7-3 區(qū)間估計(jì)

1、第6.3節(jié) 參數(shù)的區(qū)間估計(jì)一. 區(qū)間估計(jì)的根本概念二. 正態(tài)總體均值與方差的區(qū)間估計(jì)三. 小結(jié) 引言 前面，我們討論了參數(shù)的點(diǎn)估計(jì), 它是用樣本算得的一個(gè)估計(jì)值去估計(jì)未知參數(shù). 但是，點(diǎn)估計(jì)值僅僅是未知參數(shù)的一個(gè)近似值，它沒(méi)有反映出這個(gè)近似值的誤差范圍，這是點(diǎn)估計(jì)的一個(gè)缺陷。下面我們將介紹的區(qū)間估計(jì)正好彌補(bǔ)了點(diǎn)估計(jì)的這個(gè)缺陷 .一、區(qū)間估計(jì)的根本概念1. 置信區(qū)間的定義關(guān)于定義的說(shuō)明 一旦有了樣本，就把 估計(jì)在區(qū)間內(nèi).由定義可見(jiàn)， 對(duì)參數(shù) 作區(qū)間估計(jì)，就是要設(shè)法找出兩個(gè)只依賴(lài)于樣本的界限(X1,Xn)(X1,Xn)2. 求置信區(qū)間的一般步驟(共3步)二、正態(tài)總體均值與方差的區(qū)間估計(jì)1.I. 單

2、個(gè)正態(tài)總體均值與方差的區(qū)間估計(jì) 推導(dǎo)過(guò)程如下:這樣的置信區(qū)間常寫(xiě)成其置信區(qū)間的長(zhǎng)度為 包糖機(jī)某日開(kāi)工包了12包糖,稱(chēng)得重量(單位:克)分別為506,500,495,488,504,486,505,513,521,520,512,485. 假設(shè)重量服從正態(tài)分布,解附表2-1例1附表2-2查表得 從以上解題過(guò)程還可以看出，求未知參數(shù)的區(qū)間 估計(jì)最關(guān) 鍵一步是，選擇適宜的函數(shù)并求出它的分布；其次是對(duì)給定的 實(shí)數(shù) ，查分位數(shù)表求出分位點(diǎn)，通過(guò)不等式變形得到參數(shù)的區(qū)間估計(jì) 。我們從以上步驟看出區(qū)間估計(jì)中，選擇函數(shù)是關(guān)鍵，給定 查數(shù)表，不等式變形即得置信區(qū)間。推導(dǎo)過(guò)程如下:解 有一大批糖果,現(xiàn)從中隨機(jī)地取

3、16袋, 稱(chēng)得重量(克)如下: 設(shè)袋裝糖果的重量服從正態(tài)分布, 試求總體均值附表3-1例2就是說(shuō)估計(jì)袋裝糖果重量的均值在500.4克與507.1克之間, 這個(gè)估計(jì)的可信程度為95%.這個(gè)誤差的可信度為95%.解附表3-2例3(續(xù)例1)如果只假設(shè)糖包的重量服從正態(tài)分布解例4推導(dǎo)過(guò)程如下:根據(jù)前面章節(jié)的定理知II.進(jìn)一步可得:注意: 在密度函數(shù)不對(duì)稱(chēng)時(shí), 習(xí)慣上仍取對(duì)稱(chēng)的分位點(diǎn)來(lái)確定置信區(qū)間(如圖). (續(xù)例2) 求例2中總體標(biāo)準(zhǔn)差 的置信度為0.95的置信區(qū)間.解代入公式得標(biāo)準(zhǔn)差的置信區(qū)間附表4-1附表4-2例52、兩個(gè)正態(tài)總體均值與方差的區(qū)間估計(jì) 討論兩個(gè)總體均值差和方差比的估計(jì)問(wèn)題.推導(dǎo)過(guò)程

4、如下:I.例6機(jī)床廠某日從兩臺(tái)機(jī)床加工的零件中,分別抽取 假設(shè)干個(gè)樣品,測(cè)得零件尺寸分別如下(單位:cm): 第一臺(tái)機(jī)器 6.2, 5.7, 6.5, 6.0, 6.3, 5.8 5.7, 6.0, 6.0, 5.8, 6.0 第二臺(tái)機(jī)器 5.6, 5.9, 5.6, 5.7, 5.8 6.0, 5.5, 5.7, 5.5 假設(shè)兩臺(tái)機(jī)器加工的零件尺寸均服從正態(tài)分布,且方差相等,試求兩機(jī)床加工的零件平均尺寸之差的區(qū)間估計(jì)解 用 X 表示第一臺(tái)機(jī)床加工的零件尺寸, 用 Y 表示第二臺(tái)機(jī)床加工的零件尺寸 由題設(shè)經(jīng)計(jì)算，得置信下限置信上限故所求 的置信度為95%的置信區(qū)間為 0.0912,0.5088

5、.推導(dǎo)過(guò)程如下:II. 根據(jù)F分布的定義, 知解例7研究由機(jī)器A和機(jī)器B生產(chǎn)的鋼管內(nèi)徑, 隨機(jī)抽取機(jī)器A生產(chǎn)的管子18只, 測(cè)得樣本方差為均未知, 求方差比區(qū)間.設(shè)兩樣本相互獨(dú)抽取機(jī)器B生產(chǎn)的管子13只,測(cè)得樣本方差為立,且設(shè)由機(jī)器A和機(jī)器B生產(chǎn)的鋼管內(nèi)徑分別服從正態(tài)分布信解例8的置甲、乙兩臺(tái)機(jī)床加工同一種零件, 在機(jī)床甲加工的零件中抽取9個(gè)樣品, 在機(jī)床乙加工的零件信區(qū)間. 假定測(cè)量值都服從正態(tài)分布, 方差分別為在置信度由所給數(shù)據(jù)算得0.98下, 試求這兩臺(tái)機(jī)床加工精度之比中抽取6個(gè)樣品,并分別測(cè)得它們的長(zhǎng)度(單位:mm),三、小結(jié) 點(diǎn)估計(jì)不能反映估計(jì)的誤差和精度, 因此，本節(jié)引入了區(qū)間估計(jì)

6、.求置信區(qū)間的一般步驟(分三步).正態(tài)總體均值與方差的區(qū)間估計(jì)當(dāng)n1,n2充分大時(shí)近似置信區(qū)間附表2-1標(biāo)準(zhǔn)正態(tài)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標(biāo)準(zhǔn)正態(tài)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