正态总体均值与方差区间估计

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第五节正态总体均值与方差的区间估计一、单个总体的情况二、两个总体的情况三、小结 一、单个总体的情况由上节例2可知:1. 包糖机某日开工包了12包糖,称得质量(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 推导过程如下: 解有一大批糖果,现从中随机地取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:根据第六章第二节定理二知2. 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5 解例6(续例1) 二、两个总体的情况讨论两个整体总体均值差和方差比的估计问题. 推导过程如下:1. 例7为比较І,ІІ两种型号步枪子弹的枪口速度,随机地取І型子弹10发,得到枪口速度的平均值为随机地取ІІ型子弹20发,得枪口速度平均值为假设两总体都可认为近似地服从正态分布,且由生产过程可认为它们的方差相等,求两总体均值差信区间.解由题意,两总体样本独立且方差相等(但未知), 解由题意,两总体样本独立且方差相等(但未知),例8为提高某一化学生产过程的得率,试图采用一种新的催化剂,为慎重起见,在试验工厂先进行体都可认为近似地服从正态分布,且方差相等,求两总体均值差试验.设采用原来的催化剂进行了次试验,得到得率的平均值又采用新的催化剂进行了次试验,得到得率的平均值假设两总 推导过程如下:2. 根据F分布的定义,知 解例9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10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,方差分别为的置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 附表2-1标准正态分布表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 z01234567891.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190.97780.98260.98640.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.99680.99770.99830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270.99450.99590.99690.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.99980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99980.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850.99990.95350.96250.97000.97620.98120.98540.98870.99130.99340.99510.99630.99730.99800.99860.99990.95450.96330.97060.97670.98170.98580.98900.99160.99360.99520.99640.99740.99810.99861.00001.96附表2-2标准正态分布表 附表3-1=0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.70624.30273.18242.77642.57062.44692.36462.30602.26222.22812.20102.17882.16042.14482.13152.119931.82076.96464.54073.74693.36493.14272.99802.89652.82142.76382.71812.68102.65032.62452.60252.583563.65749.92485.84094.60414.03223.70743.49953.35543.24983.16933.10583.05453.01232.97682.94672.9208分布表2.1315 =0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.70624.30273.18242.77642.57062.44692.36462.30602.26222.22812.20102.17882.16042.14482.13152.119931.82076.96464.54073.74693.36493.14272.99802.89652.82142.76382.71812.68102.65032.62452.60252.583563.65749.92485.84094.60414.03223.70743.49953.35543.24983.16933.10583.05453.01232.97682.94672.92082.2010附表3-2分布表 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=0.250.100.050.0250.010.005123456789101112131415161.3232.7734.1085.3856.6267.8419.03710.21911.38912.54913.70114.84515.98417.11718.24519.3692.7064.6056.2517.7799.23610.64512.01713.36214.68415.98717.27518.54919.81220.06422.30723.5423.8415.9917.8159.48811.07112.59214.06715.50716.91918.30719.67521.02622.36223.68524.99626.2965.0247.3789.34811.14312.83314.44916.01317.53519.02320.48321.92023.33724.73626.11927.48828.8456.6359.21011.34513.27715.08616.81218.47520.09021.66623.20924.72526.21727.68829.14130.57832.0007.87910.59712.83814.86016.75018.54820.27821.95523.58925.18826.75728.29929.89131.31932.80134.267分布表27.488