區(qū)間估計;75正態(tài)總體均值與方差的區(qū)間估計課件

2022/12/241上次課復習兩種求點估計的方法:矩估計法最大似然估計法在統(tǒng)計問題中往往先使用最大似然估計法,在最大似然估計法使用不方便時,再用矩估計法.2022/12/191上次課復習兩種求點估計的方法:矩估計法2022/12/242估計量的評選的三個標準無偏性有效性相合性相合性是對估計量的一個基本要求,不具備相合性的估計量是不予以考慮的.由最大似然估計法得到的估計量,在一定條件下也具有相合性.估計量的相合性只有當樣本容量相當大時,才能顯示出優(yōu)越性,這在實際中往往難以做到,因此,在工程中往往使用無偏性和有效性這兩個標準.2022/12/192估計量的評選的三個標準無偏性有效性相合2022/12/243第四節(jié)區(qū)間估計一、區(qū)間估計的基本概念二、典型例題2022/12/193第四節(jié)區(qū)間估計一、區(qū)間估計的基本2022/12/244一、區(qū)間估計的基本概念1.

置信區(qū)間的定義2022/12/194一、區(qū)間估計的基本概念1.置信區(qū)間的2022/12/245關于定義的說明2022/12/195關于定義的說明2022/12/246若反復抽樣多次(各次得到的樣本容量相等,都是n)按伯努利大數定理,在這樣多的區(qū)間中,2022/12/196若反復抽樣多次(各次得到的樣本容量相等2022/12/247例如2022/12/197例如2022/12/2482.

求置信區(qū)間的一般步驟(共3步)2022/12/1982.求置信區(qū)間的一般步驟(共3步)2022/12/2492022/12/1992022/12/2410解例1二、典型例題2022/12/1910解例1二、典型例題2022/12/24112022/12/19112022/12/2412這樣的置信區(qū)間常寫成其置信區(qū)間的長度為2022/12/1912這樣的置信區(qū)間常寫成其置信區(qū)間的長度2022/12/2413由一個樣本值算得樣本均值的觀察值則置信區(qū)間為2022/12/1913由一個樣本值算得樣本均值的觀察值則置2022/12/2414其置信區(qū)間的長度為2022/12/1914其置信區(qū)間的長度為2022/12/2415比較兩個置信區(qū)間的長度置信區(qū)間短表示估計的精度高.說明:

對于概率密度的圖形是單峰且關于縱坐標軸對稱的情況,易證取a和b關于原點對稱時,能使置信區(qū)間長度最小.2022/12/1915比較兩個置信區(qū)間的長度置信區(qū)間短表示2022/12/2416今抽9件測量其長度,得數據如下(單位:mm):142,138,150,165,156,148,132,135,160.解例22022/12/1916今抽9件測量其長度,得數據如下(單2022/12/2417第五節(jié)正態(tài)總體均值與方差的區(qū)間估計一、單個總體的情況二、兩個總體的情況2022/12/1917第五節(jié)正態(tài)總體均值與方差的區(qū)間估2022/12/2418一、單個總體的情況由上節(jié)例1可知:1.2022/12/1918一、單個總體的情況由上節(jié)2022/12/2419

包糖機某日開工包了12包糖,稱得質量(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(tài)分布,解附表2-1例12022/12/1919包糖機某日開工包了12包糖2022/12/2420附表2-2查表得2022/12/1920附表2-2查表得2022/12/2421推導過程如下:2022/12/1921推導過程如下:2022/12/24222022/12/19222022/12/2423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22022/12/1923解有一大批糖果,現(xiàn)2022/12/2424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%.2022/12/1924就是說估計袋裝糖果重量的均值在5002022/12/2425解附表3-2例3(續(xù)例1)如果只假設糖包的重量服從正態(tài)分布2022/12/1925解附表3-2例3(續(xù)例1)如果只假設2022/12/2426推導過程如下:根據第六章第二節(jié)定理二知2.2022/12/1926推導過程如下:根據第六章第二節(jié)定理二2022/12/24272022/12/19272022/12/2428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(qū)間(如圖).2022/12/1928進一步可得:注意:在密度函數不對稱2022/12/2429(續(xù)例2)求例2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)間附表4-1附表4-2例42022/12/1929(續(xù)例2)求例2中2022/12/2430解例5

(續(xù)例1)2022/12/1930解例5(續(xù)例1)2022/12/2431二、兩個總體的情況討論兩個整體總體均值差和方差比的估計問題.2022/12/1931二、兩個總體2022/12/2432推導過程如下:1.2022/12/1932推導過程如下:1.2022/12/24332022/12/19332022/12/24342022/12/19342022/12/2435例6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兩總體都可認為近似地服從正態(tài)分布,且由生產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),2022/12/1935例6為比較?,??兩種型號步槍子彈2022/12/24362022/12/19362022/12/2437解由題意,兩總體樣本獨立且方差相等(但未知),例7為提高某一化學生產過程的得率,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,求兩總體均值差試驗.設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兩總2022/12/1937解由題意,兩總體樣本獨立且方差相等2022/12/24382022/12/19382022/12/2439推導過程如下:2.2022/12/1939推導過程如下:2.2022/12/2440根據F分布的定義,知2022/12/1940根據F分布的定義,知2022/12/24412022/12/19412022/12/2442解例8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(qū)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(tài)分布信2022/12/1942解例8研究由機器A和機器B生2022/12/24432022/12/19432022/12/2444解例9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,方差分別為的置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),2022/12/1944解例9甲、乙兩臺機床加工同一種零件,2022/12/24452022/12/19452022/12/2446作業(yè)：書面：P175:16,17,21,23.熟練掌握尋求置信區(qū)間的方法;正態(tài)總體均值與方差的區(qū)間估計。

2022/12/1946作業(yè)：書面：P175:16,17,2022/12/2447附表2-1標準正態(tài)分布表z01234567890.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.50000.53980.57930.61790.65540.69150.72570.75800.78810.81590.84130.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.94740.51200.55170.59100.62930.66640.70190.73570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.73890.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.93940.95050.52390.56360.60260.64060.67720.71230.74540.77640.80510.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.93060.94300.95350.53590.57530.61410.65170.68790.72240.75490.78520.81330.83890.86210.88300.90150.91770.93190.94410.95451.6452022/12/1947附表2-1標準正態(tài)分布表z012342022/12/2448z01234567891.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190.97780.98260.98640.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.99680.99770.99830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270.99450.99590.99690.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.99980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99980.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850.99990.95350.96250.97000.97620.98120.98540.98870.99130.99340.99510.99630.99730.99800.99860.99990.95450.96330.97060.97670.98170.98580.98900.99160.99360.99520.99640.99740.99810.99861.00001.96附表2-2標準正態(tài)分布表2022/12/1948z01234567891.60.942022/12/2449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2.13152022/12/1949附表3-1=0.250.100.02022/12/2450=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2022/12/1950=0.250.100.050.022022/12/2451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6.2622022/12/1951附表4-2=0.9950.990.92022/12/2452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27.4882022/12/1952附表4-1=0.250.100.052022/12/2453上次課復習兩種求點估計的方法:矩估計法最大似然估計法在統(tǒng)計問題中往往先使用最大似然估計法,在最大似然估計法使用不方便時,再用矩估計法.2022/12/191上次課復習兩種求點估計的方法:矩估計法2022/12/2454估計量的評選的三個標準無偏性有效性相合性相合性是對估計量的一個基本要求,不具備相合性的估計量是不予以考慮的.由最大似然估計法得到的估計量,在一定條件下也具有相合性.估計量的相合性只有當樣本容量相當大時,才能顯示出優(yōu)越性,這在實際中往往難以做到,因此,在工程中往往使用無偏性和有效性這兩個標準.2022/12/192估計量的評選的三個標準無偏性有效性相合2022/12/2455第四節(jié)區(qū)間估計一、區(qū)間估計的基本概念二、典型例題2022/12/193第四節(jié)區(qū)間估計一、區(qū)間估計的基本2022/12/2456一、區(qū)間估計的基本概念1.

置信區(qū)間的定義2022/12/194一、區(qū)間估計的基本概念1.置信區(qū)間的2022/12/2457關于定義的說明2022/12/195關于定義的說明2022/12/2458若反復抽樣多次(各次得到的樣本容量相等,都是n)按伯努利大數定理,在這樣多的區(qū)間中,2022/12/196若反復抽樣多次(各次得到的樣本容量相等2022/12/2459例如2022/12/197例如2022/12/24602.

求置信區(qū)間的一般步驟(共3步)2022/12/1982.求置信區(qū)間的一般步驟(共3步)2022/12/24612022/12/1992022/12/2462解例1二、典型例題2022/12/1910解例1二、典型例題2022/12/24632022/12/19112022/12/2464這樣的置信區(qū)間常寫成其置信區(qū)間的長度為2022/12/1912這樣的置信區(qū)間常寫成其置信區(qū)間的長度2022/12/2465由一個樣本值算得樣本均值的觀察值則置信區(qū)間為2022/12/1913由一個樣本值算得樣本均值的觀察值則置2022/12/2466其置信區(qū)間的長度為2022/12/1914其置信區(qū)間的長度為2022/12/2467比較兩個置信區(qū)間的長度置信區(qū)間短表示估計的精度高.說明:

對于概率密度的圖形是單峰且關于縱坐標軸對稱的情況,易證取a和b關于原點對稱時,能使置信區(qū)間長度最小.2022/12/1915比較兩個置信區(qū)間的長度置信區(qū)間短表示2022/12/2468今抽9件測量其長度,得數據如下(單位:mm):142,138,150,165,156,148,132,135,160.解例22022/12/1916今抽9件測量其長度,得數據如下(單2022/12/2469第五節(jié)正態(tài)總體均值與方差的區(qū)間估計一、單個總體的情況二、兩個總體的情況2022/12/1917第五節(jié)正態(tài)總體均值與方差的區(qū)間估2022/12/2470一、單個總體的情況由上節(jié)例1可知:1.2022/12/1918一、單個總體的情況由上節(jié)2022/12/2471

包糖機某日開工包了12包糖,稱得質量(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(tài)分布,解附表2-1例12022/12/1919包糖機某日開工包了12包糖2022/12/2472附表2-2查表得2022/12/1920附表2-2查表得2022/12/2473推導過程如下:2022/12/1921推導過程如下:2022/12/24742022/12/19222022/12/2475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22022/12/1923解有一大批糖果,現(xiàn)2022/12/2476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%.2022/12/1924就是說估計袋裝糖果重量的均值在5002022/12/2477解附表3-2例3(續(xù)例1)如果只假設糖包的重量服從正態(tài)分布2022/12/1925解附表3-2例3(續(xù)例1)如果只假設2022/12/2478推導過程如下:根據第六章第二節(jié)定理二知2.2022/12/1926推導過程如下:根據第六章第二節(jié)定理二2022/12/24792022/12/19272022/12/2480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(qū)間(如圖).2022/12/1928進一步可得:注意:在密度函數不對稱2022/12/2481(續(xù)例2)求例2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)間附表4-1附表4-2例42022/12/1929(續(xù)例2)求例2中2022/12/2482解例5

(續(xù)例1)2022/12/1930解例5(續(xù)例1)2022/12/2483二、兩個總體的情況討論兩個整體總體均值差和方差比的估計問題.2022/12/1931二、兩個總體2022/12/2484推導過程如下:1.2022/12/1932推導過程如下:1.2022/12/24852022/12/19332022/12/24862022/12/19342022/12/2487例6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兩總體都可認為近似地服從正態(tài)分布,且由生產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),2022/12/1935例6為比較?,??兩種型號步槍子彈2022/12/24882022/12/19362022/12/2489解由題意,兩總體樣本獨立且方差相等(但未知),例7為提高某一化學生產過程的得率,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,求兩總體均值差試驗.設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兩總2022/12/1937解由題意,兩總體樣本獨立且方差相等2022/12/24902022/12/19382022/12/2491推導過程如下:2.2022/12/1939推導過程如下:2.2022/12/2492根據F分布的定義,知2022/12/1940根據F分布的定義,知2022/12/24932022/12/19412022/12/2494解例8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(qū)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(tài)分布信2022/12/1942解例8研究由機器A和機器B生2022/12/24952022/12/19432022/12/2496解例9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,方差分別為的置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),2022/12/1944解例9甲、乙兩臺機床加工同一種零件,2022/12/24972022/12/19452022/12/2498作業(yè)：書面：P175:16,17,21,23.熟練掌握尋求置信區(qū)間的方法;正態(tài)總體均值與方差的區(qū)間估計。

2022/12/1946作業(yè)：書面：P175:16,17,2022/12/2499附表2-1標準正態(tài)分布表z01234567890.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.60.50000.53980.57930.61790.65540.69150.72570.75800.78810.81590.84130.86430.88490.90320.91920.93320.94520.50400.54380.58320.62170.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.50800.54780.58710.62550.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.94740.51200.55170.59100.62930.66640.70190.73570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.51600.55570.59480.63310.67000.70540.73890.77030.79950.82640.85080.87290.89250.90990.92510.93820.94950.51990.55960.59870.63680.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.93940.95050.52390.56360.60260.64060.67720.71230.74540.77640.80510.83150.85540.87700.89620.91310.92780.94060.95150.52790.56750.60640.64430.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.53190.57140.61030.64800.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.93060.94300.95350.53590.57530.61410.65170.68790.72240.75490.78520.81330.83890.86210.88300.90150.91770.93190.94410.95451.6452022/12/1947附表2-1標準正態(tài)分布表z012342022/12/24100z01234567891.61.71.81.92.02.12.22.32.42.52.62.72.82.93.00.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.99650.99740.99810.99870.94630.95640.96480.97190.97780.98260.98640.98960.99200.99400.99550.99660.99750.99820.99900.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.99670.99760.99820.99930.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.99680.99770.99830.99950.94950.95910.96710.97380.97930.98380.98710.99040.99270.99450.99590.99690.99770.99840.99970.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.99700.99780.99840.99980.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.99710.99790.99850.99980.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.99720.99790.99850.99990.95350.96250.97000.97620.98120.98540.98870.99130.99340.99510.99630.99730.99800.99860.99990.95450.96330.97060.97670.98170.98580.98900.99160.99360.99520.99640.99740.99810.99861.00001.96附表2-2標準正態(tài)分布表2022/12/1948z01234567891.60.942022/12/24101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2.13152022/12/1949附表3-1=0.250.100.02022/12/24102=0.250.100.050.0250.010.005123456789101112131415161.00000.81650.76490.74070.72670.71760.71110.70640.70270.69980.69740.69550.69380.69240.69120.69013.07771.88561.63771.53321.47591.43981.41491.39681.38301.37221.36341.35621.35021.34501.34061.33686.31382.92002.35342.13182.01501.94321.89461.85951.83311.81251.79591.78231.77091.76131.75311.745912.70624.30273.18242.77642.57062.44