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1、3.2统计量与抽样分布统计量与抽样分布统计量的定义统计量的定义.),( ,),(,21212121计计量量是是一一个个统统则则称称不不含含未未知知参参数数中中若若的的函函数数是是的的一一个个样样本本是是来来自自总总体体设设nnnnXXXgfXXXXXXgXXXX?,),(,22321哪哪些些不不是是些些是是统统计计量量判判断断下下列列各各式式哪哪为为未未知知为为已已知知其其中中样样本本的的一一个个是是来来自自总总体体设设 NXXX,11XT ,3212XeXXT ),(313213XXXT ),max(3214XXXT ,2215 XXT).(123222126XXXT 是是不是不是例例1.)

2、,(),(,21212121的观察值是则称的样本值是相应于样本设nnnnXXXfxxxfXXXxxx2. 几个常用统计量几个常用统计量( (样本矩样本矩) )的定义的定义.,2121是是这这一一样样本本的的观观察察值值是是来来自自总总体体的的一一个个样样本本设设nnxxxXXX(1)样本均值样本均值;11 niiXnX(2)样本方差样本方差 niiXXnS122)(11它反映了总体均值它反映了总体均值的信息的信息它反映了总体方差它反映了总体方差的信息的信息(3)样本标准差样本标准差 ;11122 niiXXnSS(4) 样本样本 k 阶阶(原点原点)矩矩 nikikXn11 (5)样本样本 k

3、 阶中心矩阶中心矩 nikikXXn1)(1 2.2抽样分布抽样分布 统计量统计量 是是n n维随机变量维随机变量 的函数，作为一个随机变量，它也有自己的的函数，作为一个随机变量，它也有自己的概率分布。通常把统计量的分布称为概率分布。通常把统计量的分布称为抽样分抽样分布。布。 ),(21nXXXg),(21nXXX 设总体设总体X的分布函数已知，如果对任意容的分布函数已知，如果对任意容量为量为n的样本的样本 ，能求出给定的，能求出给定的统计量统计量 的分布函数，则称的分布函数，则称该分布函数为该分布函数为 的的精确分布。精确分布。 ),(21nXXX),(21nXXXg),(21nXXXg1

4、1样本均值的分布样本均值的分布 正态总体的几个统计量的精确分布2 2 分布分布 3 3t t分布分布 4 4F F分布分布 2 1.样本均值的分布样本均值的分布 设 ， 是来自总体X的样本，由于正态变量的线性函数仍为正态变量，故 ),(2NX),(21nXXX)1,(2nNX)1,0( NnXU ，通常称 为总体的标准误差总体的标准误差，记作 nX标准正态分布的上侧临界值标准正态分布的上侧临界值 标准正态分布的双侧临界值标准正态分布的双侧临界值 dteuUPtu2221)()|(|2uUPyxo图图 3-3 3-3 yxo图图 3-4 3-42 u2 u 2 2 u2u).(,)1, 0(),

5、(22222221221nnXXXNXXXnn 记记为为布布分分的的服服从从自自由由度度为为则则的的样样本本，是是来来自自总总体体设设 2 2 分布分布 2 000)2(21)(2122xxexnxfxnn)(222)()()(ndxxfnnP分布的上侧临界值 图 3-6oyx)(2n 2 )(2n定理定理 设 是来自正态总体 的样本，则 1o样本均值 与样本方差 相互独立； 2o ),(21nXXX),(2NXX2S)1()()1(221222 nXXSnnii 分布的性质分布的性质2 性质性质1).(,),(),(2122221222122221221nnnn 则则立立独独并且并且设设)(

6、2分布的可加性分布的可加性 (此性质可以推广到多个随机变量的情形此性质可以推广到多个随机变量的情形).(,), 2, 1(),(21212222mmiiiiinnnmin 则则独立独立相互相互并且并且设设).(,/,),(),1,0(2ntTtnnYXTYXnYNX记记为为分分布布的的服服从从自自由由度度为为则则称称随随机机变变量量独独立立且且设设定定义义 t 分布又称分布又称学生氏学生氏(Student)分布分布.学生氏资料学生氏资料 tntnnnthn,1221)(212 分布的概率密度函数为分布的概率密度函数为)(nt分布分布t2.图图分布的概率密度曲线如分布的概率密度曲线如t.0对称的

7、对称的显然图形是关于显然图形是关于 t当当n充分大时充分大时, 其图其图形类似于标准正态形类似于标准正态变量概率密度的图变量概率密度的图形形.,21)(lim22tneth 因为因为,)1 , 0(分布分布分布近似于分布近似于足够大时足够大时所以当所以当Ntn.)1 , 0(,分分布布相相差差很很大大分分布布与与但但对对于于较较小小的的Ntn定理定理2 设设 是来自正态总体是来自正态总体 的样本，则统计量的样本，则统计量 ),(21nXXX),(2NX) 1(/ntnSX定理定理3 设设 和和 是来自是来自正态总体正态总体 和和 的样本，的样本，且两者相互独立，则统计量且两者相互独立，则统计量

8、 ),(21nXXX),(221nYYY),(211N),(222N)2(11)(212121 nntnnSYX 2)1()1(21222211 nnSnSnS 和和 分别为两分别为两个总体的个总体的样本方差样本方差 22S21S称称 为样本均值差的标准误，为样本均值差的标准误，记作记作 ， 即。即。 2111nnS 21XXS21XXS2111nnS分布的上侧临界值分布的上侧临界值t)()()()(ntdttfntntP)(nt10分布的双侧临界值分布的双侧临界值t)(2nt) )(| )(| (2ntntPyxo图图 3-8 3-8)(nt yxo图图 3-9 3-92 t2 t 服从服从

9、第一自由度第一自由度为为 ，第二自由度第二自由度为为 的的 分布，记作分布，记作 2122212212/,),(),(nYnXFYXnYnX 则称随机变量则称随机变量独立独立且且设设 3.F分布分布).,(21nnFF1n2nF定义定义分布的密度函数为分布的密度函数为F 000)1()(22112211xxxnnxCxfnnn22121211)()2()2()2(nnnnnnnC F分布的分布的上侧临界值上侧临界值是指满足下式的是指满足下式的 ),(21nnF ),(212121)() ),(),(nnFdxxfnnFnnFP定理定理设设 ， 是正态总体是正态总体 的样本的样本容量和样本方差；

10、容量和样本方差； ， 是正态总体是正态总体 的样本容量和样本方差，且两个样本相互独的样本容量和样本方差，且两个样本相互独立，则统计量立，则统计量1n21S),(211 N2n22S),(222 N)1,1(2122212221nnFSS附表附表2-12-1标准正态分布表标准正态分布表z0.000.010.020.030.040.050.060.070.080.090.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.50000.53980.57930.61790.65540.69150.72570.75800.78810.81590.8413

11、0.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.94740.51200.55170.59100.62930.66640.70190.73570.76730.79670.8238

12、0.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.73890.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.93940.95050.52390.56360.60260.64060.67720.71230.74540.77640.8051

13、0.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.93060.94300.95350.53590.57530.61410.65170.68790.72240.75490.7852

14、0.81330.83890.86210.88300.90150.91770.93190.94410.95451.645附表附表2-22-2标准正态分布表标准正态分布表z0.000.010.020.030.040.050.060.070.080.091.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190.97780.98260.986

15、40.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.99680.99770.99830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270.99450.99590.996

16、90.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.96980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99980.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850.99990.95350.962

17、50.97000.97620.98120.98540.98870.99130.99340.99510.99630.99730.99800.99860.99990.95450.96330.97060.97670.98170.98530.98900.99160.99360.99520.99640.99740.99810.99861.00001.96附表附表4-14-1= 0.250.100.050.0250.010.005123456789101112131415161.3232.7734.1085.3856.6267.8419.03710.21911.38912.54913.70114.8451

18、5.98417.11718.24519.3692.7064.6056.2517.7799.23610.64512.01713.36214.68415.98717.27518.54919.81220.06422.30723.5423.8415.9917.8159.48811.07112.59214.06715.50716.91918.30719.67521.02622.36223.68524.99626.2965.0247.3789.34811.14312.83314.44916.01317.53519.02320.48321.92023.33724.73626.11927.48828.8456

19、.6359.21011.34513.27715.08616.81218.47520.09021.66623.20924.72526.21727.68829.14130.57832.0007.87910.59712.83814.86016.75018.54820.27821.95523.58925.18826.75728.29929.89131.31932.80134.267n2 分布表分布表17.535=0.9950.990.9750.950.900.75123456789101112131415160.0100.0720.2070.4120.6760.9891.3441.7352.1562.

20、6033.0743.5654.0754.6015.1420.0200.1150.2970.5540.8721.2391.6462.0882.5583.0533.5714.1074.6605.2295.8120.0010.0510.2160.4840.8311.2371.6902.1802.7003.2473.8164.4045.0095.6296.2626.9080.0040.1030.3520.7111.1451.6352.1672.7333.3253.9404.5755.2265.8926.5717.2617.9620.0160.2110.5841.0641.6102.2042.8333.

21、4904.1684.8655.5786.3047.0427.7908.5479.3120.1020.5751.2131.9232.6753.4554.2555.0715.8996.7377.5848.4389.29910.16511.03711.912n3.247附表附表4-4-2 22 分布表分布表=0.250.100.050.0250.010.0051718192021222324252627282930313220.48921.60522.71823.82824.93526.03927.14128.24129.33930.43531.52832.62033.71134.80035.887

22、36.97324.76925.98927.20428.41229.61530.81332.00733.19634.38235.56336.74137.91639.08740.25641.42242.58527.58728.86930.14431.41032.67133.92435.17236.41537.65238.88540.11341.33742.55743.77344.98546.19430.19131.52632.85234.17035.47936.78138.07639.36440.64641.92343.19444.46145.71246.97948.23249.48033.409

23、34.80536.19137.56638.93240.28941.63842.98044.31445.64246.96348.27849.58850.89252.19153.48635.71837.15638.58239.99741.40142.79644.18145.55946.92848.29049.64550.99352.33653.67255.00356.328n34.382附表附表4-4-3 32 分布表分布表附表附表3-13-1 =0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.7

24、2670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.7062 4.3027 3.1824 2.7764 2.5706 2.4

25、469 2.3646 2.3060 2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.119931.8207 6.9646 4.5407 3.7469 3.3649 3.1427 2.9980 2.8965 2.8214 2.7638 2.7181 2.6810 2.6503 2.6245 2.6025 2.583563.6574 9.9248 5.8409 4.6041 4.0322 3.7074 3.4995 3.3554 3.2498 3.1693 3.1058 3.0545 3.0123 2.9768 2.9467 2.9208nt分

26、布表分布表1.8125附表附表3-23-2 =0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.9432

27、1.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.3060 2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.119931.8207 6.9646 4.5407 3.7469 3.3649 3.1427 2.9980 2.8965 2.8214 2.7638 2.7181 2.6810 2.6503 2.6245 2.6025 2.583563.6574 9.9248 5

28、.8409 4.6041 4.0322 3.7074 3.4995 3.3554 3.2498 3.1693 3.1058 3.0545 3.0123 2.9768 2.9467 2.9208n2.1315t分布表分布表附表附表5-15-1F分布表分布表025. 0 1234567891012152024304012012345678910111213141516171819647.838.5117.4412.2210.01 8.81 8.07 7.57 7.21 6.94 6.72 6.55 6.41 6.30 6.20 6.12 6.04 5.95 5.92799.539.0016.041

29、0.65 8.43 7.26 6.54 6.06 5.71 5.46 5.26 5.10 4.97 4.86 4.77 4.69 4.62 4.56 4.51864.239.1715.44 9.98 7.76 6.60 5.89 5.42 5.08 4.83 4.63 4.47 4.35 4.24 4.15 4.08 4.01 3.95 3.90899.639.2515.10 9.60 7.39 6.23 5.52 5.50 4.72 4.47 4.28 4.12 4.00 3.89 3.80 3.73 3.66 3.61 3.56921.839.3014.88 9.36 7.15 5.99

30、5.29 4.82 4.48 4.24 4.04 3.89 3.77 3.66 3.58 3.50 3.44 3.38 3.33937.139.3314.73 9.20 6.98 5.82 5.12 4.65 4.23 4.07 3.88 3.73 3.60 3.50 3.41 3.34 3.28 3.22 3.17948.239.3614.62 9.07 6.85 5.70 4.99 4.53 4.20 3.95 3.76 3.61 3.48 3.38 3.29 3.22 3.16 3.10 3.05956.739.3714.54 8.98 6.76 5.60 4.90 4.43 4.10

31、3.85 3.66 3.51 3.39 3.29 3.20 3.12 3.06 3.01 2.96963.339.3914.47 8.90 6.68 5.52 4.82 4.36 4.03 3.78 3.59 3.44 3.31 3.21 3.12 3.05 2.98 2.93 2.88968.639.4014.42 8.84 6.62 5.46 4.76 4.30 3.96 3.72 3.53 3.37 3.25 3.15 3.06 2.99 2.92 2.87 2.82976.739.4114.34 8.75 6.52 5.37 4.67 4.20 3.87 3.62 3.43 3.28

32、3.15 3.05 2.96 2.89 2.82 2.77 2.72984.939.4314.25 8.66 6.43 5.27 4.57 4.10 3.77 3.52 3.33 3.18 3.05 2.95 2.86 2.79 2.72 2.67 2.62993.139.4514.17 8.56 6.33 5.17 4.47 4.00 3.67 3.42 3.23 3.07 2.95 2.84 2.76 2.68 2.62 2.56 2.51997.239.4614.12 8.51 6.28 5.12 4.42 3.59 3.61 3.37 3.17 3.02 2.89 2.79 2.70

33、2.63 2.56 2.50 2.45100139.4614.08 8.46 6.23 5.07 4.36 3.89 3.56 3.31 3.12 2.96 2.84 2.73 2.64 2.57 2.50 2.44 2.39100639.4714.04 8.41 6.18 5.01 4.31 3.84 3.51 3.26 3.06 2.91 2.78 2.67 2.59 2.51 2.44 2.38 2.33101439.4913.95 8.31 6.07 4.90 4.20 3.73 3.39 3.14 2.94 2.79 2.66 2.55 2.46 2.38 2.32 2.26 2.2

34、0101839.5013.90 8.26 6.02 4.85 4.14 3.67 3.33 3.08 2.88 2.72 2.60 2.49 2.40 2.32 2.25 2.19 2.132n1n4.90 1234567891015202430406012012345678910111213141516171819161.418.5110.13 7.71 6.61 5.99 5.59 5.32 3.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38199.519.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26

35、4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 5.55 3.52215.719.16 9.28 6.59 5.41 4.76 4.35 4.07 3.81 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13224.619.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90230.219.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74234.019.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63236.819.35 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54