6-3參數(shù)的區(qū)間估計(jì)課件可編輯

第三節(jié)參數(shù)的區(qū)間估計(jì)一、區(qū)間估計(jì)的基本概念二、正態(tài)總體均值與方差的區(qū)間估計(jì)三、內(nèi)容小結(jié)第六章一、區(qū)間估計(jì)基本概念1.問(wèn)題的提出點(diǎn)估計(jì)法：不足之處：例1問(wèn)：很小較大區(qū)間估計(jì)解決了上述問(wèn)題，從而克服了點(diǎn)估計(jì)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(shuō)明因此定義中下述表達(dá)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(dāng)抽樣次數(shù)充分大時(shí)，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(jì)在區(qū)間內(nèi).這里有兩個(gè)要求: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(構(gòu)造統(tǒng)計(jì)量)(X1,…Xn)(X1,…Xn)2.估計(jì)的精度要盡可能的高.如要求區(qū)間長(zhǎng)度盡可能短，或能體現(xiàn)該要求的其它準(zhǔn)則.1.要求以很大的可能被包含在區(qū)間內(nèi)，就是說(shuō)，概率要盡可能大.即要求估計(jì)盡量可靠.可靠度與精度是一對(duì)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(jià)變形二、正態(tài)總體均值與

方差的區(qū)間估計(jì)1.I.單個(gè)總體的情況4°作等價(jià)變形簡(jiǎn)寫(xiě)成其置信區(qū)間的長(zhǎng)度為例2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,解查表得4°作等價(jià)變形簡(jiǎn)寫(xiě)成例3解有一大批糖果,現(xiàn)從中隨機(jī)地取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%.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(guò)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(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).例5

(續(xù)例3)求例3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Ⅱ.兩個(gè)總體的情況討論兩個(gè)總體均值差和方差比的估計(jì)問(wèn)題.推導(dǎo)過(guò)程如下:1.為比較?,??兩種型號(hào)步槍子彈的槍口速度,隨機(jī)地取?型子彈10發(fā),得到槍口速度的平均值為隨機(jī)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(rèn)為近似地服從正態(tài)分布,且由生產(chǎn)過(guò)程可認(rèn)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(dú)立且方差相等(但未知),例6解由題意,兩總體樣本獨(dú)立且方差相等(但未知),例7為提高某以化學(xué)生產(chǎn)過(guò)程的得率,試圖采用一種新的催化劑,為慎重起見(jiàn),在試驗(yàn)工廠(chǎng)先進(jìn)行體都可認(rèn)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(yàn).設(shè)采用原來(lái)的催化劑進(jìn)行了次試驗(yàn),得到得率的平均值又采用新的催化劑進(jìn)行了次試驗(yàn),得到得率的平均值假設(shè)兩總說(shuō)明:由于所得置信區(qū)間包含零,在實(shí)際中我們認(rèn)為采用這兩種催化劑所得的得率的均值沒(méi)有顯著差別.推導(dǎo)過(guò)程如下:2.根據(jù)F分布的定義,知解例8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9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)的精度,故而本節(jié)引入了區(qū)間估計(jì).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(jià)變形正態(tài)總體均值與方差的區(qū)間估計(jì)附表2-1標(biāo)準(zhǔn)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