应用数理统计第3章

33.13.23.33.4Bayes.1.(TestofHypothesis)2.(1)K.Pearson1900(2)R.A.Fisher1920(3)J.NeymanE.S.Pearson1928(4)A.Wald1950(5)BayesT.Bayes(1702~1761)TeaMilkTMMTTMMT8TMMTTMR.A.FisherH0H084TM—=—0.014TM11C8470(1)H0(2)H00.014Fisher()H0H02%133(H0)————=—0.24325%H0H01+C43C4117C8470.3.1.115.1mm0.05mm614.95mm()1.X~N(m0.05)m=15.1m15.1()H0m=m0(=15.1)ÛH1mm0(15.1)2.m(m=15.1)0.05~(,)6XNm6(15.1)~(0,1)0.05XzN-=3.()m(m=15.1)|15.1|X-(m=15.1)z0ua/2|15.1|X-|z|=z06|15.1|0.05X-/26|15.1|P{}0.05Xuaa-=4.H0m=m0(=15.1)ÛH1mm0(15.1)a{1.6432ua/2}a=0.056|14.9515.1|||1.64320.05z-=»(1)a=0.051.64z0=1.96(p0.05)(2)a=0.151.64z0=1.43(p0.15)aH0a.Neyman–Pearson1.XLX{FqqQ}Q0QQ1Q0QH0qQ0ÛH1qQ1H0H1nullhypothesisalternativehypothesisRemark“”2.()L0()L1()3.“”qQ0Q1“”qQ1Q0N-PNeyman(1934)4.bf(q)bf(q)=Pq{H0}qQ==bf(q)Q0Q1bf(q)qQ00qQ10qQ01-bf(q)qQ1a0“”aNeymanPearson3.1.21077.pH0p=0.5X~B(10p)p=0.5P{X7}P{X7}=P{X=7}+…+P{X=10}=0.1172+0.0439+0.0098+0.0010=0.171917%()“8”0.0547Pp=0.7{X6}0.73.1.3X1…XnX~N(ms02)s02H0mm0ÛH1mm03.1.1CH0Q=R1XnCXC0()()P{}1()fmmbmFs-==-F(x)N(01)bf(m)=1-F(————)mQH0mm0H1mm0m0Q0mm0Q1mm0n1/2(C-m)s01.bf(m)mmQ0(mm0)bf(m)m=m0abf(m0)=aC=m0+——afH0uas0n1/2uXn00asm+nnuu0000()()()1()()faammmmbmFFss--=-+=-2.m0(1).mQ1(mm0)bf(m)mm0mQ1(mm0)m01-anu00()()()fammbmFs-=-m0(2).auabf(m)auabf(m)anu00()()()fammbmFs-=-bmm0(mQ1)bf(m)1-bmm0bf(m)1-bmm1(m1m0)m0m1bf(m)mbf(m1)1-bmm1bf(m)1-bn——————am1bnu1010()()()1fammbmFbs-=-³-s02(ua+ub)2(m1-m0)23.1.4X1…XnX~N(m2.62)H0m12ÛH1m=13(1)an=100p1p2(2)np20.01.afH02.612uXna´+n=100a=0.0512.4264.X00()10(12)()()(1.64);2.6nufammmbmFFs--=-=-(1)aff11210(12)(1.64)|(1.64)0.05;2.6pmmFF£-=-£-=f210(1312)1(1.64)1(2.206)0.01369.2.6pFF-=--=-=(2)0.012(1312)1(1.64)0.01,2.6npF-=--£(1.64)0.99,2.6nF-³1.642.3263,2.6n-³106.6n³n107.100(106)50.10.002(z=2)a3.21.2.3..1.1a.H0m=m0ÛH1mm01.1b.H0mm0ÛH1mm01.1c.H0mm0ÛH1mm0X1…XnX~N(ms02)1.1s02(u)1.1.1a.H0m=m0ÛH1mm0aH0m=m0nXN0()~(0,1)ms-nXu0/20||P{}amas-=nXu0/20||ams-1.1b.H0mm0ÛH1mm03.1.3anXu00()ams-1.1c.H0mm0ÛH1mm0anXu00()ams--p-(p-value)p-p-p-p-p-3.2.1X~N(m10.12)10059.22H0m=60ÛH1m60.az=———————-uaz=-0.77u0.05=1.64u0.20=0.84u0.25=0.680.20.2510(59.22-60)10.1EXCELSPSSzp-p=P{z-0.77}=0.22065p-aRemarkH1m60|z|ua/20.40.50p-p=P{|z|0.77}=0.44131.2s2(t)1.3.1nXtnS()~(1)m--1.2a.H0m=m0ÛH1mm01.2b.H0mm0ÛH1mm01.2c.H0mm0ÛH1mm0X1…XnX~N(ms02)1.2a.H0m=m0ÛH1mm0anXtnS0/2||(1)am--1.2b.H0mm0ÛH1mm0a1.2c.H0mm0ÛH1mm0anXtnS0()(1)am--nXtnS0()(1)am---1.3s12s22(u)X1…Xn1Y1…Yn2X~N(m1s12)Y~N(m2s22)1.3a.H0m1-m2=dÛH1m1-m2d1.3b.H0m1-m2dÛH1m1-m2d1.3c.H0m1-m2dÛH1m1-m2dXNYNnn22121212~(,),~(,)ssmmXYNnn12221212()()~(0,1)mmss---+H0m1-m2=dz~N(01)XYznn221212()dss--=+1.3a.H0m1-m2=dÛH1m1-m2da1.3b.H0m1-m2dÛH1m1-m2da1.3c.H0m1-m2dÛH1m1-m2daXYunn/2221212||adss--+XYunn221212adss--+XYunn221212adss---+1.4s12s22(t)1.3.2s12s22Behrens-FisherH0m1-m2=dWnSnSSnn222112212(1)(1)2-+-=+-WXYtnnSnn121212()()~(2)11mm---+-+WXYtnnSnn1212~(2)11d--+-+1.4a.H0m1-m2=dÛH1m1-m2da1.4b.H0m1-m2dÛH1m1-m2da1.4c.H0m1-m2dÛH1m1-m2daWXYtnnSnn/21212||(2)11ad--+-+WXYtnnSnn1212(2)11ad--+-+WXYtnnSnn1212(2)11ad---+-+2.1m0(c2)2.1a.H0s2=s02ÛH1s2s022.1b.H0s2s02ÛH1s2s022.1c.H0s2s02ÛH1s2s02nkknXXNorn220021()()~(0,1)~()?mmcss=--å2.2.1a.H0s2=s02ÛH1s2s02aH0s2=s02nkkXn220210()~()mcs=-ånnkkkkXXnorn222200/21/2221100()()()()aammccss-==--åå2.1b.H0s2s02ÛH1s2s02ankkXn220210()()amcs=-å2.1c.H0s2s02ÛH1s2s02ankkXn2201210()()amcs-=-å2.2m(c2)1.3.12.2a.H0s2=s02ÛH1s2s022.2b.H0s2s02ÛH1s2s022.2c.H0s2s02ÛH1s2s02nkkXXn2221()~(1)cs=--å2.2a.H0s2=s02ÛH1s2s02aH0s2=s02nnkkkkXXXXnorn2222/21/2221100()()(1)(1)aaccss-==----åånkkXXn22210()~(1)cs=--å2.2b.H0s2s02ÛH1s2s02ankkXXn22210()(1)acs=--å2.2c.H0s2s02ÛH1s2s02ankkXXn221210()(1)acs-=--å2.3m1m2(F)X1…Xn1Y1…Yn2X~N(m1s12)Y~N(m2s22)2.3a.H0s12=s22ÛH1s12s222.3b.H0s12s22ÛH1s12s222.3c.H0s12s22ÛH1s12s22S12S22s12s222.3a.H0s12=s22ÛH1s12s22S12/S222.3b.H0s12s22ÛH1s12s22S12/S222.3c.H0s12s22ÛH1s12s22S12/S221.3.2————~F(n1-1,n2-1)S12/S22s12/s222.3a.H0s12=s22ÛH1s12s22a2.3b.H0s12s22ÛH1s12s22a2.3c.H0s12s22ÛH1s12s22aS12/S22Fa(n1-1,n2-1)S12/S22F1–a(n1-1,n2-1)S12/S22Fa/2(n1-1,n2-1)S12/S22F1–a/2(n1-1,n2-1)3.2.253.253.273.243.263.240.013.25.H0m=3.25ÛH1m3.25tXtS0.01/25|3.25|(4)-3.2520.01304t0.005t0.005(4)=4.60410.3434.6041t0.25(4)=0.74070.5p-p=0.748870.053.2.3·(MarkTwain)(Snodgrass)8103MarkTwainSnodgrass0.2250.2620.2170.2090.2050.1960.2100.2400.2300.2290.2020.2070.2240.2230.2350.2170.2200.201.2.3a.FH0s12=s22ÛH1s12s22F=S12/S22=——————=2.2725FF0.025(79)FF0.975(79)2.1212510-49.334410-5F0.025(79)=4.20F0.975(79)FF1–a(mn)=1/Fa(nm)F0.975(79)=1/F0.025(97)=1/4.82=0.2075F(=2.2725)4.200.2075p-p=0.1253.2.4·30.05.1.4a.t(d=0)H0m1=m2ÛH1m1m2|T|t0.025(16)=2.1199WXYTS11810-=+4.2.3T|T|=3.8679t0.025(16)=2.1199wnSnSSnn224112212(1)(1)1.4531100.01212--+-==´=+-xy0.2319,0.2097==t0.005(16)=2.92080.01p-p=0.0014Remark·“”tA.B.A()B03.3.2.510()1.90.81.10.10.14.45.51.64.63.40.7–1.6–0.2–1.2–0.13.43.70.80.02.0.(-)1.22.41.31.30.211.80.84.61.4X~N(ms2)1.2a.H0m=m0ÛH1mm0t1.61.2028t0.025(9)=2.2622|T|=4.20662.26220.05t0.005(9)=3.24980.01XTtS0.02510||||(9)=p-p=0.00228Remark10103.2.41.4aH0m1=m2ÛH1m1m2T|T|=1.8981t0.025(18)=2.10090.05p-p=0.0738