; 16/0; 0232 1; 962

C 1 D 977

c. Xc nh tin cy 1

t12D "

rn

f .1 f / D 0; 022s

800

0; 16.1 0; 16/ D 1; 69

Tra bng A.2 ta tnh c 12D 0; 4545. T suy ra 1 D 0; 909

7.4 Bi tp chng 7

Bi tp 7.1. Kim tra ngu nhin 25 bng n ca mt hng in t,thy tui th trung bnh l 5000 gi, lch chun ca mu c hiuchnh l 200 gi. Gi s tui th ca bng n c phn phi chun. Tnhkhong tin cy tui th trung bnh ca loi bng n trn vi tin cy95%. p s: (4917,44 gi; 5082,56 gi)


Bi tp 7.2. Kim tra ngu nhin 25 bng n ca mt hng in t,thy lch chun ca mu c hiu chnh l 200 gi. Gi s tui thca bng n c phn phi chun. S dng mu trn c lng tuith trung bnh ca loi bng n trn vi chnh xc l 73,12 gi thm bo tin cy bao nhiu? p s: 92%

Bi tp 7.3. Thm d 25 ngi ang s dng in thoi di ng v stin phi tr trong 1 thng, thy s tin trung bnh mt ngi phi trl 200 ngn ng, lch chun ca mu c hiu chnh l 50 ngn ng.Gi s s tin phi tr trong mt thng c phn phi chun. Vi tincy l 95% tnh khong tin cy s tin trung bnh mt ngi s dngin thoi di ng phi tr. p s: (179,36 ngn ng; 220,64 ngnng)


Bi tp 7.4. Thm d 25 ngi ang s dng in thoi di ng v stin phi tr trong 1 thng, thy lch chun ca mu c hiu chnhl 50 ngn ng. Gi s s tin phi tr trong mt thng c phn phichun. Vi chnh xc l 19,74 ngn ng th tin cy bao nhiu?p s: 94%

Bi tp 7.5. Bit chiu di ca mt loi sn phm l bin ngu nhinc phn phi chun. o ngu nhin 10 sn phm loi ny th c chiudi trung bnh l 10,02m v lch chun ca mu cha hiu chnh l0,04m. Tnh khong tin cy chiu di trung bnh ca loi sn phm nyvi tin cy 95%. p s: (9,9898m; 10,0502m)

Chng 8

Kim nh gi thit

Mc lc chng 88.1 Bi ton kim nh gi thit . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.2 Kim nh gi thit v trung bnh . . . . . . . . . . . . . . . . . . . . . . 113

8.3 Kim nh gi thit v t l . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.4 So snh hai gi tr trung bnh . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.5 So snh hai t l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.6 Bi tp chng 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.1 Bi ton kim nh gi thit

8.1.1 Gi thit khng, i thit

Trong chng ny chng ta s cp n bi ton thng k lin quann tham s , vi gi tr ca n khng bit thuc khng gian tham :Tuy nhin chng ta s gi s c th c phn chia thnh hai tptch bit 0 v 1 v nhim v ca ngi lm thng k phi quyt nhxem thuc 0 hay 1:

Chng ta t H0 k hiu gi thit 2 0; v H1 k hiu gi thit 2 1: Bi v 0 v 1 tch bit v 0 \ 1 D ; chnh xc ch c githit H0 hoc H1 l ng. Chng ta phi quyt nh chp nhn H0 bc bH1 hoc ngc li. Bi ton thuc dng ny c gi l kim nhgi thit.

n y, chng ta thy vai tr ca gi thit H0 v H1 c bn gingnhau. Trong hu ht cc bi ton kim nh, hai gi thit ny hi khc. phn bit gia hai gi thit ny ta gi H0 gi l gi thit khng v

Chng 8. Kim nh gi thit

H1 gi l i thit. Chng ta s dng cc thut ng ny trong phn cnli ca chng.

8.1.2 Min ti hn

Ta xt bi ton vi gi thit c dng nh sau:Gi thit khng H0 W 2 0i thit H1 W 2 1

Gi s trc khi chng ta quyt nh gi thit no s c chp nhn,chng ta c mu ngu nhin X1; : : : ; Xn c trch t phn phi ca ctnh X vi tham s cha bit. Chng ta k hiu l khng gian mu, cha tt c cc kt qu c th xy ra khi ly mu ngu nhin.

Trong qu trnh kim nh, chng ta s chia thnh hai tp con.Mt tp cha tt c cc gi tr ca X sao cho ta chp nhn H0; v tpcn li cha tt c cc gi tr ca X sao cho ta bc b H0 v chp nhnH1: Tp cc gi tr ca X H0 b bc b gi lmin ti hn, k hiu C:

Vi mi gi tr 2 ta t hm lc lng ./ l xc sut dn nbc b H0; ngc li 1 ./ l xc sut dn n chp nhn H0: Nu khiu C l min ti hn ca kim nh, hm ./ c xc nh bi quanh

./ D P .X 2 C j/ ; 8 2 Bi v ./ l xc sut ng vi mi th H0 b bc b, trong trng

hp l tng hm ./ D 0 vi mi 2 0 v ./ D 1 vi mi 2 1:Nu hm ./ c cc gi tr ny th bt chp gi tr thc t no talun c kt lun ng vi xc sut 1.

8.1.3 Hai loi sai lm

Khi chn mt trong hai quyt nh trn s ny sinh ra hai sai lm:

Sai lm loi I: Bc b H0 khi H0 ng, xc sut sai lm loi I lP .C jH0/ D P ..X1; : : : ; Xn/ 2 C jH0/

Sai lm loi II: Chp nhn H0 khiH0 sai, xc sut sai lm loi II lP NC jH1 D P ..X1; : : : ; Xn/ C jH1/

8.2 Kim nh gi thit v trung bnh Trang 113

V d 8.1. Cn nghin cu tc dng ph ca mt loi thuc mi vac nghin cu ta t gi thit v i thit nh sau

Gi thit H0 W Thuc c tc dng phi thit H1 W Thuc khng c tc dng ph

Kt lun

Thc t Thuc c tc dng ph Thuc khng c tc dng ph

Chp nhn H0 Kt lun ng Sai lm loi IIBc b H0 Sai lm loi I Kt lun ng

Vic t gi thit nh trn khi sai lm loi I xy ra l tai hi hn sailm loi II (thuc c tc dng ph m kt lun thuc khng c tc dngph).

L t nhin l ta chn min C sao cho cc tiu c hai xc sut phmsai lm. Song khng th cc tiu ng thi c hai sai lm khi c mu cnh, bi v hai xc sut trn hin h nhau bi:

P .C jH0/C P NC jH0 D 1IP .C jH1/C P NC jH1 D 1:

Do C cc tiu P .C jH0/ cha chc cc tiu P NC jH1

8.1.4 Phng php chn min ti hn

Ta c nh mt loi xc sut sai lm v tm min C sao cho xc sutphm sai lm kia t gi tr nh nht. Thng thng ta c nh xcsut sai lm loi I: P .C jH0/ , ta s chn min C sao cho P

NC jH1t cc tiu hay P .C jH1/ cc i, ngha l tim C sao cho:

P .C jH0/ P .C jH1/ t cc i hay

./ vi 2 0./ t cc i vi 2 1 (8.1)

Ta gi l mc ngha ca kim nh, khi c nh v c hm lclng ./;8 2 1 ln nht th qui tc ny gi l qui tc mnh nht.

8.2 Kim nh gi thit v trung bnh

Gi s (cha bit) l trung bnh ca bin ngu nhin X , cn kimnh

Gi thit H0 W D 0i thit H1 W 0


VarX

C mun > 30 n 30;X N.I 2/

Bit 2 t D j Nx 0j

pn t D j Nx 0j

pn

t12(Bng A.2) t1

2(Bng A.2)

Khng bit 2 t D j Nx 0js

pn t D j Nx 0j

s

pn

(t12(Bng A.2) tn1 (Bng A.3)

Kt lun

Chp nhn gi thit H0 khi t t12hoc

t tn1

Bc b gi thit H0 khi t > t1

2hoc

t > tn1

V d 8.2. Cn th 15 con g ty 1 tri chn nui khi xut chung tatnh c Nx D 3; 62kg. Cho bit 2 D 0; 01.

a. Gim c tri tuyn b trng lng trung bnh ca g ty l 3; 5kgth c tin c khng vi mc ngha D1%.

b. Gi s ngi ta dng thc n mi v khi xut chung trng lngtrung bnh ca g ty l 3,9 kg. Cho kt lun v loi thc n nyvi mc ngha D 1%:

Gii.

a. Gi cn nng trung bnh ca g khi xut chung. Cn kim nh:Gi thit H0 W D 3; 5kgi thit H1 W 3; 5kg

t D j Nx 0j

pn D j3; 62 3; 5j

0; 1

p15 D 4; 6 v t1

2D 2; 58

t > t1

2

nn bc b gi thit. Vy gim c bo co sai.

8.3 Kim nh gi thit v t l Trang 115

b. Gi cn nng trung bnh ca g ty khi xut chung (trc khi sdng thc n mi)

Gi thit H0 W D 3; 9kgi thit H1 W 3; 9kg

t D jxn 0j

pn D j3; 62 3; 9j

0; 1

p15 D 10; 84

t > t1

2

nn bc b gi thit. Vy thc n mi c tc dng tt.

8.3 Kim nh gi thit v t l

Gi s p(cha bit) l t l phn t loi A, cn kim nhGi thit H0 W p D p0i thit H1 W p p0

Qui tc thc hnh nh sau: Tnh gi tr

t D jf p0jpp0.1 p0/

pn v t1

2(Bng A.2)

Trong f l t l phn t A trn muKt lun:

Chp nhn gi thit H0 khi t t12:

Bc b gi thit H0 khi t > t12:

V d 8.3. kim tra mt loi sng th thao, ngi ta cho bn 1000vin n vo bia thy c 540 vin trng mc tiu. Sau , bng ci tink thut ngi ta tnh c t l trng mc tiu l 70%. Hy cho ktlun v ci tin vi mc ngha 1%.

Gii. Gi

p l t l bn trng trc ci tin.

f l t l bn trng trn mu (trc ci tin).


Cn kim nh gi thitGi thit H0 W p D 0; 7i thit H1 W p 0; 7

Tin hnh kim tra gi thit

t D jf p0jpp0.1 p0/

pn D j0; 54 0; 7jp

0; 7:0; 3

p1000 D 11; 04

1 D 0; 99 tra bng A.2 ta c t12D 2; 58. Kt lun ci tin c tc

dng tt.

V d 8.4. Kim tra 800 sinh vin thy c 128 sinh vin gii. Trngbo co tng kt l c 40% sinh vin gii th c th chp nhn ckhng vi mc ngha 5%.

Gii. Gi

p t l sinh vin gii thc t (cha bit)

f t l sinh vin gii tnh trn mu f D 128800

D 0; 16Gi thit H0 W p D 40%i thit H1 W p 40%

Tin hnh kim tra gi thit

t D jf p0jpp0.1 p0/

pn D j0; 16 0; 4jp

0; 4:0; 6

p800 D 13; 871

1 D 0; 95 tra bng A.2 ta c t12D 1; 96. Kt lun bo co l sai s

tht, t l sinh vin gii trong thc t thp hn nhiu.

8.4 So snh hai gi tr trung bnh

Gi s X1 v X2 l hai bin ngu nhin c lp c gi tr trung bnh l1 v 2 . Cn kim nh

Gi thit H0 W 1 D 2i thit H1 W 1 2

K hiu cc c trng ca mu 1, 2 ly t tng th 1, tng th 2.

8.4 So snh hai gi tr trung bnh Trang 117

Mu C mu Trung bnh mu lch chun c hiu chnhI n1 Nx1 s1II n2 Nx2 s2

VarX1IVarX2

C mun1In2 > 30 n1 30IX1 N.1I

21 /

n2 30IX2 N.2I 22 /

Bit 21 I 22 t Dj Nx1 Nx2js21n1

C 22

n2

t D j Nx1 Nx2js21n1

C 22

n2t1

2(Bng A.2) t1

2(Bng A.2)

Khng bit 21 I 22 t Dj Nx1 Nx2jss21n1C s

22

n2

t D j Nx1 Nx2jss2

n1C s

2

n2t1

2(Bng A.2) tn1Cn22 (Bng A.3)

Trong s2 D .n1 1/s21 C .n2 1/s22

n1 C n2 2gi l phng sai gp.

Kt lun:

Chp nhn gi thit H0 khi t t12hoc

t tn1Cn22

Bc b gi thit H0 khi t > t1

2hoc

t > tn1Cn22

V d 8.5. Cn th 100 tri cy nng trng I ta tnh c Nx1 D 101; 2Is21 D 571; 7 v 361 tri cy nng trng II tnh c Nx2 D 66; 39I s22 D29; 72. So snh trng lng trung bnh ca tri cy hai nng trngvi mc ngha 1%.

Gii. Gi 1; 2 cn nng trung bnh ca tri cy nng trng I v II.Cn kim nh

Gi thit H0 W 1 D 2i thit H1 W 1 2

Mu C mu Trung bnh mu lch chun c hiu chnhI n1 D 100 Nx1 D 101; 2 s21 D 571; 7II n2 D 361 Nx2 D 66; 39 s22 D 29; 72


Tnh gi tr

t D j Nx1 Nx2jss21n1C s

22

n2

D j101; 2 66; 39jr571; 7

100C 29; 72

361

D 14; 4549

Tra bng A.2 ta c t12D t0; 495 D 2; 58: Vy t > t1

2cho nn bc

b gi thit H0 hay cn nng trung bnh ca tri cy hai a phngkhng bng nhau.

V d 8.6. o ng knh 20 trc my do my I sn xut v 22 trcmy do my II sn xut ta tnh c Nx1 D 251; 7I s21 D 52; 853 v Nx2 D249; 8I s22 D 56; 2. C th xem ng knh trung bnh ca cc trc my 2 my nh nhau vi mc ngha 1% khng?

Gii.

8.5 So snh hai t l

Gi p1Ip2 t l phn t A trn tng th 1 v 2 cha bit. Ta cn kimnh

Gi thit H0 W p1 D p2i thit H1 W p1 p2

8.5 So snh hai t l Trang 119

Tnh: f D n1f1 C n2f2n1 C n2

(T l phn t A chung ca 2 mu), trong

f1If2 t l phn t A trn mu 1, 2.

t D jf1 f2jsf .1 f /

1

n1C 1

n2

Kt lun:

Chp nhn gi thit H0 khi t t12:

Bc b gi thit H0 khi t > t12:

V d 8.7. T hai m ng tin hnh 2 mu vi n1 D 100; n2 D 120 tnhc t l phn t loi A trn mu 1, 2 ln lt f1 D 0; 2 v f2 D 0; 3.Vi mc ngha D 1% cho kt lun t l phn t A ca 2 m ng cnh nhau khng.

Gii. Tnh f D 20C 36100C 120 D 0; 255.

Gi p1; p2 (cha bit) t l phn t A trn tng th 1, 2. Cn kimnh gi thit

Gi thit H0 W p1 D p2i thit H1 W p1 p2

t D j0; 2 0; 3js0; 255:0; 745

1

100C 1

120

D 1; 695

Vi D 1% tra bng A.2 tnh c t12D 2; 58. Kt lun chp nhn gi

thit H0 hay t l phn t A trn 2 mu nh nhau.

V d 8.8. Kim tra 120 sinh vin trng A thy c 80 sinh vin gii,150 sinh vin trng B c 90 sinh vin gii. Hi t l sinh vin gii ca2 trng nh nhau khng? Bit mc ngha l 5%.

Gii.


V d 8.9. Kim tra 230 sn phm ca ca ngy thy c 4 sn phmhng. Cn kim tra 160 sn phm ca ca m thy c 3 sn phm hng.Kt lun t l sn phm hng ph thuc vo ca c ng khng vi mc ngha 1%.

Gii.

8.6 Bi tp chng 8

Bi tp 8.1. Bit chiu di ca mt loi sn phm l bin ngu nhin cphn phi chun. o ngu nhin 10 sn phm loi ny th c chiudi trung bnh l 10,02m v lch chun ca mu cha hiu chnhl 0,04m. Kim nh gi thuyt H: chiu di trung bnh ca loi snphm ny l 10,0543m c gi tr kim nh t l bao nhiu v cho kt


lun vi mc ngha 3%. p s: t = 2,5703; chiu di trung bnhca loi sn phm ny l 10,0543m vi mc ngha 3%

Bi tp 8.2. Kho st v thi gian t hc (gi/tun) ca sinh vin hchnh quy trng i hc A trong hc k ny. Tin hnh ly mu,ngi ta thu c bng s liu:

Thi gian 3 5 5 7 7 9 9 11 11 13S sinh vin 5 14 16 8 6

a. Tm khong c lng thi gian t hc trung bnh trong tun casinh vin trng A vi tin cy 95%. p s: (7,1817gi/tun;8,4917gi/tun)

b. c lng thi gian t hc trung bnh trong tun vi tin cy95% v chnh xc nh hn " D 0; 6(gi/tun) th c mu nh nhtl bao nhiu? p s: 59


c. S dng mu ban u c lng thi gian t hc trung bnhtrong tun vi chnh xc " D 0; 6(gi/tun) th m bo tincy l bao nhiu? p s: 92,82%

d. Nhng sinh vin c thi gian t hc t 9(gi/tun) tr ln gi lsinh vin chm hc. Vi tin cy 95% khong c lng t lsinh vin chm hc l bao nhiu? p s: (15,92%; 41,22%)

e. Nhng sinh vin c thi gian t hc t 9(gi/tun) tr ln gi lsinh vin chm hc. c lng t l sinh vin chm hc vi tin cy 95% v chnh xc nh hn " D 0; 12 th c mu nhnht l bao nhiu? p s: 55


f. Nhng sinh vin c thi gian t hc t 9(gi/tun) tr ln gi lsinh vin chm hc. S dng mu trn c lng t l sinhvin chm hc vi chnh xc " D 0; 12 th m bo tin cy lbao nhiu? p s: 93,71%

g. Tnh gi tr thng k t kim nh gi thuyt H: thi gian t hctrung bnh ca sinh vin trng A l 8,4(gi/tun) v cho kt lunvi mc ngha 5%. p s: t = 1,6855; thi gian t hc trungbnh ca sinh vin trng A l 8,4(gi/tun) vi mc ngha5%

h. Trong kim nh gi thuyt H: thi gian t hc trung bnh ca sinhvin trng A l 8,4(gi/tun), mc ngha ti a gi thuyt Hc chp nhn l bao nhiu? p s: 9,1%

i. Nhng sinh vin c thi gian t hc t 9(gi/tun) tr ln gi l sinhvin chm hc. Tnh gi tr thng k t kim nh gi thuyt H:


t l sinh vin chm hc trng A l 18% v cho kt lun vimc ngha 5%. p s: t = 1,9261; t l sinh vin chm hc trng A l 18% vi mc ngha 5%

j. Nhng sinh vin c thi gian t hc t 9(gi/tun) tr ln gi lsinh vin chm hc. Trong kim nh gi thuyt H: t l sinhvin chm hc trng A l 18%, mc ngha ti a gi thuytH c chp nhn l bao nhiu? p s: 5,36%

k. Trng B kho st 64 sinh vin v thi gian t hc. Ngi ta tnhc lch chun ca mu c hiu chnh l 2(gi/tun) v trungbnh mu l 8,5(gi/tun). Tnh gi tr thng k t kim nh githuyt H: thi gian t hc trung bnh trong tun ca sinh vin haitrng l nh nhau v cho kt lun vi mc ngha 5%. p s:t = 1,5893; thi gian t hc trung bnh trong tun ca sinhvin hai trng l nh nhau mc ngha 5%


l. Trng B kho st 64 sinh vin v thi gian t hc. Ngi ta tnhc lch chun ca mu c hiu chnh l 2(gi/tun) v trungbnh mu l 8,5(gi/tun). Trong kim nh gi thuyt H: thi giant hc trung bnh trong tun ca sinh vin hai trng l nh nhau,mc ngha ti a gi thuyt H c chp nhn l bao nhiu?p s: 11,18%

m. Nhng sinh vin c thi gian t hc t 9(gi/tun) tr ln gi lsinh vin chm hc. Trng B kho st 64 sinh vin v thi giant hc thy c 28 sinh vin chm hc. Tnh gi tr thng k t kim nh gi thuyt H: t l sinh vin chm hc ca hai trngl nh nhau v cho kt lun vi mc ngha 5%. p s: t =1,6546; t l sinh vin chm hc ca hai trng l nh nhauvi mc ngha 5%

n. Nhng sinh vin c thi gian t hc t 9(gi/tun) tr ln gi lsinh vin chm hc. Trng B kho st 64 sinh vin v thi giant hc thy c 28 sinh vin chm hc. Trong kim nh gi thuytH: t l sinh vin chm hc ca hai trng l nh nhau, mc ngha ti a gi thuyt H c chp nhn l bao nhiu? p s:9,7%


Bi tp 8.3. Kho st thi gian (tun) mang thai ca thai ph khnght thuc. Tin hnh ly mu, ngi ta c s liu cho nh bng sau:

Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Khong c lng thi gian mang thai trung bnh ca thai ph vi tin cy 95% l:

A. (39,1049 tun; 39,7215 tun)C. (37,1049 tun; 37,7215 tun)

B. (38,1049 tun; 38,7215 tun)D. (40,1049 tun; 40,7215 tun)


Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

c lng thi gian mang thai trung bnh ca thai ph vi tin cy95% v chnh xc nh hn " D 0; 25(tun) th c mu nh nht l:

A. 175 B. 185 C. 195 D. 165



Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

S dng mu trn c lng thi gian mang thai trung bnh ca thaiph vi chnh xc " D 0; 25(tun) th m bo tin cy:

A. 86,82% B. 87,82% C. 88,82% D. 89,82%


Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Nhng thai ph c thi gian mang thai di 36 tun l thai ph sinhnon. Vi tin cy 95% khong c lng t l thai ph sinh non:

A. (2,63%; 10,95%)C. (4,63%; 12,95%)

B. (3,63%; 11,95%)D. (1,63%; 9,95%)



Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Nhng thai ph c thi gian mang thai di 36 tun l thai ph sinhnon. c lng t l thai ph sinh non vi tin cy 95% v chnhxc nh hn " D 0; 04 th c mu nh nht l:

A. 121 B. 141 C. 151 D. 131


Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Nhng thai ph c thi gian mang thai di 36 tun l thai ph sinhnon. S dng mu trn c lng t l thai ph sinh non vi chnhxc " D 0; 04 th m bo tin cy:

A. 91,99% B. 95,99% C. 93,99% D. 97,99%



Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Gi tr thng k t kim nh gi thuyt H: thi gian mang thai trungbnh ca thai ph l 39,7 tun l:A. t = 1,8231; thi gian mang thai trung bnh ca thai ph l 39,7tun vi mc ngha 7%B. t = 1,8231; thi gian mang thai trung bnh ca thai ph l 39,7tun vi mc ngha 5%C. t = 2,8231; thi gian mang thai trung bnh ca thai ph ln hn39,7 tun vi mc ngha 5%D. t = 2,8231; thi gian mang thai trung bnh ca thai ph nh hn39,7 tun vi mc ngha 3%



Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Trong kim nh gi thuyt H: thi gian mang thai trung bnh ca thaiph l 39,7 tun, mc ngha ti a gi thuyt H c chp nhnl:

A. 6,72% B. 7,72% C. 8,72% D. 9,72%


Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Nhng thai ph c thi gian mang thai di 36 tun l thai ph sinhnon. Gi tr thng k t kim nh gi thuyt H: t l thai ph sinhnon l 12% l:A. t = 2,1037; t l thai ph sinh non thp hn 12% vi mc ngha5%B. t = 2,1037; t l thai ph sinh non ln hn 12% vi mc ngha5%C. t = 1,1037; t l thai ph sinh non cao hn 12% vi mc ngha5%D. t = 1,1037; t l thai ph sinh non l 12% vi mc ngha 5%



Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Nhng thai ph c thi gian mang thai di 36 tun l thai ph sinhnon. Trong kim nh gi thuyt H: t l thai ph sinh non l 12%,mc ngha ti a gi thuyt H c chp nhn l:

A. 3,48% B. 4,48% C. 5,48% D. 6,48%


Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Kho st thi gian mang thai ca 100 thai ph c ht thuc v tnhc thi gian mang thai trung bnh l 38,5 tun v lch chun camu c hiu chnh 3,5 tun. Gi tr thng k t kim nh gi thuytH: Thi gian mang thai ca thai ph ht thuc v khng ht thuc lnh nhau l:


A. t = 1,3798; Thi gian mang thai ca thai ph ht thuc v khnght thuc l nh nhau vi mc ngha 5%B. t = 1,3798; Thi gian mang thai ca thai ph ht thuc nh hnvi mc ngha 5%C. t = 2,3798; Thi gian mang thai ca thai ph ht thuc ln hnvi mc ngha 5%D. t = 2,3798; Thi gian mang thai ca thai ph ht thuc nh hnvi mc ngha 5%


Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Kho st thi gian mang thai ca 100 thai ph c ht thuc v tnhc thi gian mang thai trung bnh l 38,5 tun v lch chun camu c hiu chnh 3,5 tun. Trong kim nh gi thuyt H: Thi gianmang thai ca thai ph ht thuc v khng ht thuc l nh nhau,mc ngha ti a gi thuyt H c chp nhn l

A. 2,74% B. 3,74% C. 1,74% D. 4,74%



Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Nhng thai ph c thi gian mang thai di 36 tun l thai ph sinhnon. Kho st thi gian mang thai ca 100 thai ph c ht thuc v tnhc thi gian mang thai thy c 16 thai ph sinh non. Gi tr thng kt kim nh gi thuyt H: t l sinh non ca thai ph c ht thucv khng ht thuc l nh nhau l:A. t = 2,4753; t l sinh non ca thai ph khng ht thuc ln hnvi mc ngha 5%B. t = 2,4753; t l sinh non ca thai ph c ht thuc ln hn vimc ngha 5%C. t = 1,4753; t l sinh non ca thai ph khng ht thuc ln hnvi mc ngha 5%D. t = 1,4753; t l sinh non ca thai ph c ht thuc ln hn vimc ngha 5%


Thi gian 34 36 36 38 38 40 40 42 42 44S thai ph 7 10 59 41 4

Nhng thai ph c thi gian mang thai di 36 tun l thai ph sinhnon. Kho st thi gian mang thai ca 100 thai ph c ht thuc v tnh

Ph lc . Kim nh gi thit

c thi gian mang thai thy c 16 thai ph sinh non. Trong kim nhgi thuyt H: t l sinh non ca thai ph c ht thuc v khng htthuc l nh nhau, mc ngha ti a gi thuyt H c chp nhnl:

A. 1,32% B. 2,32% C. 3,32% D. 4,32%

p n cu hi trc nghim

8.3 A

8.4 B

8.5 C

8.6 D

8.7 D

8.8 C

8.9 B

8.10 A

8.11 A

8.12 A

8.13 D

8.14 C

8.15 B

8.16 A

Ph lc A

Cc bng gi tr xc sut

Ph lc A. Cc bng gi tr xc sut

A.1 Bng gi tr f .z/

z

f .z/

O

z 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,090,0 0,3989 0,3989 0,3989 0,3988 0,3986 0,3984 0,3982 0,3980 0,3977 0,39700,1 0,3970 0,3965 0,3961 0,3956 0,3951 0,3945 0,3939 0,3932 0,3925 0,39110,2 0,3910 0,3902 0,3894 0,3885 0,3876 0,3867 0,3857 0,3847 0,3836 0,38150,3 0,3814 0,3802 0,3790 0,3778 0,3765 0,3752 0,3739 0,3725 0,3712 0,36840,4 0,3683 0,3668 0,3653 0,3637 0,3621 0,3605 0,3589 0,3572 0,3555 0,35220,5 0,3521 0,3503 0,3485 0,3467 0,3448 0,3429 0,3410 0,3391 0,3372 0,3334

0,6 0,3332 0,3312 0,3292 0,3271 0,3251 0,3230 0,3209 0,3187 0,3166 0,31250,7 0,3123 0,3101 0,3079 0,3056 0,3034 0,3011 0,2989 0,2966 0,2943 0,28990,8 0,2897 0,2874 0,2850 0,2827 0,2803 0,2780 0,2756 0,2732 0,2709 0,26630,9 0,2661 0,2637 0,2613 0,2589 0,2565 0,2541 0,2516 0,2492 0,2468 0,24221,0 0,2420 0,2396 0,2371 0,2347 0,2323 0,2299 0,2275 0,2251 0,2227 0,2181

1,1 0,2179 0,2155 0,2131 0,2107 0,2083 0,2059 0,2036 0,2012 0,1989 0,19441,2 0,1942 0,1919 0,1895 0,1872 0,1849 0,1826 0,1804 0,1781 0,1758 0,17161,3 0,1714 0,1691 0,1669 0,1647 0,1626 0,1604 0,1582 0,1561 0,1539 0,14991,4 0,1497 0,1476 0,1456 0,1435 0,1415 0,1394 0,1374 0,1354 0,1334 0,12971,5 0,1295 0,1276 0,1257 0,1238 0,1219 0,1200 0,1182 0,1163 0,1145 0,1111

1,6 0,1109 0,1092 0,1074 0,1057 0,1040 0,1023 0,1006 0,0989 0,0973 0,09421,7 0,0940 0,0925 0,0909 0,0893 0,0878 0,0863 0,0848 0,0833 0,0818 0,07911,8 0,0790 0,0775 0,0761 0,0748 0,0734 0,0721 0,0707 0,0694 0,0681 0,06571,9 0,0656 0,0644 0,0632 0,0620 0,0608 0,0596 0,0584 0,0573 0,0562 0,05412,0 0,0540 0,0529 0,0519 0,0508 0,0498 0,0488 0,0478 0,0468 0,0459 0,0441

2,1 0,0440 0,0431 0,0422 0,0413 0,0404 0,0396 0,0387 0,0379 0,0371 0,03562,2 0,0355 0,0347 0,0339 0,0332 0,0325 0,0317 0,0310 0,0303 0,0297 0,02842,3 0,0283 0,0277 0,0270 0,0264 0,0258 0,0252 0,0246 0,0241 0,0235 0,02242,4 0,0224 0,0219 0,0213 0,0208 0,0203 0,0198 0,0194 0,0189 0,0184 0,01762,5 0,0175 0,0171 0,0167 0,0163 0,0158 0,0154 0,0151 0,0147 0,0143 0,0136

2,6 0,0136 0,0132 0,0129 0,0126 0,0122 0,0119 0,0116 0,0113 0,0110 0,01042,7 0,0104 0,0101 0,0099 0,0096 0,0093 0,0091 0,0088 0,0086 0,0084 0,00792,8 0,0079 0,0077 0,0075 0,0073 0,0071 0,0069 0,0067 0,0065 0,0063 0,00602,9 0,0060 0,0058 0,0056 0,0055 0,0053 0,0051 0,0050 0,0048 0,0047 0,00443,0 0,0044 0,0043 0,0042 0,0040 0,0039 0,0038 0,0037 0,0036 0,0035 0,0033

A.1 Bng gi tr f .z/ Trang 137

z 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,093,1 0,0033 0,0032 0,0031 0,0030 0,0029 0,0028 0,0027 0,0026 0,0025 0,00243,2 0,0024 0,0023 0,0022 0,0022 0,0021 0,0020 0,0020 0,0019 0,0018 0,00173,3 0,0017 0,0017 0,0016 0,0016 0,0015 0,0015 0,0014 0,0014 0,0013 0,00123,4 0,0012 0,0012 0,0012 0,0011 0,0011 0,0010 0,0010 0,0010 0,0009 0,00093,5 0,0009 0,0008 0,0008 0,0008 0,0008 0,0007 0,0007 0,0007 0,0007 0,0006

3,6 0,0006 0,0006 0,0006 0,0005 0,0005 0,0005 0,0005 0,0005 0,0005 0,00043,7 0,0004 0,0004 0,0004 0,0004 0,0004 0,0004 0,0003 0,0003 0,0003 0,00033,8 0,0003 0,0003 0,0003 0,0003 0,0003 0,0002 0,0002 0,0002 0,0002 0,00023,9 0,0002 0,0002 0,0002 0,0002 0,0002 0,0002 0,0002 0,0002 0,0001 0,0001

Bng A.1: Gi tr f .z/

Ph lc A. Cc bng gi tr xc sut

A.2 Bng gi tr '.x/'.x/

O x

x 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,090,0 0,0000 0,0040 0,0080 0,0120 0,0160 0,0199 0,0239 0,0279 0,0319 0,03590,1 0,0398 0,0438 0,0478 0,0517 0,0557 0,0596 0,0636 0,0675 0,0714 0,07530,2 0,0793 0,0832 0,0871 0,0910 0,0948 0,0987 0,1026 0,1064 0,1103 0,11410,3 0,1179 0,1217 0,1255 0,1293 0,1331 0,1368 0,1406 0,1443 0,1480 0,15170,4 0,1554 0,1591 0,1628 0,1664 0,1700 0,1736 0,1772 0,1808 0,1844 0,18790,5 0,1915 0,1950 0,1985 0,2019 0,2054 0,2088 0,2123 0,2157 0,2190 0,2224

0,6 0,2257 0,2291 0,2324 0,2357 0,2389 0,2422 0,2454 0,2486 0,2517 0,25490,7 0,2580 0,2611 0,2642 0,2673 0,2704 0,2734 0,2764 0,2794 0,2823 0,28520,8 0,2881 0,2910 0,2939 0,2967 0,2995 0,3023 0,3051 0,3078 0,3106 0,31330,9 0,3159 0,3186 0,3212 0,3238 0,3264 0,3289 0,3315 0,3340 0,3365 0,33891,0 0,3413 0,3438 0,3461 0,3485 0,3508 0,3531 0,3554 0,3577 0,3599 0,3621

1,1 0,3643 0,3665 0,3686 0,3708 0,3729 0,3749 0,3770 0,3790 0,3810 0,38301,2 0,3849 0,3869 0,3888 0,3907 0,3925 0,3944 0,3962 0,3980 0,3997 0,40151,3 0,4032 0,4049 0,4066 0,4082 0,4099 0,4115 0,4131 0,4147 0,4162 0,41771,4 0,4192 0,4207 0,4222 0,4236 0,4251 0,4265 0,4279 0,4292 0,4306 0,43191,5 0,4332 0,4345 0,4357 0,4370 0,4382 0,4394 0,4406 0,4418 0,4429 0,4441

1,6 0,4452 0,4463 0,4474 0,4484 0,4495 0,4505 0,4515 0,4525 0,4535 0,45451,7 0,4554 0,4564 0,4573 0,4582 0,4591 0,4599 0,4608 0,4616 0,4625 0,46331,8 0,4641 0,4649 0,4656 0,4664 0,4671 0,4678 0,4686 0,4693 0,4699 0,47061,9 0,4713 0,4719 0,4726 0,4732 0,4738 0,4744 0,475 0,4756 0,4761 0,47672,0 0,4772 0,4778 0,4783 0,4788 0,4793 0,4798 0,4803 0,4808 0,4812 0,4817

2,1 0,4821 0,4826 0,4830 0,4834 0,4838 0,4842 0,4846 0,4850 0,4854 0,48572,2 0,4861 0,4864 0,4868 0,4871 0,4875 0,4878 0,4881 0,4884 0,4887 0,48902,3 0,4893 0,4896 0,4898 0,4901 0,4904 0,4906 0,4909 0,4911 0,4913 0,49162,4 0,4918 0,4920 0,4922 0,4925 0,4927 0,4929 0,4931 0,4932 0,4934 0,49362,5 0,4938 0,4940 0,4941 0,4943 0,4945 0,4946 0,4948 0,4949 0,4951 0,4952

2,6 0,4953 0,4955 0,4956 0,4957 0,4959 0,4960 0,4961 0,4962 0,4963 0,49642,7 0,4965 0,4966 0,4967 0,4968 0,4969 0,4970 0,4971 0,4972 0,4973 0,49742,8 0,4974 0,4975 0,4976 0,4977 0,4977 0,4978 0,4979 0,4979 0,4980 0,49812,9 0,4981 0,4982 0,4982 0,4983 0,4984 0,4984 0,4985 0,4985 0,4986 0,49863,0 0,4987 0,4987 0,4987 0,4988 0,4988 0,4989 0,4989 0,4989 0,4990 0,4990

A.2 Bng gi tr '.x/ Trang 139

x 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,093,1 0,4990 0,4991 0,4991 0,4991 0,4992 0,4992 0,4992 0,4992 0,4993 0,49933,2 0,4993 0,4993 0,4994 0,4994 0,4994 0,4994 0,4994 0,4995 0,4995 0,49953,3 0,4995 0,4995 0,4995 0,4996 0,4996 0,4996 0,4996 0,4996 0,4996 0,49973,4 0,4997 0,4997 0,4997 0,4997 0,4997 0,4997 0,4997 0,4997 0,4997 0,49983,5 0,4998 0,4998 0,4998 0,4998 0,4998 0,4998 0,4998 0,4998 0,4998 0,4998

3,6 0,4998 0,4998 0,4999 0,4999 0,4999 0,4999 0,4999 0,4999 0,4999 0,49993,7 0,4999 0,4999 0,4999 0,4999 0,4999 0,4999 0,4999 0,4999 0,4999 0,49993,8 0,4999 0,4999 0,4999 0,4999 0,4999 0,4999 0,4999 0,4999 0,4999 0,49993,9 0,5000 0,5000 0,5000 0,5000 0,5000 0,5000 0,5000 0,5000 0,5000 0,5000

Bng A.2: Gi tr '.x/

Tran

g140

PhlcA.C

cbn

ggi

trxc

sut

A.3 Bng gi tr tn

tn-tn

=2 =2

O

PjT j > tn D

n

0,14 0,13 0,12 0,11 0,10 0,09 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01

1 4,474 4,829 5,242 5,730 6,314 7,026 7,916 9,058 10,579 12,706 15,895 21,205 31,821 63,6572 2,383 2,495 2,620 2,760 2,920 3,104 3,320 3,578 3,896 4,303 4,849 5,643 6,965 9,9253 1,995 2,072 2,156 2,249 2,353 2,471 2,605 2,763 2,951 3,182 3,482 3,896 4,541 5,8414 1,838 1,902 1,971 2,048 2,132 2,226 2,333 2,456 2,601 2,776 2,999 3,298 3,747 4,6045 1,753 1,810 1,873 1,941 2,015 2,098 2,191 2,297 2,422 2,571 2,757 3,003 3,365 4,032

6 1,700 1,754 1,812 1,874 1,943 2,019 2,104 2,201 2,313 2,447 2,612 2,829 3,143 3,7077 1,664 1,715 1,770 1,830 1,895 1,966 2,046 2,136 2,241 2,365 2,517 2,715 2,998 3,4998 1,638 1,687 1,740 1,797 1,860 1,928 2,004 2,090 2,189 2,306 2,449 2,634 2,896 3,3559 1,619 1,666 1,718 1,773 1,833 1,899 1,973 2,055 2,150 2,262 2,398 2,574 2,821 3,25010 1,603 1,650 1,700 1,754 1,812 1,877 1,948 2,028 2,120 2,228 2,359 2,527 2,764 3,169

11 1,591 1,636 1,686 1,738 1,796 1,859 1,928 2,007 2,096 2,201 2,328 2,491 2,718 3,10612 1,580 1,626 1,674 1,726 1,782 1,844 1,912 1,989 2,076 2,179 2,303 2,461 2,681 3,05513 1,572 1,616 1,664 1,715 1,771 1,832 1,899 1,974 2,060 2,160 2,282 2,436 2,650 3,01214 1,565 1,609 1,656 1,706 1,761 1,821 1,887 1,962 2,046 2,145 2,264 2,415 2,624 2,97715 1,558 1,602 1,649 1,699 1,753 1,812 1,878 1,951 2,034 2,131 2,249 2,397 2,602 2,947

A.3Bn

ggi

trtn

Tran

g141

Bng A.3: Bng gi tr tn (tip theo)

n

0,14 0,13 0,12 0,11 0,10 0,09 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01

16 1,553 1,596 1,642 1,692 1,746 1,805 1,869 1,942 2,024 2,120 2,235 2,382 2,583 2,92117 1,548 1,591 1,637 1,686 1,740 1,798 1,862 1,934 2,015 2,110 2,224 2,368 2,567 2,89818 1,544 1,587 1,632 1,681 1,734 1,792 1,855 1,926 2,007 2,101 2,214 2,356 2,552 2,87819 1,540 1,583 1,628 1,677 1,729 1,786 1,850 1,920 2,000 2,093 2,205 2,346 2,539 2,86120 1,537 1,579 1,624 1,672 1,725 1,782 1,844 1,914 1,994 2,086 2,197 2,336 2,528 2,845

21 1,534 1,576 1,621 1,669 1,721 1,777 1,840 1,909 1,988 2,080 2,189 2,328 2,518 2,83122 1,531 1,573 1,618 1,665 1,717 1,773 1,835 1,905 1,983 2,074 2,183 2,320 2,508 2,81923 1,529 1,570 1,615 1,662 1,714 1,770 1,832 1,900 1,978 2,069 2,177 2,313 2,500 2,80724 1,526 1,568 1,612 1,660 1,711 1,767 1,828 1,896 1,974 2,064 2,172 2,307 2,492 2,79725 1,524 1,566 1,610 1,657 1,708 1,764 1,825 1,893 1,970 2,060 2,167 2,301 2,485 2,787

26 1,522 1,564 1,608 1,655 1,706 1,761 1,822 1,890 1,967 2,056 2,162 2,296 2,479 2,77927 1,521 1,562 1,606 1,653 1,703 1,758 1,819 1,887 1,963 2,052 2,158 2,291 2,473 2,77128 1,519 1,560 1,604 1,651 1,701 1,756 1,817 1,884 1,960 2,048 2,154 2,286 2,467 2,76329 1,517 1,558 1,602 1,649 1,699 1,754 1,814 1,881 1,957 2,045 2,150 2,282 2,462 2,75630 1,516 1,557 1,600 1,647 1,697 1,752 1,812 1,879 1,955 2,042 2,147 2,278 2,457 2,750

40 1,506 1,546 1,589 1,635 1,684 1,737 1,796 1,862 1,936 2,021 2,123 2,250 2,423 2,70460 1,496 1,535 1,577 1,622 1,671 1,723 1,781 1,845 1,917 2,000 2,099 2,223 2,390 2,66080 1,491 1,530 1,572 1,616 1,664 1,716 1,773 1,836 1,908 1,990 2,088 2,209 2,374 2,639100 1,488 1,527 1,568 1,613 1,660 1,712 1,769 1,832 1,902 1,984 2,081 2,201 2,364 2,6261000 1,477 1,515 1,556 1,600 1,646 1,697 1,752 1,814 1,883 1,962 2,056 2,173 2,330 2,581

Bng A.3: Gi tr tn

Ti liu tham kho

[1] inh Vn Gng. (1999). L thuyt xc sut v thng k ton. NXBGio dc.

[2] T Anh Dng. (2007). L thuyt xc sut v thng k ton. NXBHQG TP.HCM.

[3] Nguyn Bc Vn. (1999). Xc sut v x l s liu thng k. NXBGio dc.

[4] ng Hn. (1986). Xc sut thng k. NXB Thng k.

[5] Sheldon M. Ross. (1987). Introduction to probability and statisticsfor engineers and scientists. A John Wiley & Sons Publication.

[6] F.M. Dekking. (2005). A modern introduction to Probability andStatistics. Springer Publication.

[7] T.T. Song. (2004). Fundamentals of probability and statistics forengineers. A John Wiley & Sons Publication.

[8] Ronald N. Forthofer. (2007). Biostatistics: Aguide to design, analy-sis, and discovery. Academic Press.

[9] Y. Suhov. (2005). Volume I: Basic probability and statistics. Cam-bridge University Press.

[10] Michaelr. Chernick. (2003). Introductory biostatistics for the healthsciences. A John Wiley & Sons Publication.

[11] E.L. Lehmann. (2005). Testing statistical hypotheses: Third Edi-tion. Springer Publication.

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Bi tp chng 1

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Bi tp chng 2

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Bi tp chng 5

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Ti liu tham kho

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