1、第6.3節 參數的區間估計一. 區間估計的根本概念二. 正態總體均值與方差的區間估計三. 小結 引言 前面，我們討論了參數的點估計, 它是用樣本算得的一個估計值去估計未知參數. 但是，點估計值僅僅是未知參數的一個近似值，它沒有反映出這個近似值的誤差范圍，這是點估計的一個缺陷。下面我們將介紹的區間估計正好彌補了點估計的這個缺陷 .一、區間估計的根本概念1. 置信區間的定義關于定義的說明 一旦有了樣本，就把 估計在區間內.由定義可見， 對參數 作區間估計，就是要設法找出兩個只依賴于樣本的界限(X1,Xn)(X1,Xn)2. 求置信區間的一般步驟(共3步)二、正態總體均值與方差的區間估計1.I. 單

2、個正態總體均值與方差的區間估計 推導過程如下:這樣的置信區間常寫成其置信區間的長度為 包糖機某日開工包了12包糖,稱得重量(單位:克)分別為506,500,495,488,504,486,505,513,521,520,512,485. 假設重量服從正態分布,解附表2-1例1附表2-2查表得 從以上解題過程還可以看出，求未知參數的區間 估計最關 鍵一步是，選擇適宜的函數并求出它的分布；其次是對給定的 實數 ，查分位數表求出分位點，通過不等式變形得到參數的區間估計 。我們從以上步驟看出區間估計中，選擇函數是關鍵，給定 查數表，不等式變形即得置信區間。推導過程如下:解 有一大批糖果,現從中隨機地取

3、16袋, 稱得重量(克)如下: 設袋裝糖果的重量服從正態分布, 試求總體均值附表3-1例2就是說估計袋裝糖果重量的均值在500.4克與507.1克之間, 這個估計的可信程度為95%.這個誤差的可信度為95%.解附表3-2例3(續例1)如果只假設糖包的重量服從正態分布解例4推導過程如下:根據前面章節的定理知II.進一步可得:注意: 在密度函數不對稱時, 習慣上仍取對稱的分位點來確定置信區間(如圖). (續例2) 求例2中總體標準差 的置信度為0.95的置信區間.解代入公式得標準差的置信區間附表4-1附表4-2例52、兩個正態總體均值與方差的區間估計 討論兩個總體均值差和方差比的估計問題.推導過程

4、如下:I.例6機床廠某日從兩臺機床加工的零件中,分別抽取 假設干個樣品,測得零件尺寸分別如下(單位:cm): 第一臺機器 6.2, 5.7, 6.5, 6.0, 6.3, 5.8 5.7, 6.0, 6.0, 5.8, 6.0 第二臺機器 5.6, 5.9, 5.6, 5.7, 5.8 6.0, 5.5, 5.7, 5.5 假設兩臺機器加工的零件尺寸均服從正態分布,且方差相等,試求兩機床加工的零件平均尺寸之差的區間估計解 用 X 表示第一臺機床加工的零件尺寸, 用 Y 表示第二臺機床加工的零件尺寸 由題設經計算，得置信下限置信上限故所求 的置信度為95%的置信區間為 0.0912,0.5088

5、.推導過程如下:II. 根據F分布的定義, 知解例7研究由機器A和機器B生產的鋼管內徑, 隨機抽取機器A生產的管子18只, 測得樣本方差為均未知, 求方差比區間.設兩樣本相互獨抽取機器B生產的管子13只,測得樣本方差為立,且設由機器A和機器B生產的鋼管內徑分別服從正態分布信解例8的置甲、乙兩臺機床加工同一種零件, 在機床甲加工的零件中抽取9個樣品, 在機床乙加工的零件信區間. 假定測量值都服從正態分布, 方差分別為在置信度由所給數據算得0.98下, 試求這兩臺機床加工精度之比中抽取6個樣品,并分別測得它們的長度(單位:mm),三、小結 點估計不能反映估計的誤差和精度, 因此，本節引入了區間估計

6、.求置信區間的一般步驟(分三步).正態總體均值與方差的區間估計當n1,n2充分大時近似置信區間附表2-1標準正態分布表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標準正態分布表附表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