7-3 区间估计

1、第6.3节 参数的区间估计一. 区间估计的根本概念二. 正态总体均值与方差的区间估计三. 小结 引言 前面，我们讨论了参数的点估计, 它是用样本算得的一个估计值去估计未知参数. 但是，点估计值仅仅是未知参数的一个近似值，它没有反映出这个近似值的误差范围，这是点估计的一个缺陷。下面我们将介绍的区间估计正好弥补了点估计的这个缺陷 .一、区间估计的根本概念1. 置信区间的定义关于定义的说明 一旦有了样本，就把 估计在区间内.由定义可见， 对参数 作区间估计，就是要设法找出两个只依赖于样本的界限(X1,Xn)(X1,Xn)2. 求置信区间的一般步骤(共3步)二、正态总体均值与方差的区间估计1.I. 单

2、个正态总体均值与方差的区间估计 推导过程如下:这样的置信区间常写成其置信区间的长度为 包糖机某日开工包了12包糖,称得重量(单位:克)分别为506,500,495,488,504,486,505,513,521,520,512,485. 假设重量服从正态分布,解附表2-1例1附表2-2查表得 从以上解题过程还可以看出，求未知参数的区间 估计最关 键一步是，选择适宜的函数并求出它的分布；其次是对给定的 实数 ，查分位数表求出分位点，通过不等式变形得到参数的区间估计 。我们从以上步骤看出区间估计中，选择函数是关键，给定 查数表，不等式变形即得置信区间。推导过程如下:解 有一大批糖果,现从中随机地取

3、16袋, 称得重量(克)如下: 设袋装糖果的重量服从正态分布, 试求总体均值附表3-1例2就是说估计袋装糖果重量的均值在500.4克与507.1克之间, 这个估计的可信程度为95%.这个误差的可信度为95%.解附表3-2例3(续例1)如果只假设糖包的重量服从正态分布解例4推导过程如下:根据前面章节的定理知II.进一步可得:注意: 在密度函数不对称时, 习惯上仍取对称的分位点来确定置信区间(如图). (续例2) 求例2中总体标准差 的置信度为0.95的置信区间.解代入公式得标准差的置信区间附表4-1附表4-2例52、两个正态总体均值与方差的区间估计 讨论两个总体均值差和方差比的估计问题.推导过程

4、如下:I.例6机床厂某日从两台机床加工的零件中,分别抽取 假设干个样品,测得零件尺寸分别如下(单位:cm): 第一台机器 6.2, 5.7, 6.5, 6.0, 6.3, 5.8 5.7, 6.0, 6.0, 5.8, 6.0 第二台机器 5.6, 5.9, 5.6, 5.7, 5.8 6.0, 5.5, 5.7, 5.5 假设两台机器加工的零件尺寸均服从正态分布,且方差相等,试求两机床加工的零件平均尺寸之差的区间估计解 用 X 表示第一台机床加工的零件尺寸, 用 Y 表示第二台机床加工的零件尺寸 由题设经计算，得置信下限置信上限故所求 的置信度为95%的置信区间为 0.0912,0.5088

5、.推导过程如下:II. 根据F分布的定义, 知解例7研究由机器A和机器B生产的钢管内径, 随机抽取机器A生产的管子18只, 测得样本方差为均未知, 求方差比区间.设两样本相互独抽取机器B生产的管子13只,测得样本方差为立,且设由机器A和机器B生产的钢管内径分别服从正态分布信解例8的置甲、乙两台机床加工同一种零件, 在机床甲加工的零件中抽取9个样品, 在机床乙加工的零件信区间. 假定测量值都服从正态分布, 方差分别为在置信度由所给数据算得0.98下, 试求这两台机床加工精度之比中抽取6个样品,并分别测得它们的长度(单位:mm),三、小结 点估计不能反映估计的误差和精度, 因此，本节引入了区间估计

6、.求置信区间的一般步骤(分三步).正态总体均值与方差的区间估计当n1,n2充分大时近似置信区间附表2-1标准正态分布表z0.000.010.020.030.040.050.060.070.080.090.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.50000.53980.57930.61790.65540.69150.72570.75800.78810.81590.84130.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.761

7、10.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.94740.51200.55170.59100.62930.66640.70190.73570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.738

8、90.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.93940.95050.52390.56360.60260.64060.67720.71230.74540.77640.80510.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.715

9、70.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.93060.94300.95350.53590.57530.61410.65170.68790.72240.75490.78520.81330.83890.86210.88300.90150.91770.93190.94410.95451.645z0.000.010.020.030.040.0

10、50.060.070.080.091.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190.97780.98260.98640.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99

11、410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.99680.99770.99830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270.99450.99590.99690.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.96

12、980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99980.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850.99990.95350.96250.97000.97620.98120.98540.98870.99130.99340.99510.99630.99730.99800.99860.99990.95450.96330.97060.97670.98

13、170.98530.98900.99160.99360.99520.99640.99740.99810.99861.00001.96附表2-2标准正态分布表附表3-1 =0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35

14、621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.7062 4.3027 3.1824 2.7764 2.5706 2.4469 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.

15、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 2.9208分布表2.1315 =0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69

16、240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.3060 2.2622 2.2281 2.2010 2.1788 2.1604 2.

17、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 2.92082.2021附表3-2分布表附表4-1=0.250.100.050.0250.010.005123456789101

18、112131415161.3232.7734.1085.3856.6267.8419.03710.21911.38912.54913.70114.84515.98417.11718.24519.3692.7064.6056.2517.7799.23610.64512.01713.36214.68415.98717.27518.54919.81220.06422.30723.5423.8415.9917.8159.48811.07112.59214.06715.50716.91918.30719.67521.02622.36223.68524.99626.2965.0247.3789.34811.14312.83314.44916.01317.53519.02320.48321.92023.33724.73626.11927.48828.8456.6359.21011.34513.27715.08616.81218.47520.09021.66623.20924.72526.21727.68829.14130.57832.0007.87910.59712.83814