A/A ALFONSO GARCÍA PÉREZ. NOMBRE DEL CURSO: ESTADÍSTICA APLICADA A LAS CIENCIAS DE LA SALUD. NOMBRE ALUMNO: MANUEL FERNÁNDEZ MARTÍN. DNI: 52.563.220-D DIRECCION: C/ MÉNDEZ NÚÑEZ,-!"- MARC#ENA- $ SE%ILLA &.
P'()*+ !. S+ *1 +* N+* + C(*+4+'(*, + *'(4 7(' +8*'( + 4'+ /L, + 20 7+'4(4 ()++( *(4 49++4 '+49*(4: 200 !50 2!0 20 2;0 00 !<5 2!< 225 250 2;5 !6; 225 23 2; 3!0 ! 225 250 2;5 R+*' 9 =*44 +48'7( + +4(4 (4 8*89*( *9 + + 7(481, *9 + 47+'41, >8+( *9 '+7'+4+81 '=?8 + *(4 (4, +8. S(*981: !5 0
!6;
!<5
!
!
!
!
!
•
20 0 !
2 !0
2!<
225
23
20
250
2 ;0
2;5
2 ;
3!0
00
!
!
3
!
!
2
!
2
!
!
!
MEDIDAS DE P@SICIN: !50 + !6; + ... + 2;5 M+ B B
20
20
B 20,5
M+ B $22523&/2 B 232. M( M( B 225 7('9 7('9++ +4 +* *( *('' 9+ 9+ =4 =4 4+ '+7+ '+7+.. •
MEDIDAS DE DISPERSIN: !50 2 + !6; 2 + ... + 3!0 2 + 00 2 − 20,5 2 B %' B 20
B
!2!5<
D+481 78 B •
20
− 5;<0,25 B 60;,;
25:, :5
− 5;<0,25
B 25,5
B 5,35
REPRESENTACIN RÁFICA: C *(' 9+ 7'+8+ 9 4(* +, * 4+' 20 ()4+'8(+4, '+7'+4+ +* 5G. L(4 *('+4 250 H 2;5 7'+8+ (4 +8+4, 7(' *( 9+ '+7'+4+ +* !0G. +* *(' 225 7'+8+ '+4 +8+4, +* !5G.
NIVEL DE COLESTEROL 20
s a n 15 o s r e p e d 10 e j a t n e c 5 r o P 0 150
167
185
199
200
210
218
225
239
240
250
270
275
297
310
400
Nivel de colesterol
P'()*+ 2. S+ 9+'+ +4', ++ 9 +'*( + 8(?, +* +7( 9+ '489''+ +4+ * 4'81 + * 7'+' (44 + 9 9+ 89 8(' * >+74 J, >4 9+ 4+ 7'(98+ + +* 9( 9 '=48 4981 +* +* + 89+'7(4 8(' * +8( +?+'+, '+9'+( K4+ 9 9+ (44 + '+89+'(. P' * ? 4+ +*1 9 9+4' *+(' 47*+ + 0 9(4 + * 7()*81 + +49( + *(4 9+ 4+ ()4+'1 +* +7( '489''( +4+ * 4'81 + * 89 >4 * 4981 + *(4 89+'7(4, ()+K(4+ 9 + 9+4'* + 35 4. D++'' +* +8(( +'*( + 8(? 7' 9 8(+?8++ + 8(? + 0,5, 4)+( 9+ + *4 894 + 7( 4K8(, 8(( * 9 9*, +4 4)*+ 497(+' 9 +481 78 + ; 4. S(*981: N( 4+ 79+ 4+9'' 9+ * 4')981 + * 9+4' *+(' 4+ 9 4')981 ('*, 7+'( 8(( +* ( 9+4'* +*+( +4 '()*+++ '+ 7(+(4 8(4+'' 9+ * + 9+4'* x 49+ 9 4')981 ('*. P(' (, 4+ 7+ ++'' +* +'*( + 8(? 7' * + 7()*8(* + 9 4')981 + +481 78 8((8 S = ; .
C(( +* 8(+?8++ + 8(? 7+( +4 + 0,5, +* *(' + α 4+'= ! − α = 0,5
⇒ α = ! − 0,5 = 0,05
A4, +* +'*( + 8(? +4
I = x − z
S
α
2
n
S
, x + z
α
2
n
= 35 7('9+ +4 * + 9+4'*, z !,6 7('9+ 7' 9 8(+?8++ + 8(? + 0,5 4+ ()++ 2 +4+ *(' ()4+'( * )* + * 4')981 ('* $)* 3 +* *)'( E448 7*8: 8(8+7(4 )=48(4& n = 0 7('9+ +4 +* ( + * 9+4'
4+(,
x
α
=
P(' (,
;
0
I = 35 −!,6
, 35 +!,6
;
= [32,<3 , 3;,!;]
0
P'()*+ 3. S+ +4= +49( +* +7( + +'+ *(4 78++4 9 ++' +?+'+. A * ? 4+ +*+'( * ' !00 ?8>4 + 78++4 ?**+8(4 7(' * +?+'+ + +49(, ()+K(4+ 9 + 9+4'* + ;0 4 H 9 894+481 78 9+4'* + 32 4. OP9++ 4''4+ 7' *(4 78++4 + * +?+'+ + 89+41 9 +7( +( + 497+'(' ;30 4 S(*981: S+ 79++ 9*' * + 9+4'* x 7' +4' +* +7( +( + µ , 7('9+ +* ( 9+4'* +4 *( 49?8++++ '+ $Q30&. D+ +4 ?(', 7(+(4 +8' 9+
x − µ ≈ N ( 0 ,!) S n
H 8(( +=4 7(' * 9 +* ( 9+4'* )K 7(+(4 4949' * ' 7(' 9 +4(' 49H( D+ +4 ?(' ++(4 9+ 7(' (,
x − µ ≈ N ( 0 ,!) S n
x = ;30 n = !00 S = 32
( x − µ ) !0 P {( x − µ ) < !0} = ! − P {( x − µ ) > !0} = ! − P > = ! − P { Z > 3,!25} = S n 3,2 = ! − 0,000 = 0,!
A4 79+4, * 7'())* +4 +* ,!G 7(' *( 9+ 4+ 79++ '4+ 7' *(4 78++4 + +4 +?+'+ 9 +7( + 497+'(' ;30 4.
Problema 4. Se !"ere a#er"$!ar %" &re% &"'o% (e ("e&a% A) B * C 're%e+&a+ (",ere+-"a% %"$+","-a&"#a% e+ -!a+&o a %!% e,e-&o% %obre el a!me+&o (e 'e%o. Co+ e%&e 'ro'%"&o %e el"$"ero+ al aar 01 "+("#"(!o% a la% !e %e %ome&"ero+ alea&or"ame+&e a !+a (e la% &re% ("e&a%) ob&e+"2+(o%e lo% %"$!"e+&e% a!me+&o% (e 'e%o: D"e&a A B C
A!me+&o (e 'e%o 3 33 3 31 35 36 47 31 37 5 33 33 36 56 31
A la #"%&a (e e%&o% (a&o%) 8'!e(e "+,er"r%e !e e9"%&e+ (",ere+-"a% %"$+","-a&"#a% e+&re la% &re% ("e&a% a +"#el α = 0,05 S(*981: P' '+4(*+' +* 7'()*+ 7(+(4 '+*' 9 A=*44 + * %' 7' 9 ?8(' + 9 4+( 8(7*+++ *+('(. C( K* 4+ 9+'+ 8('4' * >71+44 9* + 9* + *(4 +?+8(4 +(4 + *4 '+4 +4, H 0 = µ A = µ B = µ C , ?'++ * *+' + ( 4+' *4 '+4 9*+4. E 7'+' *9' 8(4'9(4 * )* + A=*44 + * %': F. + '81
S9 + 89'(4 r
D+4
SST i = ∑ i =!
T i
2
ni
−
T i
.*. r −!
n
8. +(4 SST i r − !
E448( SST i
$ r !& −
SSE $ n r & −
R+49* T(*
n − r
SSE = SST − SST i
r
ni
2
SST = ∑∑ xij − i =! j =!
SSE n − r
T 2 n
n −!
C*89*(4 * 49 + *(4 89'(4 4+ * ?1'9*:
SST i
7'' + * )* + (4 +* +98(,
ni
= ∑ xij
T i
j =!
D+ A J C
3 36 33
A9+( + 7+4( 3 35 35 30 36 26
33 0 33
T i
32 2 35
!;< !;0 !63
T = 5!!
E* +'( (* + ()4+'8(+4 +4 !5. L 49 + 89'(4 +) *4 +4 4+'=, 7(' (, r
T i 2
i =!
ni
SST i = ∑
−
T 2 n
= 22,533
S94 '(4 + *)+' 4( 9* * +'( + '+(4 +(4 9(, r − ! = 3 − ! = 2 r
L 49 (* + 89'(4 4+'=
ni
SST = ∑∑ x − 2 ij
i =! j =!
T 2 n
= 20<,33
S94 '(4 + *)+' 4( +* +'( + ()4+'8(+4 +(4 9(, L 49 '+49* + 89'(4 4+'=
n − ! = !5 − ! = !
SSE = SST − SST i = 20<,33 − 22,533 = !<6,
S94 '(4 + *)+' 4+ 8*89* )K 8(( ?+'+8 + *(4 '(4 + *)+' + *4 (4 494 + 89'(4 9+ 4'+'( 7' ()++' SSE ( n −!) − ( r −!) = ! − 2 = !2
C( +4( H ++(4 8*89*4 *4 494 + 89'(4 H *(4 '(4 + *)+' + *4 '+4 ?9++4 + '81 9+ ?(' * )* + A=*44 + %'. L(4 89'(4 +(4 8(''+47(++4 8 ?9++ + '81, 4+ ++' >(', 47*+++, +( 8 49 + 89'(4 7(' 494 '(4 + *)+': SST i r − ! SSE n − r
=
22,533
=
!<6,
2 !2
= !!,266
= !5,533
SST i
E* +448( 9*' + +* 8('4+ 4+'= F =
$r − !&
SSE $n − r &
= 0,;3
R+49(4 + * 49++ )* *(4 '+49*(4 ()+(4: F. + '81
S9 + 89'(4
.*.
8. +(4
E448(
D+4
SST i = 22,533
2
!!,266
F = 0,;3
R+49*
SSE =!<6,
!2
T(*
SST = 20<,33
!
!5,533
S +4 8+' * >71+44 9* + 9* + *(4 +?+8(4 +(4 + *(4 '+4 ?+'*+4, +* +448( F + S++8(', 8( '(4 + *)+' 9* * 7' ?('( 7(' *(4 '(4 + *)+' 8(''+47(++4 *4 ?9++4 + '81 D+4 H R+49*, +4 ++'(4 ( r −! , n −!) = ( 2 , !2) , 7(' *( 9+ 7' ++'' +* 79( 8'8(, +* + 4?881 α = 0,05 , )948'+(4 + * )* + * F + S++8(' +* *(' F ( 2 , !2) S 0,05 = 3,<<53 . D( 9+ F = 0,;3 +4 +(' 9+ 8>( 79( 8'8(, 7(+(4 8+7' H 0 , 8(8*9H+( 8( * ( +4+8 + ?+'+84 4?84 +'+ *4 '+4 +4. T)K 7(+(4 ()++' 9 8(81 +* 7-*(': 7-*(' B P F ( 2 , !2) > 0,;3 > P F ( 2 , !2) > 2,<06< = 0,! E4 8(81 +4 49?8++ 8(( 7' 7(+' 8(?'' * +841 (.
Problema 1. Se real" !+ e9'er"me+&o 'ara e%&!("ar el e,e-&o (e !+ (e&erm"+a(o me("-ame+&o e+ la ("%m"+!-"+ (e lo% a&a!e% (e -ora+. La #ar"able "+(e'e+("e+&e ,!e la (o%"% (e (ro$a e+ m"l"$ramo%) ;) * la #ar"able (e'e+("e+&e la re(!--"+ e+ lo% a&a!e% al -ora+) al -om'ararlo% -o+ !+ $r!'o -o+&rol) <. Lo% (a&o% re-o$"(o% ,!ero+ lo% %"$!"e+&e%: ; 7)17 7)=1 0)77 0)51 0)17 0)=1 5)77 5)51 5)17 5)=1 3)77 3)51 3)17
< 07 > 05 05 04 05 06 0> 0= 57 0> 57 50
De&erm"+ar la re-&a (e re$re%"+ * a+al"ar %" e% %"$+","-a&"#a.
S(*981: L '+8 + '+'+41 + 4()'+ +4
V Y
α +β X
=
α = Y − b X ∑$ X i − X &$Y i −Y & β = $ X i − X & 2
(+
R+*( *(4 8=*89*(4 **+(4 E* C(+?8++ + D++'81 +4
+ ,0<;!2 x y = ;,055 R = 0,50;!
R+*'+(4 9 +4 + * >71+44 9* + 9* 8+'( +* 8(+?8++ 7' 8(7'()' 4 * R+8 + R+'+41 ()+ +4 4?8, +4 +8', 4 +7*8 8(+++++ * ')*+ +7+++ + ?981 + * +7+++. L )* + A=*44 + * %' $AN@%A& +4 F. (e #ar"a-"+
S9 +
.*.
8. +(4
E48(
F
89'(4
Re$re%"+ L"+eal S"m'le
SSE
!
SSE
Re%"(!al
SSNEBSST-SSE
-2
SSNE / $-2&
?o&al
SST
-!
S9 +
.*.
8. +(4
E48(
!03,!!5;
P' +4+ 7'()*+ +4 F. (e #ar"a-"+
89'(4
Re$re%"+ L"+eal S"m'le
!0,0<;!2!
!
!0,0<;!2!
Re%"(!al
20,2!;<022
!!
!,<3
?o&al
2!0,30;623
!2
S 9+'+(4 8('4' -
S+ 8+7
-
S+ '+8>
H 0 : β = 0
H 0
?'++
4 t < t n −2 Sα 2
H 0 4 t
≥ t n−2 S α
2
H ! : β ≠ 0
S+( +* +448( +* 8('4+ V2 σ 2 b
S
= SSNEX = n −2 =
σ V 2 SSEX
t =
20,2!;<022 !!
β V 2
S b
=
SSEX $n − 2& SSNEX
= !,<3
20 ,2!;<022 =
V β
!0 ,0<;!2!
:,0<;!2 2
=
0,!6!5665
t =!0,!6!6; > 2,20! = t !!S 0, 025 , 7(' (, 7(' *( 9+ 4+ '+8> * >71+44 9* H 0 : β = 0 . E4 +8', * '+8 + '+'+41 +4 =* 7' +7*8' * ')*+ +7+++ + ?981 + * +7+++ .
P'()*+ 6. S+ +4= +49( * 4')981 + *(4 89'( '97(4 49+(4 @, A, J, AJ + 9 7()*81. E' 9 9+4' *+(' + +** 4+ ()9+'( *(4 49++4 (4: '97( 49+( @ A J AJ
?'+89+8 !2! !20 ; 33
U (+*( +1'8( 4 *4 49++4 7'())*+4 8 9( + *(4 '97(4: '97( 49+( @ A J AJ
?'+89+8 r 2 p 2 + 2 pr q 2 + 2 pr
2 pq
8( 7 ' B ! A 7'' + *(4 (4 + * 9+4' 4+ ()9+'( *4 49++4 V = 0,!;32 V = 0,265 H q p +48(+4 + *(4 7'=+'(4: C('4' * >71+44 + 9+ *(4 (4 4+ W94 * (+*( +1'8(.
S(*981:
E* (* + (4 +4 353 S+ +* (+*( +1'8(
p + q + r = !
⇒ ⇒
r =! − p − q r =! − 0,!;32 − 0,265 = 0,5<03
L )* + ?'+89+84 4+'=: '97( S9+( @ A J AJ
F'+89+8 ()4+'
F'+89+8 +47+'
ni
n X p i
!2! !20 ; 33
353 X 0,5<03 2 =!!<,<;
353 X $ p + 2 pr & = !22,3 2
353 X $ q 2 + 2qr & =
= 30,!
E4 +8': @ --------------------------------------- !!<.<; A --------------------------------------- !22.3 J --------------------------------------- -89'( Y B [ $@)4+'( - E47+'(&Y/E47+'( \ 94'+(4 - ] -! '(4 + *)+' 7' +* +448( 8>-89'( B $?*4& ]B2 $7'+'(4 +4(4& '(4 B -2-! B ! Y B $!2! -!!<.<;2! &Y/!!<.<;2! $!20 -!22.3)9* 5G + 8(? +4 8B 3.<! C(( +* *(' 8*89*( Y B 0.3;2 +4 =4 7+9+( 9+ +* *(' 8'8( 8B 3.<!, ( 7(+(4 '+8>' 9+ *(4 (4 4+ W94 * (+*( +1'8(, H 7(' *( ( 8(8*9(4 9+ *(4 (4 4+ W94 * (+*( 7'(79+4(.
P'()*+ ;. S+ 9+'+ *' 4 79++ 8(4+''4+ 4?8++ +7+++4 ( ( +* ++' 7'()*+4 '+47'('(4 + * (*+48+8 H +* ++' 9 >4('* + )'(94 + * ?8. P' +**( 4+ 4+*+88('( * ' !000 (*+48++4 H 4+ *+4 8*4?81 4+ 9+' + * 89* 7'()*+4 '+47'('(4 ( ( H 4+ 49 >4('* 8*8( + )'(94 + * ?8. L(4 '+49*(4 ()+(4 ?9+'( *(4 49++4: #4('* + )'(94 P'()*+4 '+47'('(4
S N( T(*
S 25 200
N( 0 ;35
T(* !000
OP9++ '+8>'4+ * >71+44 9* + +7++8 +'+ )4 ')*+4
S(*981: S+ ' + *' * 7(4)*+ +7++8 +'+ (4 8'8+'+4 ()4+'(4 + *(4 9(4 + * 7()*81. L4 >71+44 8('4' 4( H 0 : os #ara#teres ^ Historia de bronq"itis^ y ^ P' obe!as respirator ios^ son indep
?'++ H ! : os #ara#teres ^ Historia de bronq"itis^ y ^ P' obe!as respirator ios^ no son in
#4('* + )'(94 P'()*+4 '+47'('(4
S N( T(*
S 26 200
N( 0 ;35
T(* 65 35 !000
E(8+4 +* 8('4+ 17( 9*' 8(44+ + A8+7' H 0 4 λ < X $2a !& $b !& S R+8>' H 0 4 λ ≥ X $2a !& $b !& S 4+( a H b +* +'( + (*+4 + 8 8'=8+' $2 H 2, '+47+8++&. −
α
−
−
−
α
ni n j n
!,625 2!0,3;5
nij −n n
50,3;5 ;2,625
2
i j
n
!0;,6062 5 !0;,6062 5
!0;,60625 !0;,60625
2
n − n n n ij
i
j
n n n i
j
;,36002; 0,5!!660; 2
2,!36;<66 0,!<5666
2 λ = !0,!5;036; > 3,<! = X !S 0, 05 E* +448( + P+'4( ( +* *(' 7(' *( 9+ '+8>(4 * >71+44 9* + +7++8 + *(4 8'8+'+4 #4('* + )'(94 H P'()*+4 '+47'('(4, +* α = 0,05
P'()*+ <. E 9 +49( 4()'+ * 8'+4 +* + (4 + 89'( (4 +('=?84 8( 4(4 +*+4 + ?*(' + +* 9, 4+ (1 9 9+4' + !20 (4 + 8 ( + * 9+ 4+ ()9+'( *(4 49++4 (4: Z( A J C D
N(4 4 8'+4 < !< 0 5
OS+ 79++ 8+7' 8(( +9*++4 *4 89'( (4 +('=?84 '+47+8( * 7'+4+8 + 8'+4
S(*981: D+)+(4 '+*' 9 8('4+ + >((++ + *4 9+4'4 7' +'9' 4 +4+ ( ( ?+'+84 4?84 +'+ *4 (4. T+'+(4 * 49++ )* + ?'+89+84 )4(*94:
Zo+a A B C D ?o&al
N"@o% %"+ -ar"e% < !< 0 5 !60
D+)+(4 8('4', +*
N"@o% -o+ -ar"e% ;2 !02 <0 66 320
ni
!20 !20 !20 !20 <0
α = 0,05 * >71+44 9*
H 0 : son >( o%&neas as poba#ione s de as q"e se $an obtenido as !"estras
?'++ * *+' + ( >((++ + * 7()*8(+4. S +4
λ =
r
s
∑∑
$ nij −ni ! j / n & 2
ni ! j / n
i =! j =!
, +(8+4 +* 8('4+ 17( 8(44+ + A8+7' H 0 4 λ < X $2r −!& $ s −a & S R+8>' H 0 4 λ ≥ X $2r !& $ s a & S
α
−
ni ! j n
0 0 0 0
nij −n ! i
<0 <0 <0 <0
2
j
n
6 < 0 !6
6 < 0 !6
2
n − n ! n ij
i
j
n ! n i
j
!,6 !2,! 0 ,
0,< 6,05 0 2,5
−
α
2 λ = 2;, > ;,!5 = X 3 S 0, 05 E* +448( + P+'4( ( +* *(' 7(' *( 9+ '+8>(4 * >71+44 9* + >((++ + *4 89'( (4. E4+ ?+'+8 4?8 +'+ *4 89'( (4 + 89( * +4+8 + 8'+4 +*.
