Scarica esercizi di matematica e più Esercizi in PDF di Matematica solo su Docsity!
H+mMB +QKTBiB /B T`2+2/2MiB TT2HHB
:X J `+?2iiB
kyRe p2`X R
AM/B+
*R QKTBiQ k ;2MM BQ kyRk R
*k QKTBiQ Rj 72##` BQ kyRk k
*j QKTBiQ /2H k b2ii2K#`2 kyR8 k
*9 QKTBiQ 6 /2H d ;Bm;MQ kyRk d
*8 QKTBiQ > /2H d ;Bm;MQ kyRk RR
R *QKTBiQ k ;2MM BQ kyRk
RkX aB X mM p B #BH2 + bm H2 /BbiB#mBi +QK2 mM MQK H2 +QM K2/B _μ_ 2 /2pB xBQM2 bi M/ / σ X aB MQ
X 1 2
X 2 H2 K2/B2 /B /m2 + KTBQMB /B /BK2MbBQM2 BbT2iiBp K2Mi2 T B Re 2 k8 mMBi¨ b2H2xBQM iB
+ bm HK2Mi2 2/ BM/BT2M/2Mi2K2Mi2 / HH TQTQH xBQM2 /2b+Bii / HH pX+X _X_ X aB +QMbB/2BMQ H2 b2;m2MiB
2bT`2bbBQMB,
(1) P (
X 1 < μ ) , (2) P (
X 2 < μ )
Zm H2 /2HH2 b2;m2MiB z2K xBQMB ĕ p2 \
V LQM ĕ TQbbB#BH2 bi #BHB2 mM 2H xBQM2 i H2 /m2 2bT2bbBQMBX
"V AH p HQ2 /2HH 2bT2bbBQM2 R ĕ m;m H2 H p HQ2 /2HH 2bT2bbBQM2 k X
*V AH p HQ2 /2HH 2bT2bbBQM2 R ĕ K ;;BQ2 /2H p HQ2 /2HH 2bT`2bbBQM2 kX
.V AH p HQ2 /2HH 2bT2bbBQM2 k ĕ K ;;BQ2 /2H p HQ2 /2HH 2bT`2bbBQM2 RX
aQHmxBQM
X
1
∼ N ( μ, σ
2 / 16) 2
X
2
∼ N ( μ, σ
2 / 25)X ZmBM/B
P (
X
1
< μ ) = P ( Z < 0) = 0_._ 5 P (
X
2
< μ ) = P ( Z < 0) = 0_._ 5
S2`+Bǁ "VX
RjX a2 X ĕ mM p B #BH2 + bm H2 #BMQKB H2 +QM _n_ = 20- 2 _p_ = 0_._ 5 - HHQ SUs 4 kyV 4 RXyX0 o2`Q Q 6 HbQ\
aQHmxBQM
P ( X = 20) =
20 · 0_._ 5
0 = 0_._ 5
20 " = 1
[mBM/B 6 HbQX
R8X AM mM TB++QH mMBp2bBi¨ +B bQMQ dk3 K iB+QH2 2- i HQQ- kRR mb MQ aFvT2X aB +QMbB/2`B mM + KTBQM2 /B
e8 K iB+QH2X G TQ# #BHBi¨ +?2 H TQTQxBQM2 + KTBQM B bB BM72BQ2 yXj ĕ Umb 2 9 +B7`2 /2+BK HBV
V yX9k9N
"V yX8dR
*V yX9j
.V yX83ee
R
aQHmxBQM2 aB +QMbB/2 mM TQTQH xBQM2 /B+QiQKB+ +QM mM TQTQ`xBQM2 /B bm++2bbB U+?B mb aFvT2V p =
211 / 728 X . iQ +?2 BH + KTBQM2 /B e8 K iB+QH2 ĕ ## bi Mx ; M/2 U2 np (1− p ) = 13_._ 379 > 9 V H TQTQxBQM
P /B bm++2bbB M2H + KTBQM2 ĕ TTQbbBK iBp K2Mi2 MQK H2 +QM K2/B p = 211 / 728 2 +QM /2pB xBQM2 bi M/ `/
σ =
p (1 − p ) /n = 0_._ 05627279 X
G TQ# #BHBi¨B+?B2bi ĕ +?2 H TQTQxBQM2 + KTBQM `B bB < 0_._ 3 +BQĕ
P (
P < 0_._ 30) = P [(0_._ 30 − p ) /σ )] = P ( Z < 0_._ 18) = 0_._ 5714_._
ZmBM/B H `BbTQbi ĕ "VX
*k QKTBiQ Rj 72##` BQ kyRk
ReX L2;HB mHiBKB Ry MMB- B 2M/BK2MiB /2HHQ aiQ+F bQMQ bi iB BM K2/B /2HHǶ3X9W +QM /2pB xBQM2 bi M/ /
kXRW 2 [m2HHB /2HHQ aiQ+F " bQMQ bi iB BM K2/B /2H jXeW +QM /2pB xBQM2 bi M/ `/ yXNWX Zm H2 /2HH
b2;m2MiB z2K xBQMB ĕ p2 \
V GQ aiQ+F " ? mM p B #BHBi¨2H iBp TBɍ TB++QH /2HHQ aiQ+F X
"V A /m2 biQ+F T2b2Mi MQ H bi2bb p B #BHBi¨ `2H iBp X
*V GQ aiQ+F ? mM p B #BHBi¨2H iBp TBɍ TB++QH /2HHQ aiQ+F "X
.V LQM + ĕ bm{+B2Mi2 BM7QK xBQM2 T2 /B2 [m H2 /2B /m2 biQ+F T2b2Mi H p B #BHBi¨2H iBp TBɍ TB++QH X
aQHmxBQM2 G p B #BHBi¨2H iBp bB KBbm +QM BH +Q2{+B2Mi2 /B p B xBQM2 CV = σ/μ X ZmBM/B
CV
A
= 2. 1 / 8. 4 = 0. 25 , CV
B
2 [mBM/B 2 " ? MMQ H bi2bb p B #BBi¨2H iBp X "VX
*j QKTBiQ /2H k b2ii2K#`2 kyR
RX lM + KTBQM2 + bm H2 /B /BK2MbBQM2 R8 2bi iiQ / mM TQTQH xBQM2 /BbiB#mBi MQK HK2Mi2 7QMBb+
mM K2/B + KTBQM B /B d8 2/ mM p B Mx + KTBQM B /B k8X AH HBKBi2 bmT2BQ2 /B mM BMi2p HHQ /B
+QM}/2Mx H N8W T2H K2/B + KTBQM B b ¨ T B ,
V dkXdkd
"V ddXdeN
*V dkXkjR
.V ddXkdj
aQHmxBQM
GBKBi2 bmT2BQ2 4 ¯ x + t 14 ,α/ 2
s/
n = 75 + 2_._ 145
(25 / 15) = 77. 769. "V
kX aB Z ∼ N ( μ = 0 , σ
2 = 1)X * H+QH `2 P (− 0_._ 44 < Z < 1_._ 2)X
aQHmxBQM2 SQbiQ F ( z ) = P ( Z ≤ z ) Ui pQH R /2HH MQK H2- 7mMxBQM2 /BBT `iBxBQM2V
(− 0. 44 < Z < 1. 2) = F (1. 2) − [1 − F (0. 44)] = 0. 5549.
jX GǶ xB2M/ TQ/miiB+2 /B mM MmQpQ K ++?BM BQ bb2Bb+2 +?2 [m2biQ T`Q/m+2 BM K2/B HK2MQ kN mMBi¨
H ;BQMQ BM TBɍ /2H K ++?BM BQ iim HK2Mi2 BM mbQ BM mMǶBM/mbiB K Mm7 iimB2 X AH K M ;2 /2+B/2 /B
+[mBbi 2 R8 MmQpB K ++?BM B 2 Qbb2p +?2 H TQ/mxBQM2 K2/B ;BQM HB2 mK2Mi bQHQ /B ke mMBi¨
+QM mM /2pB xBQM2 bi M/ / /B eXkX 6Bbb iQ _α_ = 0_._ 05 - [m H2 /Qp2##2 2bb22 H 2;QH /B /2+BbBQM2\
V aB `B}mi >y b2 H bi iBbiB+ i ĕ < − 2_._ 145 X
"V aB `B}mi >y b2 H bi iBbiB+ i ĕ > 2_._ 145 X
*V aB `B}mi >y b2 H bi iBbiB+ i ĕ < − 1_._ 761 X
k
aQHmxBQM2 a2 X ∼ N ( μ, σ
2 ) HHQ H K2/B M2HHǶmMBp2bQ /2B + KTBQMB ĕ
X ∼ N ( μ, σ
2 /n )X ZmBM/B [m HmM[m
bB n
X 2 MQK H2X o2QX
3X . i mM p B #BH2 + bm H2 MQK H2 X +QM K2/B dy 2 /2pB xBQM2 bi M/ / Rk- BH p HQ2 /2HH p `B #BH
+ bm H2 MQK H2 bi M/ /Bxx i Z +Q``BbTQM/2Mi2 s 4 3k ĕ TBɍ ; M/2 /B x2QX o2`Q UhV Q 6 HbQ U6V\
aQHmxBQM2 Z bB QiiB2M2 / X +QM H i b7QK xBQM
Z =
X − μ
σ
X − 70
[mBM/B b2 X = 82
Z =
ZmBM/B ĕ p2`QX
NX lM 2+2Mi2 BM/ ;BM2 ? bim/B iQ H +QM/BxBQM2 H pQ iBp /2HH2 +QTTB2 +?2 +QbiBimBb+QMQ mM 7 KB;HB X
A / iB KQbi MQ +?2 M2HH 33W /2HH2 +QTTB2 HK2MQ mMQ /2B K2K#B H pQ` X L2H kyW /2HH2 +QTTB2 BM +mB
H /QMM MQM H pQ - M2KK2MQ HǶmQKQ H pQ X AMQHi2- M2H 9yW /2HH2 +QTTB2 BM +mB H mQKQ MQM H pQ -
M2KK2MQ H /QMM H pQ X Zm H ĕ H TQ# #BHBi¨ +?2 Mû HǶmQKQ Mû H /QMM H pQ`BMQ\
aQHmxBQM
aB MQ M 2/ F ;HB 2p2MiB M = HǶmQKQ H pQ 2 _F_ 4 H /QMM H pQ X .mM[m2 P ( M ∪ F ) = 0_._ 88 X AMQHi`
P (
M |
F ) = 0. 20 2 P (
F |
M ) = 0. 4.
HHQ`
P (
M ∩
F ) = 1 − P ( M ∪ F ) = 1 − 0. 88 = 0. 12.
:HB HiB / iB MQM b2pQMQX
RyX G2 p2M/Bi2 / T i2 /B mM ;QbbBbi /B BKTB MiB B/ mHB+B pp2M;QMQ i KBi2 bT2/BxBQM2 QTTm2 TQMi
+QMb2;M X AM mM K2b2 H2 p2M/Bi2 T`QMi +QMb2;M ? MMQ mM K2/B /B 0-RykNdk +QM mM /2pB xBQM
bi M/ / /B 0-Rj8kjX .m Mi2 HQ bi2bbQ K2b2 H2 bT2/BxBQMB ? MMQ mM K2/B /B 0-k9kj89 +QM mM
/2pB xBQM2 bi M/ / /B 0-k9N8eX amTTQMB KQ +?2 H2 p2M/Bi2 i KBi2 bT2/BxBQM2 2 T`QMi +QMb2;M
bB MQ BM/BT2M/2MiBX * H+QH 2 H K2/B 2 H /2pB xBQM2 bi M/ / /2H iQi H2 /2HH2 p2M/Bi2 K2MbBHBX
aQHmxBQM
aB X 4 p2M/Bi2 TQMi +QMb2;M 2 _Y_ 4 p2M/Bi2 T2 bT2/BxBQM2X a TTB KQ +?
μ X
= 102972 μ Y
= 242354; σ X
= 13523 , σ Y
G K2/B 2 H /2pB xBQM2 bi M/ `/ /2HH2 p2M/Bi2 iQi HB T = X + Y bQMQ
μ T
= μ X
2- TQB+?û X 2 Y bQMQ BM/BT2M/2MiB 2 [mBM/B σXY = 0-
σ T
σ
2
X
2
Y
RRX *QMbB/2 H b2;m2Mi2 /BbiB#mxBQM2 /2H MmK2Q /B bim/2MiB b2+QM/Q BH MmK2Q /B bb2Mx2 b+QH biB+?2X
Zm MiB bim/2MiB ? MMQ 7 iiQ mM bQH bb2Mx \
JQ/X 62[m2Mx S2+2Mim H2 *mKmH i
y kyy kyXyy
R ∗ 8yXyy
k j8y 38Xyy
j Ryy N8Xyy
9 8y RyyXyy
V k8y
"V jyy
*V 8y
.V R8y
aQHmxBQM
" bi iQp 2 H2 72[m2Mx22H iBp2 T2+2Mim HB 2 7 2 mM TQTQxBQM2 T2H2 TBK2 /m2 `B;?2X
JQ/X 62[m2Mx S2+2Mim H
y kyy kyXyy
R x jyXyy
S2`+Bǁ
200 : 20_._ 00 = x : 30_._ 00
+BQĕ x = 200 · 30 / 20 = 300X
RkX G TQ# #BHBi¨ +?2 mM T2bQM +[mBbiB mM +QbimK2 / # ;MQ /m Mi2 H2 p + Mx2 6Qi2 /2B J `KB
ĕ yX9X aB b2H2xBQM MQ + bQ Ry T2bQM2X Zm H ĕ H /2pB xBQM2 bi M/ / /2H MmK2Q /B T2bQM2 +?
+QKT2 MMQ mM +QbimK2\
V RXRk
"V RXk
*V RX89N
.V RXke
aQHmxBQM2 a2 S 4 MmK2Q /B T2bQM2 +?2 +QKT2 MMQ BH +QbimK2 bm Ry b2H2xBQM iB + bm HK2Mi2 +QM `BT2@
iBxBQM2 S = X 1 + · · · + X 10 ? /Bbi`B#mxBQM2 #BMQKB H2 +QM K2/B μS = np = 10 · 0_._ 4 = 4 2 /2pB xBQM
bi M/ `/
σ =
np (1 − p ) =
RjX lM bi iBbiB+ + KTBQM B i H2 +?2 H K2/B /2B p HQB +?2 Tmǁ bbmK22 M2HH mMBp2bQ /2B + KTBQMB ĕ
m;m H2 H p HQ2 /2H T K2iQ /2HH TQTQH xBQM2 +?2 pmQH2 biBK 2 ĕ mMQ biBK iQ2 +Q``2iiQX o2Q UhV
Q 6 HbQ U6V\
aQHmxBQM
aŢ ĕ H /2}MxBQM2 2b ii X
R9X a2 H +Qp B Mx i /m2 p B #BHB + bm HB _X_ 2 _Y_ ĕ TQbBiBp - HHQ var( X − Y ) ĕ TBɍ ;` M/2 /B var( X + Y )X
o2`Q UhV Q 6 HbQ U6V\
aQHmxBQM2 SQB+?û σ XY
var( X − Y ) = σ
2
X
2
Y
− 2 σ XY
≤ σ
2
X
2
Y
= var( X + Y )
ZmBM/B 6 HbQX
R8X . mMǶBM/ ;BM2 +QM/Qii bm 9dk QT2 B- ĕBbmHi iQ +?2 BH ejW T2Mb /B /2B2 / mMQ b+BQT2`Q bBM/ + H
M2B TQbbBKB i2 K2bBX Zm HB /2B b2;m2MiB BMi2p HHB TT2b2Mi mM BMi2p HHQ /B +QM}/2Mx H N3W T2`
H TQTQxBQM2 /2;HB QT2 B +?2 ? MMQ TB MB}+ iQ /B /2B2 / mMQ b+BQT2Q M2B TQbbBKB i2 K2bB\
V 0. 63 ± 0. 042
"V 0. 63 ± 0. 052
*V 0. 63 ± 0. 047
.V 0. 63 ± 0. 057
aQHmxBQM2 AH K ;BM2 /B 2``Q2 UT2; M/B + KTBQMBV ĕ
M E = z 0_._ 01
ˆ p (1 − ˆ p ) /n = 2_._ 326
LQi +?2 BH HBp2HHQ /B +QM}/2Mx ĕ 1 − α = 0_._ 98 / +mB α/ 2 = 0_._ 01
P (4. 0 ≤ X ≤ 4. 2) = P
≤ Z ≤
= P (− 0. 24 ≤ Z ≤ 0. 24) = F (0. 24)−(1− F (0. 24)) = 0. 1896.
*9 QKTBiQ 6 /2H d ;Bm;MQ kyRk
RX GǶ xB2M/ TQ/miiB+2 /B b ++?B /B +2K2MiQ z2`K +?2 +B b+mM b ++Q +QMiB2M2 HK2MQ 8yXR F; /B +2K2MiQX
G /2pB xBQM2 bi M/ / /2HH [m MiBi¨ /B +2K2MiQ +QMi2Mmi BM +B b+mM b ++Q ĕ RXk F;X G 2;QH /B
/2+BbBQM2 /Qii i / HHǶ xB2M/ ĕ /B K2ii22 BM K Mmi2MxBQM2 mM K ++?BM B2KTBi`B+2 b2 H K2/B
+ KTBQM B /2HH [m MiBi¨ /B +2K2MiQ BM mM + KTBQM2 /B 9y b ++?B ĕ BM72BQ`2 9NXdX Zm H ĕ H
TQ# #BHBi¨ /B +QKK2ii22 mM 2``Q2 /2H TBKQ iBTQ\
V yXyk
"V yXyRd
*V yXyjy
.V yXyje
aQHmxBQM
z2`K xBQM2, μ ≥ 50_._ 1 /Qp2 μ = [m MiBi¨ K2/B /B +2K2MiQX Zm2bi ĕ HǶBTQi2bB H 0
T2`+?û +QMiB2M2 BH b2;MQ
/B m;m ;HB Mx X G /2pB xBQM2 bi M/ / _σ_ = 1_._ 2 ĕ T2 H TQTQH xBQM2 T2+?û M+Q MQM ĕ bi iQ 2bi` iiQ
M2bbmM + KTBQM2X G 2;QH /B /2+BbBQM2 ĕ,B}mi U2 K2iiB BM K Mmi2MxBQM2 H K ++?BM V b
X < 49_._ 7 BM mM + KTBQM2 /B 9y b ++?BX
G TQ# #BHBi¨ /B +QKK2ii22 2``Q`2 /2H A iBTQ ĕ BH T@p Hm2 bbQ+B iQ HH K2/B 49_._ 7 X
G bi iBbiB+ i2bi T2_H_ 0 : _μ_ ≥ 50_._ 1 +QMiQ H 1 : μ < 50_._ 1 ĕ
Z =
X − 50. 1
∼ N (0 , 1)
P` b
X = 49. 7
Z = (49. 7 − 50. 1) / (1. 2 /
AH T@p Hm2 ĕ
P ( Z ≤ − 2. 11) = 1 − P ( Z < 2. 11) = 1 − 0. 9826 = 0. 0174.
kX G TQ# #BHBi¨ +?2 mM T2bQM T2M/ BH z2//Q2 /m Mi2 HǶBMp2MQ ĕ yX9X aB b2H2xBQM MQ + bQ Ry
T2bQM2X Zm H ĕ H TQ# #BHBi¨ +?2 2b ii K2Mi2 9 /B HQQ T2M/2 MMQ BH z2//Q2\
V yXk9k
"V yXk8R
*V yX8yk
.V yXd8R
aQHmxBQM2 aQMQ Ry TQp2 /B "2MQmHHB BM/BT2M/2MiB +B b+mM +QM T`Q# #BHBi¨ p = 0_._ 4 2 q = 0_._ 6 X a2 X ĕ BH
MmK2`Q /B dzbm++2bbBǴ
P ( X = 4) =
p
4 q
6 = 210 · (0_._ 4
4 )(0_._ 6
6 ) = 0_._ 251_._
jX AH 2bTQMb #BH2 /2HH HQii2B M xBQM H2 ? /B+?B iQ +?2 BH2//BiQ K2/BQ 7 KBHB 2 MMmQ /B +QHQQ
+?2 ;BQ+ MQ HH HQii2B ĕ bmT2BQ2 jdyyy 2mQX aB bmTTQM2 +?2 H /BbiB#mxBQM2 /2H2//BiQ /B i H
+ i2;QB /B BM/BpB/mB bB /BbiB#mBb+ MQK HK2Mi2 +QM /2pB xBQM2 bi M/ / /B 8d8e 2m`QX aB bmTTQM;
+?2 BM mM + KTBQM2 /B k8 7 KB;HB2 +?2 ;BQ+ MQ HH HQii2B bB bB BH2p iQ mM 2//BiQ K2/BQ MMmQ T B
jek9j 2mQXZm H ĕ H bi iBbiB+ i2bi Qbb2p i \
V w 4 − 0_._ 66
d
"V w 4 1_._ 92
*V i 4 0_._ 66
.V i 4 1_._ 92
aQHmxBQM2 z2K xBQM2, _μ >_ 37000 X Ĕ HǶBTQi2bB Hi2M iBp T2`+?û MQM +QMiB2M2 HǶm;m ;HB Mx X ZmBM/B
H 0 : μ ≤ 37000 X G TQTQH xBQM2 ĕ N ( μ, σ = 5756)X G K2/B + KTBQM `B ĕ
X = 36243 bm k8 Qbb2`p xBQMBX
G bi iBbiB+ i2bi ĕ
Z = (36243 − 37000) / (57 /
9X a2 bB B}mi HǶBTQi2bB MmHH +QMiQ HǶBTQi2bB Hi2M iBp / mM HBp2HHQ /B bB;MB}+ iBpBi¨ /2H 8W - HHQ - +QM
;HB bi2bbB / iB /2p2 2bb22B}mi i M+?2 / mM HBp2HHQ /B bB;MB}+ iBpBi¨ /2HHǶRWX o2`Q Q 6 HbQ\
aQHmxBQM2 X 6 HbQ T2+?û b2 bBB}mi H HBp2HHQ /2H 8W pmQH /B2 +?2 TQbbB KQ +QKK2ii22 2``Q2 /2H TBKQ
iBTQ M+?2 8 pQHi2 bm RyyX AM i H + bQ pBQH22KKQ BH +Bi2BQ /B mM i2bi +?2B+?B2/2 mM HBp2HHQ /B 2``Q2 KBMQ
/2HHǶRWX
8X GǶlXaX SQbi H a2pB+2 z2K +?2 HK2MQ BH ejX9W /B mM +2`iQ Tm##HB+Bi¨ TQbi H2 pB2M2 H2ii / B /2biB@
M i BX lM ;mTTQ K#B2Mi HBbi pmQH2 p2B}+ 2 i H2 z2K xBQM2X AMi2pBbi MQ mM + KTBQM2 + bm H
/B kky 7 KB;HB2 2/ Qbb2p MQ +?2 bQHQ BH 83XdW H2;;2 H Tm##HB+Bi¨X .Qp2KKQ +2/22 HHǶ z2`K xBQM
/2HHǶlXaX SQbi H a2`pB+2 / mM HBp2HHQ /B bB;MB}+ iBpBi¨ /2H RyW\
aQHmxBQM
z2K xBQM2, H TQTQxBQM2 /B bm++2bbB ĕ _p_ ≥ 63_._ 4%X Ĕ HǶBTQi2bB MmHH T2+?û +QMiB2M2 HǶm;m ;HB Mx X .mM[m
BH bBbi2K /B BTQi2bB ĕ H 0
: p > 0_._ 634 +QMi`Q H 1
: p < 0_._ 634 X
G bi iBbiB+ i2bi mb HǶ TTQbbBK xBQM2 MQK H
Z =
G 2;BQM2 +BiB+ ĕ Z < − z 0_._ 1
= − 1_._ 282 X SQB+?û BH p HQ2 Qbb2p iQ − 1_._ 447 + /2 M2HH 2;BQM2 +BiB+ bB `B}mi
HǶ z2`K xBQM2X
eX . i H b2;m2Mi2 /BbiB#mxBQM2 /2HH2 7 KB;HB2 T2 +H bbB /B bT2b K2MbBH2 UBM KB;HB B /B 2mQV- /Bb2;M
mM BbiQ;` KK X
*H bbB /B bT2b 0 * 0_._ 5 0_._ 5 * 1 1 * 1_._ 5 1_._ 5 * 2 2 * 2_._ 5 2_._ 5 * 3 3 * 3_._ 5
W 7 KB;HB2 20 30 25 15 7 2 1
aQHmxBQM
0
10
20
30
40
50
60
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.
dX G +Qp B Mx i /m2 p `B #BHB X 2 Y ,
V ĕ +QKT2b i − 1 2 +1X
"V Tmǁ 2bb2`2 TQbBiBp Q M2; iBp X
*V ĕ b2KT`2 TQbBiBp X
V P (
X < μ + σ/
n ) < P (
X < μ − σ/
n )
"V P (
X < μ + σ/ (2 n )) < P (
X < μ + σ/n )
*V P (
X < μ + σ/
n ) = P (
X < μ − σ/
n )
.V P (
X < μ − σ/ (2 n )) < P (
X < μ − σ/n )
aQHmxBQM
aB++QK2 Z =
X − μ ) / ( σ/
n ) ∼ N (0 , 1) H2 z2K xBQMB bB TQbbQMQ i /m``2 +QK2 b2;m
V P ( Z < 1) < P ( Z < −1), 6 Hb
"V P ( Z <
n/ (2 n ) < P ( Z <
n/n ), o2`
*V P ( Z < 1) = P ( Z < −1), 6 Hb
.V P ( Z < −
n/ (2 n )) < P ( Z < −
n/n ), 6 Hb X
1b2`+BxBQ /B{+BH
RjX GǶ2``Q2 /B biBK ĕ H /Bz22Mx i BH p HQ2 /B mM bi iBbiB+ /2i2`KBM i bm mM + KTBQM2 2/ BH
+Q``BbTQM/2Mi2 p HQ2 /2H T K2iQ /2i2KBM iQ M2HH TQTQH xBQM2X o2`Q Q 6 HbQ\
aQHmxBQM2 X o2Q- ĕ _T_ − _θ_ X S2 2b2KTBQ T2` H K2/B ĕ
X − μ X
R9X lM i2bi BbTQbi KmHiBTH ? 8 /QK M/2- Q;MmM +QM 8 TQbbB#BHBBbTQbi2 / 1X amTTQMB /B iB
+ bQ T2imii2 H2 /QK M/2X Zm H ĕ H TQ# #BHBi¨ /B BbTQM/22 +Q``2ii K2Mi2 imii2 H2 /QK M/2\
V yXky
"V yX8y
*V yXyyyjk
.V yXyjRk
aQHmxBQM
Ĕ H TQ# #BHBi¨ /B p22 8 bm++2bbB bm 8 TQp2 +B b+mM /2HH2 [m HB ? TQ# #BHBi¨ p = 1 / 5 = 0_._ 2 X ZmBM/B / iQ
+?2 H2 T`Qp2 bQMQ BM/BT2M/2MiB bB ? p
5 = 0_._ 2
5 = 0_._ 00032 X
R8X AH ; }+Q /2HH p B #BH2 + bm H2 #BMQKB H2 ĕ b2KT2 bBKK2iB+QX o2`Q Q 6 HbQ\
aQHmxBQM2 6 HbQX AH /B ; KK # ``2 ĕ bBKK2iB+Q bQHQ b2 H T`Q# #BHBi¨ /B bm++2bbQ ĕ p = 5X
ReX lM TQi 7Q;HBQ +QKT2M/2 ky xBQMB G6 2 jy xBQMB "1hX AH T2xxQ /2HH2 xBQMB G6 ĕ mM p B #BH
+ bm H2 +QM K2/B Ry 2 p B Mx N- BH T2xxQ /2HH2 xBQMB "1h ĕ mM p `B #BH2 + bm H2 +QM K2/B k8 2
p B Mx ReX A T2xxB /2HH2 /m2 xBQMB bQMQ +Q2H iB M2; iBp K2Mi2 +QM mM +Q2{+B2Mi2 /B +Q2H xBQM
HBM2 2 T B − 0_._ 4 X * H+QH 2 BH p HQ2 ii2bQ 2 H p B Mx /2H p HQ2 /2H TQ`i 7Q;HBQX
aQHmxBQM
AH p HQ2 /2H TQi 7Q;HBQ ĕ X = 20 A + 30 B X AH p HQ`2 ii2bQ ĕ
E ( X ) = 20 E ( A ) + 30 E ( B ) = 20 · 10 + 30 · 25 = 200 + 750 = 950.
G p `B Mx ĕ
σ
2
[20 E ( A ) + 30 E ( B )] = 20
2
σ
2
( A ) + 30
2
σ
2
( B ) + 2 · 20 · 30 σ ( A, B )
TQB+?û H +Qp B Mx i A 2 B ĕ
σ ( A, B ) = ρ ( A, B ) σ ( A ) σ ( B ) = − 0_._ 4 · 3 · 4 = − 4_._ 8
p`2KQ
σ
2
[20 E ( A ) + 30 E ( B )] = 400 · 9 + 900 · 16 + 1200 · (− 4_._ 8) = 12240_._
RdX lM b[m / /B K Mmi2MxBQM2 ĕ2bTQMb #BH2 /B mM +2iQ i iiQ /B QH2Q/QiiQ HmM;Q 9 +?BHQK2i`BX G
/Bbi Mx UBM +?BHQK2iBV HH [m H2 Tmǁ p2B}+ bB mM ;m biQ Tmǁ 2bb22 TT2b2Mi iQ / mM p `B #BH
+ bm H2 +QMiBMm mMB7QK2X aT2+B}+ 2 H 7mMxBQM2 /B /2MbBi¨ 2 + H+QH 2 H TQ# #BHBi¨ +?2 bB p2`B}+?B
mM ;m biQ i` y 2 R FKX
Ry
aQHmxBQM2 G /2MbBi¨ mMB7QK2 ĕ _f_ ( _x_ ) = 1 _/_ 4 T2 x ∈ [0 , 4]X G T`Q# #BHBi¨ ĕ
Pr(0 < X < 1) = 1 / 4
R3X aB X ∼ N (2 , 1)X G /Bz22Mx BMi2[m iBH2 ĕ T B
V yXkkee
"V kXd
*V − 4
.V RXj
aQHmxBQM2 G /Bz22Mx BMi2[m `iBH2 /B mM N (0 , 1) ĕ Q 3
− Q
1
/Qp2 P ( Z < Q 3
) = 0. 75 2 P ( Z < Q
1
) = 0. 25 X
. HH2 i pQH2 bB p2/2 +?2 P ( Z < 0_._ 67) = 0_._ 75 X ZmBM/B P ( Z < − 0_._ 67) = 0_._ 25 X 6 i2 mM /Bb2;MQ
HHQ H /Bz22Mx BMi2[m iBH2 ĕ IQR = 0_._ 67 − (− 0_._ 67) = 2 · 0_._ 67 = 1_._ 34 X
G /Bz22Mx BMi2[m iBH2 /2HH MQK H2 N (2 , 1) ĕ H /BbT2bBQM2 /B mM MQK H2 Qii2Mmi bTQbi M/Q H
MQK H2 bi M/ / /2bi /B kX ZmBM/B H /BbT2bBQM2 2bi H bi2bb 2/ ĕ m;m H2 1_._ 34 X oQH2M/Q p2B}+ `
T2+Bb K2Mi2- # bi MQi 2 +?2 b2 X ∼ N (2 , 1) P ( X < 2 + 0_._ 67) = 0_._ 75 2 P ( X < 2 − 0_._ 67) = 0_._ 25 X ZmBM/B
HQ b+ iQ BMi2[m `iBH2 ĕ IQR = (2 + 0_._ 67) − (2 − 0_._ 67) = 1_._ 34 X
RNX AH i2KTQ +?2 ;HB bim/2MiB /2/B+ MQ HHQ bim/BQ b2;m2 mM /BbiB#mxBQM2 MQK H2 +QM /2pB xBQM2 bi M/ `/
/B 3 Q2X aB 2bi 2 mM + KTBQM2 + bm H2 /B 9 bim/2MiBX G TQ# #BHBi¨ +?2 H K2/B + KTBQM B /Bz2`Bb+
/ HH K2/B /2HH TQTQH xBQM2 T2TBɍ /B 9Q2 ĕ
V yXkN3d
"V yXjy3y
*V yXjRd
.V yXjy
aQHmxBQM2 a2 X ∼ N ( μ, σ = 8) H K2/B BM mM + KTBQM2 /B n = 4 ? /Bbi`B#mxBQM
X ∼ N ( μ, 8 / 2 = 4)X
S2+Bǁ H TQ# #BHBi¨ +?2 H K2/B + KTBQM B /Bz2Bb+ / μ T2` TBɍ /B 9 ĕ
P (|
X − μ | > 4) = P (|
X − μ | / 4 > 1) = P (| Z | > 1)
/Qp2 Z N (0 , 1)X
ZmBM/B bB ? 2(1 − P ( Z < 1)) = 2(1 − 0_._ 8413) = 0_._ 3174 X _BbTQbi *X
kyX G K2/B ĕ mM KBbm /B i2M/2Mx +2Mi H2 KB;HBQ2 /2HH K2/B M [m M/Q +B bQMQ p HQB MQK HBX
o2`Q Q 6 HbQ\
aQHmxBQM2 Ĕ 2b ii K2Mi2 BH +QMi BQX 6 HbQX
*8 QKTBiQ > /2H d ;Bm;MQ kyRk
RX lM p B #BH2 ĕ +H bbB}+ i +QK2 Q/BM H2 b2,
V A / iB bB Qii2M;QMQ +QM mM TQ+2bbQ /B KBbm xBQM2 +QMiBMmQ
"V 1bBbi2 mM Q/BM K2MiQ M im H2 /2HH2 bm2 KQ/ HBi¨
*V Pbb2pB KQ H p B #BH2 T2mM T2BQ/Q /B i2KTQ
.V LQM 2bBbi2 mM Q/BM K2MiQ M im H2 /2HH2 bm2 KQ/ HBi¨
aQHmxBQM
lM p B #BH2 bB /B+2 Q/BM H2 b2 2bBbi2 mM Q/BM K2MiQ M im H2 /2HH2 KQ/ HBi¨X S22b2KTBQ H 2HB;BQM2 MQM
ĕ Q/BM H2- K2Mi2 BH HBp2HHQ /B BbimxBQM2 bŢX ZmBM/B H BbTQbi ĕ "VX
kX Zm H2 /2HH2 b2;m2MiB z2K xBQMB ĕ p2 \
V G K2/B ĕ b2KT`2 TBɍ TB++QH /2HH K2/B M X
RR
eX . mM BM/ ;BM2 +QM/Qii bm 9dk /B2iiQB /2H T2bQM H2- ĕBbmHi iQ +?2 BH ejW T2Mb /B bbmK2`2 MmQpQ
T2bQM H2 M2B TQbbBKB i2 K2bBX Zm HB /2B b2;m2MiB BMi2p HHB TT2b2Mi mM BMi2`p HHQ /B +QM}/2Mx /2H
N3W T2H TQTQxBQM2 /2B /B2iiQB /2H T2bQM H2 +?2 ? MMQ TB MB}+ iQ /B bbmK22 MmQpQ T2bQM H
M2B TQbbBKB i2 K2bB\
V 0. 63 ± 0. 042
"V 0. 63 ± 0. 047
*V 0. 63 ± 0. 057
.V 0. 63 ± 0. 052
aQHmxBQM2 a2 BH HBp2HHQ ĕ 1 − α = 0_._ 98 α = 0_._ 02 2 α/ 2 = 0_._ 01 X
AH K ;BM2 /B 2``Q2 ĕ
M E = z α/ 2
p ˆ(1 − p ˆ) /n = z 0_._ 01
dX AM mM 2+2Mi2 BM/ ;BM2 bm eyy /mHiB- BH ReX9W ? /B+?B iQ /B 2bb2bB //QK2Mi iQ HK2MQ mM pQHi
/B 7QMi2 HH i2H2pBbBQM2 M2H K2b2 b+QbQX Zm H ĕ BH HBp2HHQ /B +QM}/2Mx bbQ+B iQ HH BMi2`p HHQ (RkX33W-
RNXNkW)\
aQHmxBQM
G TQTQxBQM2 biBK i ĕ BH TmMiQ +2Mi H2 /2HHǶBMi2p HHQ +BQĕ
p ˆ = (12_._ 88 + 19_._ 92) / 2 = 16_._ 4% = 0_._ 164_._
GǶ2``Q2 bi M/ / /2HH biBK ĕ ES =
p ˆ(1 −
0 AH K ;BM2 /B 2``Q2 ĕ
M E = 1 / 2 Ampiezza = (19_._ 92 − 12_._ 88) / 2 = 3_._ 52%.
SQB+?û M E = z α/ 2
ES `BbmHi +?
z α/ 2
= M E/ES = 0. 0352 / 0. 0151164 = 2. 33.
ZmBM/B P ( Z > 2_._ 33) = α/ 2 X . HH2 i pQH2 /2HH MQ`K H2 P ( Z > 2_._ 33) = 0_._ 01 2 [mBM/B α = 0_._ 02 X aB++QK2 BH
HBp2HHQ /B +QM}/2Mx ĕ 1 − α [m2biQ `BbmHi 1 − 0_._ 02 = 0_._ 98 X
3X lM BKT2M/BiQ2 ? BMbi HH iQ mM MmQp 7QM +2 2M2;2iB+ K2Mi2 TBɍ 2{+B2Mi2X aB biBK +?2 BM mM
MMQ H MmQp 7QM +2B/m``¨ BH +QbiQ /2HH 2M2;B T2 mM BKTQiQ +?2 Tmǁ 2bb22 pBbiQ +QK2 mM
p B #BH2 + bm H2 +QM K2/B 1mQ ke8 2 /2pB xBQM2 bi M/ / /B 1mQ 88X amHH # b2 /B BTQi2bB +?
Bi2M2i2 M2+2bb B2 7QKmH 2- iQp 2 H K2/B 2 H /2pB xBQM2 bi M/ / /2HH B/mxBQM2 /2H +QbiQ iQi H
/2HH 2M2;B BM mM T2BQ/Q /B 8 MMBX
aQHmxBQM
_B/mxBQM2 /2H +QbiQ 4 X +QM E ( X ) = 265 2 σ ( X ) = 55X AM 8 MMB H `B/mxBQM2 /2H +QbiQ ĕ 5 X /Qp2 bB bmTTQM
+?2 H `B/mxBQM2 /B +QbiQ bB +Qbi Mi2 Q;MB MMQX
ZmBM/B H K2/B /2HH `B/mxBQM2 /2H +QbiQ iQi H2 ĕ E (5 X ) = 5 E ( X ) = 5 · 265 = 1325_._
G p `B Mx ĕ σ
2 (5 X ) = 25 σ
2 ( X ) = 25 · 55
2
- K2Mi
2 H /2pB xBQM2 bi M/ / ĕ 5 · 55 = 275X
NX lM ;mTTQ HQ+ H2 /B TB MB}+ xBQM2 /2B i bTQiB ĕ T2Q++mT iQ T2H K M+ Mx /B + @b? BM; T2
B T2M/QH BX h2KQMQ +?2 H TQTQxBQM2 /B miBHBxx iQB HQ+ HB /B + @b? BM; bB BM72BQ2 HH K2/B
M xBQM H2- T B H kyWX lM BM/ ;BM2 bm j8e ;mB/ iQB HQ+ HB BH2p +?2 BH R3XdW /B HQQ mb BH + @b? BM;X
Zm HB bQMQ H2 im2 +QM+HmbBQMB- +QMbB/2` M/Q mM HBp2HHQ /B bB;MB}+ iBpBi¨ /2H 8W\
V LQM +Ƕĕ 2pB/2Mx +?2 H TQTQxBQM2 /B miBHBxx iQB HQ+ HB /B + @b? BM; MQM bB BM72BQ`2 HH K2/B
M xBQM H2X
"V *Ƕĕ 2pB/2Mx +?2 H TQTQxBQM2 /B miBHBxx iQB HQ+ HB /B + @b? BM; bB BM72BQ`2 HH K2/B M xBQM H2X
*V LQM +Ƕĕ 2pB/2Mx +?2 H TQTQxBQM2 /B miBHBxx iQB HQ+ HB /B + @b? BM; bB BM72BQ`2 HH K2/B M xBQM H2X
Rj
.V *Ƕĕ 2pB/2Mx +?2 H TQTQxBQM2 /B miBHBxx iQB HQ+ HB /B + @b? BM; MQM bB BM72BQ`2 HH K2/B M xBQM H2X
aQHmxBQM
GǶ z2K xBQM2B;m / H TQTQxBQM2 _p_ /B miBHBxx iQB 2 bB Tmǁ b+Bp22 +QK2 p < 0_._ 2 X Zm2bi BTQi2bB p
+QMi` TTQbi p ≥ 0_._ 2 +?2 p2M/Q BH b2;MQ /B m;m ;HB Mx ĕ HǶBTQi2bB H 0 ,
H 0 : p ≥ 0_._ 2 +QMi`Q H 1 : p < 0_._ 2
G bi iBbiB+ i2bi ĕ
z = (0_._ 187 − 0_._ 2) /
G 2;BQM2 +BiB+ H HBp2HHQ /2H 8W ĕ Z < − 1_._ 645 [mBM/B MQM bB `B}mi H 0
+BQĕ MQM +Ƕĕ 2pB/2Mx bm{+B2Mi
T2
X
H+mM2 /QK M/2 +QMi2;QMQ 2``Q`B +QM+2iim HBX
V LQM +Ƕĕ 2pB/2Mx +?2 p MQM bB < 0_._ 2 X GǶ2pB/2Mx Tmǁ 2bb22 bQHQ +QMiQ H 0
MQM 7 pQ`2 /B H 0
X
"V *Ƕĕ 2pB/2Mx +?2 p < 0_._ 2 - +BQĕ +QMi`Q H 0
X 6 HbQX
*V LQM +Ƕĕ 2pB/2Mx +?2 T I yXkX Zm2biQ 2Ƕ o1_P T2`+?û BM7 iiB ##B KQ ++2ii iQ H 0
X
.V *Ƕĕ 2pB/2Mx +?2 T MQM bB I yXkX o2/B VX
G `BbTQbi +Q``2ii ĕ *VX
RyX L2HH b2;m2Mi2 /BbiB#mxBQM2 /B 72[m2Mx - [m H ĕ H 7`2[m2Mx +mKmH i K M+ Mi2 M2HH TQbBxBQM
Q++mT i / HHǶ bi2`Bb+Q\
JQ/ HBi¨ 62[m2Mx S2+X +mKmH i
V RyyW
"V eyW
*V 3yW
.V NyW
aQHmxBQM2 a2 x ĕ H T2+2Mim H2 +mKmH i +?2 K M+ ĕ +?B Q +?
x = (6 + 6 + 5 + 1) / (2 + 6 + 6 + 5 + 1) = 90%
+BQĕ .VX
RRX HHǶ mK2Mi 2 /2HH /BK2MbBQM2 /2H + KTBQM2 HǶ2``Q2 bi M/ / /2HH K2/B + KTBQM B MQM + K#B X
o2`Q Q 7 HbQ\
aQHmxBQM2 GǶ2``Q2 bi M/ / /2HH K2/B ĕ σ/
n [mBM/B ĕ +?B Q +?2 b2 mK2Mi H /BK2MbBQM2 + KTBQM B
n HǶ1a /BKBMmBb+2X S2+Bǁ HǶ z2K xBQM2 ĕ 6GaX
RkX amTTQMB KQ /B H M+B 2 /m2 / /BX aB +QMbB/2B H bQKK /2B /m2 / /B, bB A HǶ 2p2MiQ dzbB Qbb2`p mM
MmK2Q T BǴX Zm H ĕ BH +QKTH2K2Mi `2 /2HHǶ2p2MiQ A \
V { 1 , 3 , 5 , 7 , 9 , 11 }
"V { 3 , 5 , 7 , 9 , 11 }
*V { 8 , 10 , 11 , 12 }
.V { 8 , 9 , 10 , 11 , 12 }X
R
RNX a2 X 2 Y bQMQ /m2 p B #BHB + bm HB i HB +?2 _Y_ = _a_ + _bX_ - HHQ μY = a + bμX X o2`Q Q 7 HbQ\
aQHmxBQM2 a2 Y = a + bXE ( Y ) = E ( a + bX ) = a + bE ( X )X Ĕ mM TQTB2i¨ /2H p HQ`2 ii2bQX o1_PX
kyX a2 Z
N (0 , 1)- HHQ` P (− 1_._ 25 < Z < − 0_._ 75) ĕ,
V yXeed
"V yXRkRy
*V yXkkee
.V yXRy8eX
aQHmxBQM
P (− 1. 25 < Z < − 0. 75) = P (0. 75 < Z < 1. 25) = P ( Z < 1. 25) − P ( Z < 0. 75) = 0. 89435 − 0. 773373 = 0. 1210.
_BbTQbi "VX
Re
