2011 c f - ustcstaff.ustc.edu.cn/~zwp/teach/math-stat/lec9.pdf · 2011-04-11 · i=...
TRANSCRIPT
Lec9: b�u�
Ü�²
2011c 4� 11F
b�u�=¦^��é¤'%�b�?1íä. b�u�¯K��©�ü�a:
1. ëê.b�u�: =oN�©Ù/ª®�(X��!�ê!��©Ù�) , oN©Ù�6
u��ëê(½ëê�þ) θ, �u��´k'��ëê�b�. XX ∼ N(a, σ2), a��, u�
H0 : a = a0 ←→ H1 : a 6= a0 ½ H0 : a ≤ a0 ←→ H1 : a > a0
2. �ëê.b�u�: XJoN©Ù/ª��, d�ÒI�k�«�oN©Ùx�äNê
Æ/ªÃ'�ÚO�{, ¡��ëê�{. Xu��1êâ´Ä5g,�®��oN, Òáuù
a¯K.
1 b�u��eZ�Vg
�!u�¯K�J{
�`²b�u�¯K�J{, �e¡�~f.
~~~5.1.1 ,ó�)����1�¬, �ñ�ûA. U5½g¬ÇpØ��L0.01,83Ù¥
Ä�100�, ²u�k3�g¬, ¯ù1�¬�ÄÑ�?
'uù�¯K, 3·�¡c�3ü«�U5:
`: 0 < p ≤ 0.01; ¯: 0.01 < p < 1.
·��ÏLlù1�¬¥Ä�5û½`, ¯ü«�U5¥=�¤á.
ù�¯K~±eã�ªJÑ: Ú?��“b�”
H0 : 0 < p ≤ 0.01
§��"b� (null hypothesis) ½�b�, k��{¡�b�. ,���U´
H1 : 0.01 < p < 1
��éáb�½�Jb�(alternative hypothesis).
·��8�´�ÏL��û½�ÉH0, �´áýH0. �±/�/r¯K�¤
H0 : 0 < p ≤ 0.01←→ H1 : 0.01 < p < 1
5¿ù�J{¥òH0�3¥% �, §´u��é�. H0ÚH1� �Ø�6�. lù�~
f�òb�u�¯K��z, J{Xe:
�këê©Ùx{Fθ, θ ∈ Θ},d?Θ�ëê�m. X1, · · · , Xn´lþã©Ùx¥Ä�
�{ü�Å��. 3ëêb�u�¯K¥, ·�a,��´θ´Äáuëê�mΘ�
,�ýf8Θ0, K·KH0 : θ ∈ Θ¡�"b�½�b�, Ù(�¹Â´: �3�
�θ0 ∈ Θ0¦�X�©Ù�Fθ0 . PΘ1 = Θ−Θ0,K·KH1 : θ ∈ Θ1¡�H0�éáb
�½�Jb�(3~5.1.1¥,Θ = (0, 1), Θ0 = (0, 0.01], Θ1 = (0.01, 1)) . Kb�u�
¯KL�
H0 : θ ∈ Θ0 ←→ H1 : θ ∈ Θ1, (1.1)
3(5.1.1)ª¥, eΘ0½Θ1��¹ëê�mΘ¥���:, K¡�{üb� (simple hypothe-
sis);ÄK, ¡�EÜb� (composite hypothesis). ~X, ��ÄgN(a, σ20), σ2
0®�, Këê�m
�Θ = {a : −∞ < a < +∞}. -H0 : a = a0 ←→ H1 : a 6= a0,KH0�{üb�, H1�EÜb�.
2X, Óþ¯K, -H′
0 : a ≤ a0 ←→ H′
1 : a > a0, K"b�H′
0Úéáb�H′
1��EÜb�.
�!b�u����V�n
3~5.1.1¥, duù1�¬êþé�, �ePX�Ä��100��¬¥�g¬ê, K�±Cq
@�X ∼ B(100, p). XJ"b�0 < p ≤ 0.01´�(�, K
P (X ≥ 3) ≤ 1−2∑i=0
Ci1000.01i0.99100−i = 0.079
=XJ@�ù1�¬´Ü��, K100��¬¥k3�g¬½ö�õg¬��U5�k7.9%, ù
�VÇ'��, Uì�VÇ�n, Ø��U3�g¢�¥Òu), �·� *ÿ�. Ïdk
nd~¦"b�´Ø�(�.
A^�VÇ�n�U�NþL�·�é"b�´Ä¤á���íä.
n!Ľ�!u�¼êÚu�ÚOþ
·�EÏL~f5`²ù�Vg.
~~~5.1.2 �X = (X1, · · · , Xn)�loNX ∼ N(a, 1)¥Ä���Å��. �Äu�¯K:
H0 : a = a0 ←→ H1 : a 6= a0, (1.2)
d?, a0��½�~ê.
ù«u���«�*þ��{´: k¦a����Oþ,·���X = 1n
∑ni=1Xi´a���
`û�O. e|X − a0|��,·�Ò��uĽH0;��, XJ|X − a0|��, ·�Ò@�Ä�(
2
J�H0��C,Ï ��u�ÉH0. äN/`,·��(½��êA,é��X = (X1, . . . , Xn),�
ÑX, �|X − a0| > A�ÒĽH0;�|X − a0| ≤ A�Ò�ÉH0.·�¡
D = {X = (X1, . . . , Xn) : |X − a0| > A} (1.3)
�Ľ�,½��áý� (reject region).=,Ľ�´d���m X¥��¦|X − a0| > A�
@��X = (X1, . . . , Xn)�¤. kĽ�,�duò���m X ©¤Ø���üÜ
©X1 = X −DÚX2 = D,��k��X,�X ∈ X1�,Ò�ÉH0; �X ∈ X2 = D�,ÒĽH0.
·�¡X1��É� (acceptance region).��A½e5,KĽ�(½�É�) �Ò(½. Ï
d, 3d¯K¥u��À�Xe��«{K:
T :
{� |X − a0| > A�, áý H0
� |X − a0| ≤ A�, �É H0
þª¥�T,�½�«{K,��k��,·�Ò�±3�ÉH0½Ä½H0 ùü�(Ø¥ÀJ
��. ·�¡ù��«{KT�u�¯K(1.2)���u�.
�BuêÆþ?n,·�Ú\Xeu�¼êϕ(x)�Vg,ϕ(x)�u�T ´��éA�.3
~5.1.2¥
ϕ(x) =
{1 � |X − a0| > A�
0 � |X − a0| ≤ A�(1.4)
·�kXe½Â:
½½½ÂÂÂ5.1.1 d(5.1.4)�Ñ�u�¼ê ϕ(x)´½Â3���m Xþ, ��u[0,1]�¼ê.
§L«�k��X�,ĽH0�VÇ.
d½Â��,eϕ(x) = 1,K±VÇ�1ĽH0,�ϕ(x) = 0, K±VÇ�0ĽH0(=±VÇ
�1�ÉH0).eϕ(x)��0,1ùü��, Kù«u�¡���Åzu� (non-randomized test).d
�,Ľ���^u�¼êL«Xe: D = {X = (X1, . . . , Xn) : ϕ(x) = 1}.
eé,��X,k0 < ϕ(x) < 1,K¡ϕ(x)��Åzu� (randomized test).X3~5.1.1¥,�X =
(X1, . . . , Xn)���,�∑100i=1Xi < c �@�ù1�¬Ü�,�ÉH0;�
∑100i=1Xi > c�,@�ØÜ
�, áýH0.�∑100i=1Xi = c�,e5½áýH0,��ú��áý��U5�, ¯º. ��, e
�ÉH0,ï�(ûA)�ÉØÜ��¬��U5�,�ú�¯º.3V�ð±Øe��¹e,e�ò
¥�Y´V�Ñ�±�É�: ½e��ê0 < r < 1,5½�∑100i=1Xi = c�,±VÇ�r��gÁ
�,�âÁ�(J5û½áý�´�Éù1�¬. X�r = 1/2,K�ÏL��qM15û½.5½
eÑy�¡KáýH0, ÄK�ÉH0.ù�,�Ñy∑100i=1Xi = c,V�Ñk1/2��U,�ÑégCØ
|�û½,V�Ñú�Ün, �±�É.XJ�r = 1/3,Á��±ÏL3k2�x¥Ú1�ç¥�Ý
f¥¹¥5û½, e¹�ç¥(u)�VÇ�1/3)KáýH0,e¹�x¥(u)�VÇ�2/3) K�
ÉH0.ù«�Åzu�¼ê�L�
ϕ(x) =
1 e
∑100i=1Xi > c
r e∑100i=1Xi = c
0 e∑100i=1Xi < c.
(1.5)
3
3~5.1.2¥�(½u�,7L½Ñ(5.1.3)½(5.1.4)ª¥�A, d?A ¡��.� (critical
value).�½ec��I�é�u�ÚOþ�©Ù. 3d~¥u�ÚOþ´T = X. Ó�3
~5.1.1¥, u�¼ê(5.1.5) ¥�c¡��.�,u�ÚOþ´T =∑100i=1Xi.(½u�ÚOþ�©
Ù´)ûb�u�¯K�'�. �u�ÚOþ�°(©ÙéJé��, eÙ4�©Ù'�{
ü,·��^4�©Ù�O°(©Ù,¼�b�u�¯K�Cq).
o!üa�Ø�õ�¼ê
ÚOíä´±����â�,du����Å5,·�ØU�yÚOíä�{�ýé�(5,
�U±�½�VÇ��yù«íä���5.3b�u�¯K¥�UÑye�ü«�/¬�
�Ø:
PPPPPPPPPPb�
ûüáýH0 �ÉH0
H0�ý �� Ø��
H1�ý Ø�� ��
1. "b� H0�5´é�, du����Å5, ��*�á\Ľ�D, �Ø/òH0Ľ
, ¡�ïý. ù����Ø¡�1�a�Ø (Type I error).
2."b� H0�5Øé, du����Å5, ��*�á\�É�D, �Ø/òH0�É,
¡���. ù����Ø¡�1�a�Ø( Type II error).
X3~5.1.1¥(½��Åu�Xe:
ϕ(x) =
{1 e
∑100i=1Xi > 3;
0 e∑100i=1Xi ≤ 3.
XJoN�ý¢g¬Ç�p = 0.005 < 0.01,du����Å5,Ä�(Jw«∑100i=1Xi = 5,=�
�á\Ľ�,ù�·��1�a�Ø. ��k�UoN�ý¢g¬Çp = 0.03 > 0.01, du��
��Å5,Ä�(Jw«∑100i=1Xi = 1,=��á\�É�. ù�·��1�a�Ø.
A�5¿,3z�äN|Ü,·��¬�üa�Ø¥���. �u�(½�,�üa�Ø�V
Ç�Ò(½. ·�F"�üa�Ø�VÇ���Ð,�ù�:éJ��. 3����n�½�
cJe,�öØ�o�. ùÒXÓ«m�O¯K¥��ÝÚ°Ý�öØ�o���. @o,N��
O��üa�Ø�VÇQ?�d, ÚÑõ�¼ê�Vg.
½½½ÂÂÂ5.1.2 �ϕ(x)´H0 : θ ∈ Θ0 ←→ H1 : θ ∈ Θ1���u�¼ê,K
βϕ(θ) = Pθ{^u� ϕĽ H0
}= Eθ[ϕ(X)], θ ∈ Θ
¡�ϕ�õ�¼ê (power function),�¡��¼ê½³¼ê.
eϕ(x)���Åzu�,Ľ��D,K
βϕ(θ) = Pθ(X = (X1, . . . , Xn) ∈ D
)
4
Ïdõ�¼êL«���©Ùëê�θ�,ĽH0�VÇ. é~5.1.1,�u�¼ê��Åzu
�(1.5)�, |^∑ni=1Xi ∼ b(n, θ), 0 < θ < 1��u�õ�¼ê�
βϕ(θ) = Eθ[ϕ(x)] = P
( n∑i=1
Xi > c
)+ rP
( 100∑i=1
Xi = c
)
=
100∑k=c+1
(100
k
)θk(1− θ)100−k + r
(100
c
)θc(1− θ)100−c .
±e?Ø¥b½ϕ(x)����Åz�u�¼ê,Ø�AO�², Ø@�ϕ(x)��Åzu�¼ê.
��u�ϕ(x)�õ�¼ê�,Ò�±O��üa�Ø�VÇ. e±α∗ϕ(θ) Úβ∗ϕ(θ)©OP�
1�!�a�Ø�VÇ,K�1�a�Ø�VÇ�L«�
α∗ϕ(θ) =
{βϕ(θ) � θ ∈ Θ0
0 � θ ∈ Θ1,
�1�a�Ø�VÇ�L«�
β∗ϕ(θ) =
{0 � θ ∈ Θ0
1− βϕ(θ) � θ ∈ Θ1.
�I�`²��:´: Xc¤ã, �üa�Ø�VÇ��dõ�¼êû½, lù�:þw,
XJü�u�kÓ�õ�¼ê, Kdüu�35�þ����Ó.
o!u�Y²Ú���1�a�ØVÇ��K
c¡`L, ·�F"��u��üa�Ø�VÇÑé�, �Ø4~�/, ��`53�
½�����é?Ûu�Ñ�Ø�. ~X, �¦�1�a�Ø�VÇ~�, Ò� �áý�, ¦
�É�O�, ù7,���1�a�ØVÇO�, ��½,. Ïd, Neyman-PearsonJÑ�
^�K, Ò´���1�a�ØVÇ��K. =3�y�1�a�Ø�VÇØ�L�½ê�α
(0 < α < 1,Ï~����ê) �u�¥, Ïé�1�a�ØVǦ�U��u�. eP
Sα = { ϕ : α∗ϕ(θ) = βϕ(θ) ≤ α, �θ ∈ Θ0},
SαL«d¤k�1�a�Ø�VÇÑØ�Lα�u�¼ê�¤�a. ·���ÄSα¥�u�.
3Sα¥]À/�1�a�Ø�VǦ�U��u�0, ù«{K¡����1�a�ØVÇ�
{K.
�âNeyman-Pearson�K, 3�b�H0�ý�, ·��Ñ�Øû½(=ĽH0) �
VÇÉ���. ùL², �b�H0É��o, Ø�u�´�Ľ. ¤±3äN¯K
¥, ·� òkrº!ØU�´Ä½�·K���b�H0, rvkrº�!Ø
U�´�½�·K��éáb�. Ïd�b�H0Úéáb�H1�/ ´Ø²��,
ØU�pN�.
5
��1�a�ØVÇ�éX�,��Vg´u�Y², Ù½ÂXe:
½½½ÂÂÂ5.1.3 �ϕ´(1.1)���u�, 0 ≤ α ≤ 1. XJϕ�1�a�Ø�VÇoØ�Lα
(½�d/`, eϕ÷v: βϕ(θ) ≤ α,é��θ ∈ Θ0) , K¡α´u�ϕ���Y², ϕ¡�wÍ5
Y²�α�u�, {¡Y²�α�u�.
Uù�½Â,u��Y²Ø��.eα�u�ϕ�Y², α < α′ < 1,Kα′�´u�ϕ�Y².
�;�ù�¯K, k�¡��u����Y²�Ùý¢Y². �Ò´
u�ϕ�ý¢Y² = sup{ βϕ(θ), θ ∈ Θ0} (1.6)
�uY²�ÀJ,S.þrα��'���IOz,Xα = 0.01, 0.05, 0.10�. IOz´��
BEL.
Y²�À�, éu��5�ké�K�. ØJ),XJY²À�é$, @o·�NN�1�
a�Ø�VÇé�, ���ù�:³7�� �Ľ�, ù�ÒO\�1�a�Ø��
U5. ��,eY²À�p,KĽ�*�, ¦�É� �, l �1�a�Ø�VÇ�A�òü
$. ù�w5,Y²�ÀJØ´��êƯK, ´��7Ll¢S�Ý5�Ä�¯K. ��`5
k±eA�Ï�K�Y²�À½.
1.���u��9ü�|Ã�, Y²�À½~´V��Æ�(J. ±~5.1.1�~,ûA�ó
�?À, u�Ùg¬Ç´Ä�L0.01,eY²À�$,K�Uk�õ�g¬�ûA�É;��,eY
²½�p, Kòk�õ�Ü�¬�ûAáÂ. ÏdY²½����9ûAÚó�V�|Ã,Ad
V�û½. Xc¤ã,k���æ��Åz��{,¦V�|Ã��²ï.
2.ü«�Ø��J��35�þké��ØÓ. XJ1�a�Ø��J35�þéî, ·
�Òå¦3Ün���S¦þ~��ù«�Ø��U5,ù��A�Y²Ò���$�. ~
X,����)��«#��OÎ�£�,«;¾, Sü�Á�,�é#Î�Ô���Ñu
�. duÎ�®²�Ï�K¦^,k�½���. #�ÿ�²�Ï�K¦^, ���JØÐ�,ò
�9¾<�)·S�, E¤��J¬éî. ¤±3?1u��, ò�b�H0��”Î�Ø'
#��”,�¦u�Y²α½����, ù�¦H0�Ľ��U5��~�. ù�Ò�y:
/�b��Ľ!#���É�u�0ò´�~î��.
3.��`5,Á�ö3Á�cé¯K��¹oØ´�ä��. ¦é¯K�)¦¦é"b
� ´ÄU¤áÒk�½�w{, ù«w{�UK��¦éY²�ÀJ. '�`��ÔnÆ[
�â,«nØí½�ÅCþXAk©ÙF, ¦��òù�nØGÃu�. é²w,XJ¦éù�
nØék&%,¦ò�~��u@�b�U¤á, ù��kékå�yââ�U¦¦@�ùb�
Øé. �A/,¦òru�Y²��$�.
3¢S¯K¥, "b��Ľ, ~~¿�Xí��«nؽ^#�{5�O��¦^�
IO�{. 3�õê�¹e,<�F"ù���k�����â. lùp�±w�, Neyman-
Pearson���1�a�Ø��K,3"b��ÀJ¥ké��¢S¿Â, ûØüX´�êÆ
¯K. Ó�,�?�Ún)3b�u�¯K¥,"b�?3âÑ/ ��Ï.
���`²��:´:eY²αé�,�b�H0ج�´�Ľ. XJ��*�á\Ľ
�, ·��Ñ/Ľ�b�H00�(ØÒ'���(Ï�, d�·��¬�1�a�Ø, �ÙV
Çé�). ��, �αé��,XJ��*�á\�É�,·��Ñ/�É�b�H00�(Ø�7
��. ù�UL²:3¤À½�Y²evk¿©�â@�H0ؤá,ûØ¿�Xk¿©�â`²
§�((Ï�d�·��¬�1�a�Ø, �ÙVÇ�Ué�) .
6
Ê!¦)b�u�¯K���Ú½
1. �â¯K��¦JÑ"b�H0Ú�Jb�H1;
2. �ÑĽ��/ª, (½u�ÚOT (X), Ù¥�.�A�½.
3. À�·�Y², |^u�ÚOþ�©Ù¦Ñ�.�A.
4. d��X�Ñu�ÚOþT (X)�äN�, �\�Ľ�¥, ��.��'�, �Ñ�ɽ
öáý�b�H0�(Ø.
2 ��oNëê�b�u�
��©Ù´�~��©Ù, 'u§�ëê�b�u�´¢S¥~���¯K, Ïd�´�
���au�¯K. �!ò©e�A«�¹5?Ø��oNëê��*u��{: ü���o
Nþ�Ú���u�; ü���oNþ��Ú��'�u�; 4�©Ù���©Ù�k'���
u�.
3?Ø��©ÙoNëê�b�u�¯K�, §2.4¥�½n2.2.3ÚíØ2.4.2–íØ2.4.53¦
u�ÚOþ�©Ù¥å��©���^.
�!ü���oNþ��u�
�X = (X1, . . . , Xn)�l��oNN(µ, σ2)¥Ä��{ü�Å��, �½u�Y²α, ¦e
�nau�¯K:
(1) H0 : µ = µ0 ←→ H1 : µ 6= µ0;
(2) H ′0 : µ ≤ µ0 ←→ H ′1 : µ > µ0;
(3) H ′′0 : µ ≥ µ0 ←→ H ′′1 : µ < µ0;
Ù¥µ0Úu�Y²α�½.
·�¡u�¯K(1)�V>u� (two-side test), ¡u�¯K(2)Ú(3)�ü>u� (one-side
test).
ü���oN������u�¯K�'��®��¹�~�, ·�ò:?Øù��/.
'uþ�®���u��{, ·�ò3�¡���`².
Äk�Äu�¯K(1), =
H0 : µ = µ0 ←→ H1 : µ 6= µ0
·�^�*�{�Eu��Ľ�. ·���X = 1n
∑ni=1Xi ´µ�à �O, �ä
kûÐ5�. �*þw|X − µ0|��, H0�Ø�¤á. Ïdu��Ľ���Xe/ª:
{X = (X1, . . . , Xn) : |X − µ0| > A}, A�½. �σ2���, díØ2.4.2��,3µ = µ0^�e,
T =
√n(X − µ0)
S∼ tn−1. (2.1)
Ïd�T =√n(X−µ0)
S ��u�ÚOþ,KĽ���d/ª���{(X1, . . . , Xn) : |T | > c
}, c�½.
7
du�Y²�α,��
P (|T | > c |H0) = P(∣∣√n(X − µ0)/S
∣∣ > c∣∣H0
)= α,
��c = tn−1(α/2).ÏddĽ�
D1 ={
(X1, . . . , Xn) :∣∣√n(X − µ0)/S
∣∣ > tn−1(α/2)}
(2.2)
(½�u��u�¯K(1)�Y²�α�u�.
éu�¯K(2), �T =√n(X − µ0)/S��u�ÚOþ,Ïd�*þĽ��/ª�
{(X1, . . . , Xn) : T > c}, c�½.
�¦u�(2′)äkY²α,, =�¦
P (T > c |H ′0) = P(√n(X − µ0)/S > c
∣∣µ ≤ µ0
)≤ α, (2.3)
5¿�
P(√n(X − µ0)/S > c
∣∣µ ≤ µ0
)= P
(√n(X − µ)/S > c+
√n(µ0 − µ)
S
∣∣µ ≤ µ0
)≤ P
(√n(X − µ)/S > c,
∣∣µ ≤ µ0
)Ïd�IP
(√n(X − µ)/S > c,
∣∣µ ≤ µ0
)= αKª(2.3)¤á. ¤±c = tn−1(α).�u�¯
K(2′)�Y²�α�u��Ľ�´
D∗2 ={
(X1, . . . , Xn) : T =√n(X − µ0)/S > tn−1(α)
}. (2.4)
aq�{��u�¯K(3)�Y²�α�u��Ľ�.
D3 ={
(X1, . . . , Xn) :√n(X − µ0)/S < −tn−1(α)
}ù«Äuu�ÚOþÑlt©Ù�u��{¡����tu� ,T¡����tu�ÚOþ.
5555.2.1 ��®����oNþ��u��{�Xe`²: 3N(µ, σ2)¥, �σ2®�,
�H0¤á, =µ = µ0�k
U =√n(X − µ0)/σ ∼ N(0, 1) (2.5)
Ïd�U=√n(X −µ0)/σ ��u�ÚOþ, ^�����σ2���/�Ó��{(¤ØÓ�Ò´
^u�ÚOþU �O@��u�ÚOþT ) ��u�¯K(1)-(3)�Y²�α�u��Ľ�. Ä
½�¥��.�òtn−1© êU¤�A�IO��©Ù�© ê. �[(J�L5.2.1.
ù«Äuu�ÚOþÑlN(0, 1)©Ù�u��{¡����Uu�.
L5.2.1 ü���oNþ��b�u�H0 H1 u�ÚOþ9٩٠Ľ�
σ2 µ = µ0 µ 6= µ0 U =√n(X − µ0)/σ |U | > uα/2
® µ ≤ µ0 µ > µ0 U |µ = µ0 ∼ N(0, 1) U > uα
� µ ≥ µ0 µ < µ0 U < −uασ2 µ = µ0 µ 6= µ0 T =
√n(X − µ0)/s |T | > tn−1(α/2)
� µ ≤ µ0 µ > µ0 T |µ = µ0 ∼ tn−1 T > tn−1(α)
� µ ≥ µ0 µ < µ0 T < −tn−1(α)
8
~~~5.2.1 ¬�^gÄC-ÅC-Þ ¬,z-IOþ�500�,zUmóIu�Åì�
ó�G¹. 8Ä�10-,ÿ�Ùþ(ü :�)
495, 510, 505, 498, 503, 492, 502, 512, 497, 506.
b½-ÞþXÑl��©ÙN(µ, σ2),®�σ = 6.5, ¯Åì´Äó��~?(�u�Y²α =
0.05)
))) u�¯K�
H0 : µ = 500←→ H1 : µ 6= 500
�K¥σ2®�,�Ľ�dL5.2.1¥1�1�Ñ,=
D ={
(X1, . . . , Xn) : |√n(X−µ0)
σ | > µα2
}Ù¥n = 10, α = 0.05,�L�u0.025 = 1.96,d����X = 502, Ïd
|U | = |√n(X−µ0)
σ | = |√
10(502−500)6.5 | = 0.973 < 1.96
¤±3wÍ5Y²α = 0.05e@�vkvnd`²gÄC-Åó�Ø�~,��ÉH0.
~~~5.2.2 ,<�¤)��/<�|ärÝXÑl��©ÙN(µ, σ2), 8lT�)��/
<¥�ÅÄ�6¬ÿ�|ärÝXe(ü :kg/cm2)
32.56, 29.66, 31.64, 30.00, 31.87, 31.03.
¯ù�1/<�|ärÝ�Ä@�Ø$u32.50 kg/cm2? (�u�Y²α = 0.05)
))) u�¯K�
H0 : µ ≥ 32.50←→ H1 : µ < 32.50
�K¥σ2��,�æ^tu�{,Ľ�dL5.2.1¥���1�Ñ,=
D = {(X1, . . . , Xn) :√n(X−µ0)
S < −tn−1(α)},
Ù¥n = 6, α = 0.05,�L�t5(0.05) = 2.015,dêâ��X = 31.13, S = 1.123,Ïdk
T =√
6(31.13−32.50)1.123 = −3.36 < −2.015
�ĽH0,=@�/<r��32.50 kg/cm2.
�!ü���oN���u�
�X1, . . . , Xn�g��oNN(µ, σ2)¥Ä��{ü�Å��, �ã�?Øe�nau�¯
K:
(4) H0 : σ2 = σ20 ←→ H1 : σ2 6= σ2
0 ;
(5) H ′0 : σ2 ≤ σ20 ←→ H ′1 : σ2 > σ2
0 ;
(6) H ′′0 : σ2 ≥ σ20 ←→ H ′′1 : σ2 < σ2
0 ;
Ù¥σ20Úu�Y²α�½.
·�ò:?Ø, þ����ü���oN����u��{. �þ�®��XÛ?n, ·
�ò3�¡���`². Äk�Äu�¯K(4), =
H0 : σ2 = σ20 ←→ H1 : σ2 6= σ2
0
9
duþ�µ��, ·���S2 = 1n−1
∑ni=1 (Xi − X)2 ´σ2���à �O, �äkûÐ
5�. �*þwS2/σ20��½öS
2/σ20���,H0Ø�¤á. Ïdu��Ľ���Xe/ª:
{(X1, . . . , Xn) : S2/σ20 < A1 ½ S2/σ2
0 > A2}, A1, A2 �½. 3�½σ2 = σ2
0�^�e, d½
n2.2.3��
(n− 1)S2/σ20 ∼ χ2
n−1. (2.6)
��u�ÚOþ�χ2 = (n− 1)S2/σ20 . Ïd, Ľ���d/ª���
D ={
(X1, . . . , Xn) : (n− 1)S2/σ20 < c1½ (n− 1)S2/σ2
0 > c2}
Pθ = (µ, σ2),�(½c1, c2,-
α = Pθ((n− 1)S2/σ2
0 < c1½ (n− 1)S2/σ20 > c2
∣∣H0
)÷vþª�c1, c2�éfkéõ,�3�éc1, c2´�`�,�O��E,, �¦^Ø�B. (
½c1, c2���{ü¢^��{´:-
Pθ((n− 1)S2/σ2
0 < c1∣∣H0
)= α/2, Pθ
((n− 1)S2/σ2
0 > c2∣∣H0
)= α/2
dþãüªÚ(2.6)´��.�c1 = χ2n−1(1 − α/2), c2 = χ2
n−1(α/2).¤±u�¯K(4)�Y²
�α��É��
D4 ={
(X1, . . . , Xn) : χ2n−1(1− α/2) ≤ (n− 1)S2/σ2
0 ≤ χ2n−1(α/2)
}d�É��L�ª'Ľ�{ü,¦^þ��B, �d?æ^�É��OĽ�.
^��aqu(�) ¥¦u�¯K(2)!(3)��{�©O¦�u�¯K(5)Ú(6)�Y²�α�
Ľ�Xe:
D5 ={
(X1, . . . , Xn) : (n− 1)S2/σ20 > χ2
n−1(α)},
D6 ={
(X1, . . . , Xn) : (n− 1)S2/σ20 < χ2
n−1(1− α)}.
5555.2.2 �þ�µ®��, ��σ2�u��{{ãXe: �µ®��,σ2���äkûÐ5
��à �O´S2∗ = 1
n
∑ni=1 (Xi − µ)2.�σ2 = σ2
0�,díØ2.4.1 ��
nS2∗/σ
20 =
n∑i=1
(Xi − µ)2/σ20 ∼ χ2
n. (2.7)
Ïd�u�ÚOþ�χ2∗ = nS2
∗/σ20 ,^§�Oχ
2 = (n− 1)S2/σ20 .æ^��aquc¡µ���/
�?Ø�{, ��u�¯K(4)-(6)�Y²�α�Ľ�,=5¿3Ľ�¥ò(½�.��χ2©
Ù�gdÝdn− 1U¤n=�. �[(J�L5.2.2.
ù«Äuu�ÚOþÑl�½gdÝ�χ2©Ù�u��{¡�χ2u�.
L5.2.2 ü���oN���b�u�
10
H0 H1 u�ÚOþ Ľ�
9Ù©Ù
µ σ2 = σ20 σ2 6= σ2
0 χ2∗ = ns2
∗/σ20 nS2
∗/σ20 < χ2
n(1− α/2)
® ½ nS2∗/σ
20 > χ2
n(α/2)
� σ2 ≤ σ20 σ2 > σ2
0 χ2∗|σ2
0 ∼ χ2n nS2
∗/σ20 > χ2
n(α)
σ2 ≥ σ20 σ2 < σ2
0 nS2∗/σ
20 < χ2
n(1− α)
µ σ2 = σ20 σ2 6= σ2
0 χ2 = (n−1)s2
σ20
(n− 1)S2/σ20 < χ2
n−1(1− α/2)
� ½ (n− 1)S2/σ20 > χ2
n−1(α/2)
� σ2 ≤ σ20 σ2 > σ2
0 χ2|σ20 ∼ χ2
n−1 (n− 1)S2/σ20 > χ2
n−1(α)
σ2 ≥ σ20 σ2 < σ2
0 (n− 1)S2/σ20 < χ2
n−1(1− α)
~~~5.2.3 ,ó�)���«[ã|êÑl��©Ù,ÙoNIO��1.2. yl,F)�
��1�¬¥Ä�16Ã?1|êÿþ,ÿ���IO��2.1,¯ã�þ!Ý´ÄC�?(α = 0.05)
))) u�¯K�
H0 : σ2 = 1.44←→ H1 : σ2 6= 1.44.
u�����
D = {(X1, . . . , Xn) : χ2n−1(1− α/2) < (n− 1)S2/σ2
0 < χ2n−1(α/2)}
d?n = 16, α = 0.05.�L��χ215(0.975) = 6.262, χ2
15(0.025) = 27.488. d®�êâ�
�S2 = 2.12 = 4.41, σ20 = 1.22 = 1.44.Ïdk
χ2 = (n−1)S2
σ20
= 15×4.411.44 = 45.94 > 27.488.
�ĽH0,=@��ã�þ!ÝC�.
n!ü���oNþ���u�
�X = (X1, . . . , Xm)�g��oNN(µ1, σ21) ¥Ä��{ü�Å��, Y = (Y1, . . . , Yn)�
g��oNN(µ2, σ22) ¥Ä��{ü�Å��, �Ü��X1, . . . , Xm;Y1, . . . , YnÕá. �ã�?
Øe�nau�¯K:
(7) H0 : µ2 − µ1 = µ0 ←→ H1 : µ2 − µ1 6= µ0;
(8) H ′0 : µ2 − µ1 ≤ µ0 ←→ H ′1 : µ2 − µ1 > µ0;
(9) H ′′0 : µ2 − µ1 ≥ µ0 ←→ H ′′1 : µ2 − µ1 < µ0;
Ù¥µ0Úu�Y²α�½.
±þb�u�¯K�¦þ����&«m�Behrens-Fisher¯K (�§4.2,n) ��, ا
�A«AÏ�/®¼��÷)û, ���¹�8ÿvk��{ü!°(�){. e¡©A«
�¹©O?Ø.
1. �σ21Úσ
22®��, þ���u�¯K
Äk�Äü>u�¯K(7), =
H0 : µ2 − µ1 ≤ µ0 ←→ H1 : µ2 − µ1 > µ0
11
duY − X´µ2 − µ1���à �O, �äkûÐ5�. �*þw|Y − X − µ0|���,
H0�Ø�¤á, �Ľ���Xe/ª{(X1, . . . , Xm;Y1, . . . , Yn) : |Y − X − µ0| > A}, A�½.
duY − X ∼ N(µ2 − µ1,√σ2
1/m+ σ22/n), ��µ2 − µ1 = µ0�k
U =Y − X − µ0√σ2
1/m+ σ22/n
∼ N(0, 1). (2.8)
Ïd, �u�ÚOþ�U = (Y − X − µ0)/√
σ21/m+ σ2
2/n, KĽ��d/ª�{(X1, . . . , Xm;Y1, . . . , Yn) :
∣∣∣∣ Y − X − µ0√σ2
1/m+ σ22/n
∣∣∣∣ > c
}, c�½.
Pθ = µ2 − µ1, �(½c,-
α = Pθ (|U | > c|H0) = P
(∣∣∣∣ Y − X − µ0√σ2
1/m+ σ22/n
∣∣∣∣ > c∣∣∣H0
)= 2− 2Φ(c)
dd�(½�.�c = uα/2.Ïdu�¯K(3)�Y²�α�u�Ľ��
D7 =
{(X1, . . . , Xm;Y1, . . . , Yn) :
∣∣∣∣ Y − X − µ0√σ2
1/m+ σ22/n
∣∣∣∣ > uα/2
},
��aq(�)¥�¦u�¯K(2)!(3)�{, ��u�¯K(8)!(9)�Y²�α�u��Ľ
��
D8 =
{(X1, . . . , Xm;Y1, . . . , Yn) :
Y − X − µ0√σ2
1/m+ σ22/n
> uα
}D9 =
{(X1, . . . , Xm;Y1, . . . , Yn) :
Y − X − µ0√σ2
1/m+ σ22/n
< −uα}
ù«Äud(2.8)�Ñ�u�ÚOþU�u��{,¡�ü��Uu�.
2. �σ21 = σ2
2 = σ2���, þ���u�
eσ21 = σ2
2 = σ2®�, Kdc¡ff?Ø�ü��Uu���, d�u�ÚOþUC�
U =Y − X − µ0√σ2/m+ σ2/n
=Y − X − µ0
σ
√mn
m+ n. (2.9)
3σ21 = σ2
2 = σ2���, þãL�ª¥�σ2~^
S2w =
1
n+m− 2
((m− 1)S2
1 + (n− 1)S22
)=
1
n+m− 2
( m∑i=1
(Xi − X)2 +
n∑j=1
(Yj − Y )2
)(2.10)
5�O, Ù¥
S21 =
1
m− 1
m∑i=1
(Xi − X)2, S22 =
1
n− 1
n∑j=1
(Yj − Y )2
12
©O���X1, . . . , XmÚY1, . . . , Yn�����. �µ2−µ1 = µ0�,òúª(2.9)¥�σ^(2.10)¥
�sw�O, ��e��ÚOþTw,díØ2.4.3��
Tw =Y − X − µ0
Sw
√mn
m+ n∼ tn+m−2. (2.11)
Ïd·��Tw = Y−X−µ0
Sw
√mnm+n��u�ÚOþ.
�(�)¥���tu��{aq, 3ü��þ���u�¯K¥, ^u�ÚOþTw �O@�
����tu�ÚOþT,^���Ó�?Ø�ª, ��u�¯K(7)-(9)�Y²�α �Ľ�, �
�5¿3Ľ�¥ò(½�.��t©Ù�gdÝdn − 1U�n + m − 2=�. �[(J�e�
L5.2.3.
ù«Äuu�ÚOþÑltn+m−2©Ù�u��{,¡�ü��tu�.
L5.2.3 ü���oNþ���b�u�
H0 H1 u�ÚOþ Ľ�
9Ù©Ù
σ21 µ2 − µ1 = µ0 µ2 − µ1 6= µ0 U = Y−X−µ0√
σ21/m+σ2
2/n|U | > uα
2
σ22
® µ2 − µ1 ≤ µ0 µ2 − µ1 > µ0 U |µ0 ∼ N(0, 1) U > uα
� µ2 − µ1 ≥ µ0 µ2 − µ1 < µ0 U < −uασ2
1 µ2 − µ1 = µ0 µ2 − µ1 6= µ0 Tw=Y−X−µ0
Sw
√mnm+n |Tw| >
σ22 tn+m−2(α2 )
� µ2 − µ1 ≤ µ0 µ2 − µ1 > µ0 Tw|µ0 ∼ tn+m−2 T > tn+m−2(α)
� µ2 − µ1 ≥ µ0 µ2 − µ1 < µ0 S2w=
(m−1)S21+(n−1)S2
2
n+m−2 T<−tn+m−2(α)
~~~5.2.4 �ïÄ�~¤cIåÉ�ù[�²þê��O, u�,/�~¤cIf156<,
åf74<, O�Iåù[��²þêÚ��IO�©O�
I: X = 465.13� /mm3, S1 = 54.80� /mm3
å: Y = 422.16� /mm3, S2 = 49.20� /mm3
b½�~¤cIåù[�ê©OÑl��©Ù, ����Ó. u��~¤c<ù[�ê´Ä�5
Ok'. (α = 0.01)
))) �X1, . . . , Xm i.i.d. ∼ N(µ1, σ2); Y1, . . . , Yn i.i.d. ∼ N(µ2, σ
2), �b½ùü|��Õ
á. u�¯K�
H0 : µ2 − µ1 = µ0 ←→ H1 : µ2 − µ1 6= µ0
u��Ľ��
D ={
(X1, . . . , Xm, Y1, . . . , Yn) :∣∣∣ Y−XSw
√mnm+n
∣∣∣ > tn+m−2(α2 )}
d?
m = 156, n = 74, X = 465.13, Y = 422.16, S1 = 54.80, S2 = 49.20,
S2w =
1
n+m− 2
[(m− 1)S2
1 + (n− 1)S22
]= 2816.6, Sw = 53.07,
13
�L�t228(0.005) = 2.576 .d
|T | =∣∣∣ Y−XSw
√mnm+n
∣∣∣ =∣∣∣ 422.16−465.13
53.07
√156×74156+74
∣∣∣ = 5.74 > 2.576,
�ĽH0,=@��~¤c<�ù[�ê�5Ok'.
3. ���Nþm = n�þ���u�—¤é'�¯K
c¡?Ø�^uü���oNþ���u�¥, b½5gü���oN���´�
pÕá�. �3¢S¯K¥, k�ÿ�¹Øo´ù�. �Uùü���oN���´5gÓ
��oNþ�E*, §�´¤éÑy�, �´�'�. ~X�*�«S����
J, P¹n ���¾<Ñ�c�z�Z��mX1, X2, . . . , XnÚÑ^dS���z�Z��
mY1, Y2, . . . , Yn.ùp(Xi, Yi)´1i�¾<ØÑ^S��ÚÑ^S��z��Z��m. §�´
k'X�, ج�pÕá. ,��¡, X1, X2, . . . , Xn´n�ØÓ��¾<�Z��m, du�<
N�Ã�¡�^�ØÓ, ùn�*�ØU@�´5gÓ����oN���. Y1, Y2, . . . , Yn�
´��. ù��êâ¡�¤éêâ. ù��êâ^ü��tu�ÒØÜ·, Ï�XiÚYi´Ó3
1i�¾<�þ*����Z��m, ¤±Zi = Yi −Xi Ò�Ø<�N�Ã�¡��É, =
�eS����J. eS��Ã�, Zi��É=d�ÅØ�Úå, �ÅØ��@�Ñl��©
ÙN(0, σ2).��b½Z1, . . . , Zn�gN(µ, σ2)¥Ä��{ü�Å��, µÒ´S���²þ�J.
S��´Äk�, Ò8(�u�Xeb�
H0 : µ = 0←→ H1 : µ 6= 0
Ï�Z1, . . . , Zn�@�´5g��oNN(µ, σ2)�{ü��, ��^'uü���oNþ�
�tu��{. u��Ľ��
D = {(Z1, Z2, . . . , Zn) : |T | > tn−1(α2 )}
d?α�u�Y², T =√n Z/S�u�ÚOþ, Ù¥ZÚS2�Z1, Z2, . . . , Zn ���þ�Ú��
��.
~~~5.2.5 8kü�ÿþá�¥,«7á¹þ�1̤AÚB,��½§���þkÃwÍ
�É, éT7á¹þØÓ�9�á��¬?1ÿþ, ��9é*�Xe:
µ(%) : 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00.
ν(%) : 0.10, 0.21, 0.52, 0.32, 0.78, 0.59, 0.68, 0.77, 0.89.
¯�â¢�(J, 3α = 0.01e, UÄ�äùü�1̤��þkÃwÍ�É?
))) ò1̤AÚBé9��¬�ÿ½�P�X1, X2, . . . , X9,ÚY1, Y2, . . . , Y9. duù9�
�¬7á¹þØÓ, ÏdX1, X2, . . . , X9 ØUw¤5gÓ�oN. Y1, Y2, . . . , Y9���. �I^
¤é'�. P
Zi = Yi −Xi, i = 1, 2 . . . , 9.
eùü1̤�þ��, ÿþ���zéêâ��É=d�ÅØ�Úå. �ÅØ��@�Ñl
��©ÙN(0, σ2).��b½Z1, . . . , Zn�gN(µ, σ2)¥Ä���Å��, �u�
H0 : µ = 0←→ H1 : µ 6= 0, α = 0.01
dL5.2.1��du��Ľ��
{(Z1, · · · , Zn) : |T | > tn−1(α/2)}
14
d?n = 9.dK¥êâ��:
Z = 19
∑9i=1 Zi = 0.06, S2 = 1
8
∑9i=1 (Zi − Z)2 = 0.01505, S = 0.12268
�L� tn−1(α2 ) = t8(0.005) = 3.3554.du
|T | =∣∣∣√nZS ∣∣∣ = 3×0.06
0.12268 = 1.47 < 3.3554
�Ãvyâw«ü�¤ìkwÍ�É, Ïd�ÉH0.
o!ü���oN��'�u�
�X = (X1, . . . , Xm)´l��oNN(µ1, σ21)¥Ä��{ü�Å��,Y = (Y1, . . . , Yn)´l
��oNN(µ2, σ22)¥Ä��{ü�Å��, �Ü��(X,Y) = (X1, . . . , Xm;Y1, . . . , Yn)Õá.
?Øe�nab�u�¯K:
(10) H0 : σ22/σ
21 = 1←→ H1 : σ2
2/σ21 6= 1;
(11) H ′0 : σ22/σ
21 ≤ 1←→ H ′1 : σ2
2/σ21 > 1;
(12) H ′′0 : σ22/σ
21 ≥ 1←→ H ′′1 : σ2
2/σ21 < 1;
u�Y²α�½.
PXÚS21�X1, . . . , Xm���þ�Ú����; YÚS2
2�Y1, . . . , Yn���þ�Ú���
�. Ù¥
S21 = 1
m−1
∑mi=1 (Xi − X)2, S2
2 = 1n−1
∑nj=1 (Yj − Y )2.
e¡òX?ص1Úµ2�����'�u��{,�µ1Úµ2®�´��'�u�XÛ?n,
·�ò3�¡�Ñ��`².
Äk?Øu�¯K(1), =
H0 : σ22/σ
21 ≤ 1←→ H1 : σ2
2/σ21 > 1
duS21ÚS
22©O´σ
21Úσ
22�à �O,¿äkûÐ5�. �*þw, S2
2/S21��½öS
22/S
21�
��, H0Ø�¤á. ���Ľ��/ª�
{(X1, . . . , Xm;Y1, . . . , Yn) : S22/S
21 < c1½ S2
2/S21 > c2}, c1, c2�½.
3S22/S
21 = 1�^�e, díØ2.4.4��
F = S22/S
21 ∼ Fn−1,m−1. (2.12)
Ïd�u�ÚOþ�F = S22/S
21 . Pθ = (µ1, µ2, σ
21 , σ
22),�(½Ä½�¥��.�c1, c2,-
α = Pθ(S2
2/S21 < c1 ½ S2
2/S21 > c2|H0
)÷vþª�¦�c1Úc2kéõ, Ù¥�3�éc1, c2�`, �O�E,, ¦^Ø�B. (½c1, c2�
��{ü¢^��{´: -
Pθ(S2
2/S21 < c1|H0
)= α/2, Pθ
(S2
2/S21 > c2|H0
)= α/2.
dþãüªÚ(2.12)´��.�c1 = Fn−1,m−1(1 − α/2), c2 = Fn−1,m−1(α/2), ¤±u�¯
K(10)�Y²�α��É��
15
D10 = {(X,Y) : Fn−1,m−1(1− α/2) 6 S22/S
21 6 Fn−1,m−1(α/2)}
d�É��L�ª'Ľ�{ü, ¦^�B, �d?æ^�É��OĽ�.
^��aqu(�) ¥¦u�¯K(2)! (3)��{�©O¦�u�¯K(11)Ú(12)�Y²
�α�Ľ�Xe:
D11 = {(X1, . . . , Xm;Y1, . . . , Yn) : S22/S
21 > Fn−1,m−1(α)}
D12 = {(X1, . . . , Xm;Y1, . . . , Yn) : S22/S
21 < Fn−1,m−1(1− α)}
ùp�5¿�:�´: �α´���ê, Xα = 0.01, 0.05�, lF−©Ù�© êLþ�Ø�Fn−1,m−1(1− α)�ê�, �|^1�ÙSK¥®y²�¯¢
Fn,m(1− α) = 1/Fm,n(α), (2.13)
¦¯K¼�)û. ~XlLþ�Ø�F5,10(1 − 0.01)��, ����F10,5(0.01)��, |^ú
ª(2.13) ��F5,10(1− 0.01) = 1/F10,5(0.01),l �¦�¤��ê�.
5555.2.3 �µ1Úµ2®��, ��'�u��{{ãXe: �µ1Úµ2®��, σ21Úσ
22äkû
Ð5��à �O©O´
S21∗ =
1
m
m∑i=1
(Xi − µ1)2, S22∗ =
1
n
n∑i=1
(Yi − µ2)2
�σ21/σ
22 = 1�, |^íØ2.4.1ÚF©Ù�½Â, N´y²
F∗ = S22∗/S
21∗ ∼ Fn, m . (2.14)
Ïd,�u�ÚOþF∗�OF = S22/S
21 , ��aquµ1Úµ2���/�?Ø, ���u
�¯K(10)-(12)�Y²�α�Ľ�,��5¿3Ľ�¥, ò(½�.��F©Ù�gdÝ
dn− 1,m− 1©OU�n,m=�. �[(J�e�L5.2.4.
ù«Äuu�ÚOþÑlF©Ù�u��{, ¡�Fu�.
L5.2.4 ü���oN��'�b�u�
H0 H1 u�ÚOþ9٩٠Ľ�
µ1 σ22 = σ2
1 σ22 6= σ2
1 F∗ = s22∗/S
21∗ F∗ < Fn,m(1− α/2)
µ2 F∗|σ22=σ21∼ Fn,m ½ F∗ > Fn,m(α/2)
® σ22 ≤ σ2
1 σ22 > σ2
1 S21∗=
1m
∑mi=1(Xi − µ1)2 F∗ > Fn,m(α)
� σ22 ≥ σ2
1 σ22 < σ2
1 S22∗= 1
n
∑ni=1(Yi − µ2)2 F∗ < Fn,m(1− α)
µ1 σ22 = σ2
1 σ22 6= σ2
1 F = s22/S
21 F<Fn−1,m−1(1−α/2)
µ2 F |σ22=σ21∼ Fn−1,m−1 ½ F>Fn−1,m−1(α/2)
� σ22 ≤ σ2
1 σ22 > σ2
1 S21= 1
m−1
∑mi=1 (Xi − X)2 F > Fn−1,m−1(α)
� σ22 ≥ σ2
1 σ22 < σ2
1 S22= 1
n−1
∑nj=1 (Yj − Y )2 F<Fn−1,m−1(1− α)
~~~5.2.6 ÿ�ü1������6�>fìá>{�þ�X = 0.14, Y= 0.139, ��IO
�©O�SX = 0.0026, SY = 0.0024,b�ùü1ìá�>{©OÑlN(µ1, σ21), N(µ2, σ
22), þ
������, �ü|��Õá, ¯ùü1>fì��>{´Ä�Ó? (α = 0.05)
16
): ù�¯KL¡w´éü���oNþ���u�,�·�Ø��´Äkσ21 = σ2
2 ,ÏdÄ
k¦ü���oN��´Ä�Ó�u�. XJu�@�σ21 = σ2
2 , ,�2�ü��tu�. XJ²
u�Ľσ21 = σ2
2 , K·�ØU^ü���tu��{u�þ��, ùÒC¤Behrens-Fisher¯
K, ò33�!��)û.
Äk�Äe�u�¯K:
(1) H0 : σ21 = σ2
2 ←→ H1 : σ21 6= σ2
2 , α = 0.05 .
dL5.2.4��du���É�´
{(X,Y) : Fm−1,m−1(1− α/2) ≤ S2X/S
2Y ≤ Fm−1,n−1(α/2)},
d?m = n = 6, S2X/S
2Y = 0.00262/0.00242 = 1.17.dα = 0.05 , �F©ÙL�F5,5(0.025) =
7.15. du
1/7.15 = F5,5(1− 0.025) < F = S2X/S
2Y = 1.17 < F5,5(0.025) = 7.15,
�@�vkv�yâĽH0,Ïd�ÉH0.
3�Éþãu��, ·��±b½σ21 = σ2
2 ,?�Ú�Äe�u�¯K:
(2) H ′0 : µ1 = µ2 ←→ H ′1 : µ1 6= µ2, α = 0.05 .
dL5.2.3��du��Ľ��
{(X,Y) : |Tw| > tn+m−2(α/2)}
d?m = n = 6, X = 0.14, Y = 0.139, SX = 0.0026, Sy = 0.0024,Ïdk
S2w = 1
10 [5× 0.00262 + 5× 0.00242] = 6.26× 10−6, Sw = 0.0025.
dα = 0.05,�t−©ÙL�t10(0.0025) = 2.228 . du
|T | =√
nmn+m
∣∣∣ Y−XSw
∣∣∣ =√
3×∣∣ 0.14−0.139
0.0025
∣∣ = 0.6928 < 2.228
�vk¿v�ndĽü1>fì��>{��Ó,Ïd�ÉH ′0
17