6-3參數(shù)的區(qū)間估計課件可編輯

第三節(jié)參數(shù)的區(qū)間估計一、區(qū)間估計的基本概念二、正態(tài)總體均值與方差的區(qū)間估計三、內(nèi)容小結(jié)第六章一、區(qū)間估計基本概念1.問題的提出點估計法：不足之處：例1問：很小較大區(qū)間估計解決了上述問題，從而克服了點估計的不足之處.2.置信區(qū)間與置信度定義6.7關(guān)于定義的說明因此定義中下述表達式的本質(zhì)是:若反復(fù)抽樣多次(各次得到的樣本容量相等,都是n)按貝努利大數(shù)定理,當抽樣次數(shù)充分大時，在這些區(qū)間中包含

真值的頻率接近置信度1

,即例如一旦有了樣本，就把估計在區(qū)間內(nèi).這里有兩個要求:由定義可見，對參數(shù)作區(qū)間估計，就是要設(shè)法找出兩個只依賴于樣本的界限(構(gòu)造統(tǒng)計量)(X1,…Xn)(X1,…Xn)2.估計的精度要盡可能的高.如要求區(qū)間長度盡可能短，或能體現(xiàn)該要求的其它準則.1.要求以很大的可能被包含在區(qū)間內(nèi)，就是說，概率要盡可能大.即要求估計盡量可靠.可靠度與精度是一對矛盾，一般是在保證可靠度的條件下盡可能提高精度.3.求置信區(qū)間的一般步驟(共3步)3°作等價變形二、正態(tài)總體均值與

方差的區(qū)間估計1.I.單個總體的情況4°作等價變形簡寫成其置信區(qū)間的長度為例2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(shè)重量服從正態(tài)分布,解查表得4°作等價變形簡寫成例3解有一大批糖果,現(xiàn)從中隨機地取16袋,稱得重量(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%.例4解(續(xù)例2)如果只假設(shè)糖包的重量服從正態(tài)分布推導(dǎo)過程如下:根據(jù)第五章第三節(jié)定理5.8知2.方差

2的置信區(qū)間進一步可得:注意:在密度函數(shù)不對稱時,習(xí)慣上仍取對稱的分位點來確定置信區(qū)間(如圖).例5

(續(xù)例3)求例3中總體標準差

的置信度為0.95的置信區(qū)間.解代入公式得標準差的置信區(qū)間Ⅱ.兩個總體的情況討論兩個總體均值差和方差比的估計問題.推導(dǎo)過程如下:1.為比較?,??兩種型號步槍子彈的槍口速度,隨機地取?型子彈10發(fā),得到槍口速度的平均值為隨機地取??型子彈20發(fā),得槍口速度平均值為假設(shè)兩總體都可認為近似地服從正態(tài)分布,且由生產(chǎn)過程可認為它們的方差相等,求兩總體均值差信區(qū)間.解由題意,兩總體樣本獨立且方差相等(但未知),例6解由題意,兩總體樣本獨立且方差相等(但未知),例7為提高某以化學(xué)生產(chǎn)過程的得率,試圖采用一種新的催化劑,為慎重起見,在試驗工廠先進行體都可認為近似地服從正態(tài)分布,且方差相等,求兩總體均值差信區(qū)間.試驗.設(shè)采用原來的催化劑進行了次試驗,得到得率的平均值又采用新的催化劑進行了次試驗,得到得率的平均值假設(shè)兩總說明:由于所得置信區(qū)間包含零,在實際中我們認為采用這兩種催化劑所得的得率的均值沒有顯著差別.推導(dǎo)過程如下:2.根據(jù)F分布的定義,知解例8研究由機器A和機器B生產(chǎn)的鋼管內(nèi)徑,隨機抽取機器A生產(chǎn)的管子18只,測得樣本方差為均未知,求方差比區(qū)間.設(shè)兩樣本相互獨抽取機器B生產(chǎn)的管子13只,測得樣本方差為立,且設(shè)由機器A和機器B生產(chǎn)的鋼管內(nèi)徑分別服從正態(tài)分布信解例9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(qū)間.假定測量值都服從正態(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(jié)點估計不能反映估計的精度,故而本節(jié)引入了區(qū)間估計.求置信區(qū)間的一般步驟(分三步).求置信區(qū)間的一般步驟(共3步)3°作等價變形正態(tài)總體均值與方差的區(qū)間估計附表2-1標準正態(tài)分布表z01234567890.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.50000.53980.57930.61790.65540.69150.72570.75800.78810.81590.84130.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.94740.51200.55170.59100.62930.66640.70190.73570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.73890.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.93940.95050.52390.56360.60260.64060.67720.71230.74540.77640.80510.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.93060.94300.95350.53590.57530.61410.65170.68790.72240.75490.78520.81330.83890.86210.88300.90150.91770.93190.94410.95451.645附表3-1=0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.70624.30273.18242.77642.57062.44692.36462.30602.26222.22812.20102.17882.16042.14482.13152.119931.82076.96464.54073.74693.36493.14272.99802.89652.82142.76382.71812.68102.65032.62452.60252.583563.65749.92485.84094.60414.03223.70743.49953.35543.24983.16933.10583.05453.01232.97682.94672.9208分布表2.1315附表4-2=0.9950.990.9750.950.900.75123456789101112131415160.0100.0720.2070.4120.6760.9891.3441.7352.1562.6033.0743.5654.0754.6015.1420.0200.1150.2970.5540.8721.2391.6462.0882.5583.0533.5714.1074.6605.2295.8120.0010.0510.2160.4840.8311.2371.6902.1802.7003.2473.8164.4045.0095.6296.2626.9080.0040.1030.3520.7111.1451.6352.1672.7333.3253.9404.5755.2265.8926.5717.2617.9620.0160.2110.5841.0641.6102.2042.8333.4904.1684.8655.5786.3047.0427.7908.5479.3120.1020.5751.2131.9232.6753.4554.2555.0715.8996.7377.5848.4389.29910.16511.03711.912分布表6.262附表4-1=0.250.100.050.0250.010.005123456789101112131415161.3232.7734.1085.3856.6267.8419.03710.21911.38912.54913.70114.84515.98417.11718.24519.3692.7064.6056.2517.7799.23610.64512.01713.36214.68415.98717.27518.54919.81220.06422.30723.5423.8415.9917.8159.48811.07112.59214.06715.50716.91918.30719.67521.02622.36223.68524.99626.2965.0247.3789.34811.14312.83314.44916.01317.53519.02320.48321.9202