試驗(yàn)設(shè)計(jì)與數(shù)據(jù)處理方法解析09版資料

1、單因素方差分析: 用同種原料織成的布，用不同的染整工藝處理后進(jìn)行縮水率試驗(yàn)， 試用方差分析 考察不同工藝對(duì)縮水率是否有顯著影響。這里采用不同的染整工藝，每種工藝處 理四塊布，記A為染整工藝這一因子。它的五個(gè)水平記為 A,A,A,A,A試驗(yàn)結(jié)果如下：染整工藝布樣號(hào)AAAAA14.36.16.59.39.527.87.38.38.78.833.24.28.67.211.446.54.18.210.17.8解：n 5， m 4AAAsAA514.36.16.59.39.527.87.38.38.78.833.24.2:8.6：7.2M1.446.54.18.210.17.821.821.731.63

2、5.337.5147.921n mC 一Xjn m i 1 j 11093.72052475.24470.89998.561246.091406.254597.0321 nmR -Xjm i 1 j 11149.257518.4937.2142.2586.4990.2560.8453.29 :68.8975.6977.4410.2417.6473.9651.84129.9642.2516.8167.24102.0160.84131.82124.95252.34316.03358.49n mWx2i 1 j 11183.63Xjj=1093.7205n m i i j iXij j 1=1149

3、.2575n m2WXij =1183.63i 1 j 1則組內(nèi)偏差平方和：SSe W R =34.3725組間偏差平方和：SS R C =55.537總偏差平方和：SSp W C =89.9095方差來(lái)源偏差平方和自由 度方差估計(jì)值F值F0.05fA , fEF0.01fA , fE顯著性分組 因素55.537413.884256.0593.064.89*隨機(jī)誤差34.3725152.2915總和89.909519雙因素?zé)o重復(fù)試驗(yàn)設(shè)計(jì)方差分析:為了考察材質(zhì)和淬火溫度對(duì)某種鋼材淬火后的彎曲變形的影響，對(duì)四種不同材質(zhì)分別用五種不同的淬火溫度進(jìn)行試驗(yàn)，測(cè)得其淬火后試件的延伸率數(shù)據(jù)如下：材質(zhì)甲乙丙丁

4、8004.45.24.34.9820溫5.35.05.14.78405.85.54.84.9度8606.66.96.67.38808.48.38.57.9試對(duì)表中數(shù)據(jù)作方差分析來(lái)回答：不同材質(zhì)對(duì)延伸率有顯著影響嗎，不同溫度對(duì)延伸率有顯著影響嗎？解：a 4， b 5甲乙丙丁Tj28004.45.24.34.918.8353.448205.35.05.14.720.1404.018405.85.54.84.9214418606.66.96.67.327.4750.768808.48.38.57.933.11095.61T0Tbj 11 b_T02ja j 130.530.929.329.7120.

5、43044.82761.2052aT2T2Ti0930.25954.81858.49882.09i 114496.163625.641 a1 T2 匚li0 b i 1725.128T2a b724.80819.36I 27.0418.4924.0188.928.09r 25 d26.0122.09101.1933.6430.2523.0424.01110.9443.5647.6143.5653.29188.0270.5668.8972.2562.41274.11a b2Xji 1 j 1763.16a b)2T2SSpXij38.352i 1 j1ab1 a2T2SSaTi20.32b i

6、1abSSb1 bT0jT236.397a j 1abSS.SSpSSaSSb1.635方差來(lái)源偏差平方和自由度方差估計(jì)值F值F0.05 f , fEF0.01f , fE顯著性A因素0.3230.1066670.7828753.495.95不顯著B因素36.39749.0992566.783493.265.41*隨機(jī)誤差1.635120.13625總和38.35219雙因素有重復(fù)試驗(yàn)設(shè)計(jì)方差分析：試確定三種不同的材料和三種不同的使用環(huán)境對(duì)蓄電池輸出電壓的影響，為此，對(duì)每種水平組合重復(fù)測(cè)輸出電壓 4次，測(cè)得數(shù)據(jù)V 100列入下表，試分析各因素及因素之間交互作用的顯著性。溫度：C每行總計(jì)Tioo

7、1018271130， 155，34，40，20，70，973材74， 18080，5082，582150， 188，136，122，22，70，1297159， 126106， 11558，45料3138， 110，174，120，96， 104，1501168， 160150， 13982， 60每列總計(jì)Tojo173812667673771解：a 3，b 3，r 4a b rTXjk =3771i 1 j 1 k 1SSaT2 .11=4881939-14220441=11816abr12361 a 丄 T2 ,1 i00br i 11T2SSBar j 1To2oabr 1252116

8、89-3614220441=39295j 1 br iaT2i001arbTo2oj 1T2abr1829261-124881939- 5211689+丄 1422044仁111921236abr"Tj2o =475281-1 1829261 = 17966j 14方差來(lái)源偏差平方 和自由 度方差估計(jì) 值F值F0.05f , fEF0.01f , fE顯著性A因素11816259088.883.355.49*B因素39295219647.529.533.355.49*交互作用A B11192427984.202.734.11*誤差1796627665.4SSer i 12Xjki 1

9、 j 1 k 1三因素試驗(yàn)設(shè)計(jì)方差分析：某農(nóng)作物的B族維生素含量與 3個(gè)因子有關(guān)。因子 A :漂白與加工；因子 B :小、中、大3 種形狀；因子 C： Ri，R2, R33種處理方法。每種水平組合下作 n=10次試驗(yàn)，共180次試 驗(yàn)，試驗(yàn)結(jié)果數(shù)據(jù)如表所示。因子B （顆粒大小）小中大因子A漂 白 過(guò) 的Ai因子C （處理方法）因子C （處理方法）因子C （處理方法）yiRiR2R3RiR2R3RiR2R3658790944492596370658395886080811239574974828867085758180788588999687105130100131202278658070968

10、57473818868751221301211152447759873886496617195959698121125127992842956677729265635490768482172133101111(411)(817)(759)(760)(694)(900)(834)(1199)(1123)加 工 過(guò) 的A262113106107126193150112138120150112146172529710013310957171135791221221151361201351261261231381241121161251241231321259612611011813412013112

11、5110113121121991151221201171111169811512511413512412512012511612012111211615313212412611210911212013712511012516513712213499105(1258)(1112)(1226)(1241)(1292)(1226)(1357)(1094)(1151)yj。558361136758諸偏差平方和計(jì)算如下：a b c n22 y .SSt=yjki -i i j i k i i iabcn2 2=65 +87 + +i052184542 3 3 102yi.bcn2y .abcn=1628

12、82.91aSSa=i 1274972109573 3 10218454180=66508.89SSb=b2yj.j iacnabcn2 2 25583611367582184543 10180=11541.94c2yk.k 1SSc=abn2yabcn2=586162082 2 26385184542 310180=2368.41a bSSab =2yij.2y .SSa- SSbi 1 j 1cnabc n2= 198722354.2 2360218454310180=12854.81a c2yik .2ySSac =SSa- SSci 1 k 1bnabc n=2005227102.36

13、032184542310180=12177.54b c2yjk .2y .SSbc=SSb- SScj 1 k 1anabc n66508.8966508.8918454222742166921929211541.942368.41=1359.86a b cSSabc =i 1 j 1 k 12 102yijk .18011541.942368.412y .SSa- SSb- SSc- SSab -SSac- SSbcabc n2 2 241181711511021845418066508.891541.94 2368.4112854.81 12177.541359.86=8613.86SSe

14、=SSt- SSa- SSb- SSc- SSab - SSac- SSbc- SSabc=47457.60方差分析來(lái)源平方和自由度均方和F值因子A66508.89166508.89227.03因子B11541.9425770.9719.70因子C2368.4121184.214.01交互效應(yīng)AB12854.8126427.4121.94交互效應(yīng)AC12177.5426088.7720.78交互效應(yīng)BC1359.864339.971.16交互效應(yīng)ABC8613.8642153.477.35誤差47457.60162292.95總162882.91179因?yàn)?Fa>Fo.oi (1162)

15、疋 6.81 ,所以因子 A 顯著；因?yàn)?Fb, Fab , Fac>Fo.oi (2162)疋 4.74, 所以因子B、交互效應(yīng)AB、交互效應(yīng) AC都顯著；因?yàn)镕abc>Fo.01 (4, 5) 3.45，所以交 互效應(yīng)ABC顯著；又因?yàn)镕bc<F0.01 (4, 5),所以交互效應(yīng) BC不顯著。正交試驗(yàn)設(shè)計(jì)直觀分析法和方差分析法：自溶酵母提取物是一種多用途食品配料， 為探討外加中性蛋白酶的方法，需作啤 酒酵母的最適自溶條件試驗(yàn)，為此安排如下試驗(yàn)，試驗(yàn)指標(biāo)為自溶液中蛋白質(zhì)含 量（%，取含量越高越好。因素水平表如下：水平因素A （單位：C）B （pH 值）C （加酶量：%15

16、06.52.02557.02.43587.52.8試驗(yàn)結(jié)果如下，試進(jìn)行直觀分析和方差分析，找出使產(chǎn)量為最高的條件試驗(yàn)號(hào)ABC空列含量111116.25212224.97313334.54421237.53522315.54623125.507313211.408321310.90933218.95解：直觀分析試驗(yàn)號(hào)因素含量ABC空列K115.7625.1822.6520.74K218.5721.4121.4521.87K331.2518.9921.4822.97K15.258.397.556.91K26.197.147.157.29Ka10.426.337.167.66極差R5.172.060

17、.40.75主次順序A>B>C優(yōu)水平A3B1C1D3優(yōu)組合A 3B1C1方差分析n 9, r 3MSC因?yàn)镸ScSSA45.422.7, MSb6.493.23,dfA2dfB2s§c0.310.155, MSe -Se0.830.415dfc2edfe22MSe，所以因素C的偏差平方和、自由度并入誤差的偏差平方和、自由MSA處理號(hào)ABC空列試驗(yàn)結(jié)果K1j15.7625.1822.6520.74nTx 65.58i 1CT上nnQt= x2ii 1SSp Qt CT1 mQ.-K<y jijr i 1SSj Qj CTK2j18.5721.4121.4521.87K

18、3j31.2518.9921.4822.97248.38634.03513.02430.15K；j344.84458.39460.10478.30Kaj976.56360.62461.39527.62SSj45.46.490.310.83dfBdfcdfe3-1=2dfA度根據(jù)以上計(jì)算，進(jìn)行顯著性檢驗(yàn)，列出方差分析表，結(jié)果見表:方差來(lái) 源偏差平方和自由 度方差估計(jì)值F 值F0.05f , fEF0.01f , fE顯著性A因素45.40222.7079.66.9418.00*B因素6.4923.2411.46.9418.00*C因素0.3120.16誤差e0.8320.41誤差e1.1440.

19、285總和53.03因素A高度顯著，因素 B顯著，因素C不顯著。本試驗(yàn)指標(biāo)越大越好。對(duì)因素 A、B分析，確定優(yōu)水平為 A3、B1；因素C的水平改變對(duì)試驗(yàn)結(jié)果幾乎無(wú)影響，從經(jīng)濟(jì)角度考慮，選C1。優(yōu)水平組合為 a3B1C1。即溫度為58 C,pH值為6.5，加酶量為2.0%。兀回歸分析:某水稻品種在5月至8月期間播種，測(cè)定播種至齊穗的天數(shù)x和播種至齊穗的總 積溫y （單位：日度）的數(shù)據(jù)，如下表，Xi5055606570yi1441.81494.91547.11600.91646.3設(shè)y關(guān)于x有線性關(guān)系:? a bx，試用最小二乘法確定回歸系數(shù)，相關(guān)系數(shù)，并 進(jìn)行方差分析。序號(hào)kXkYk2XkXky

20、k2yk1501441.82500720902078787.242551494.9302582219.52234726.013601547.13600928262393518.414651600.94225104058.52562880.815701646.349001152412710303.6930077311825046643511980216.161n1nX 一Xj =60，yyi =1546.2ni 1n i12n21n1 2SSX= 18250- 300250i 1ni 15n1nn1SPXyxyiyiXi = 466435-7731300=2575i 1ni 1i 15n21n2

21、1ssyyiyi= 11980216.16-7731226543.96i 1ni 15SPXy =2575SS< = 250a y bx = 1546.2-10.3 60 928.2? 928.2 10.3xSPXy =26522.5, SSr里=21.46SSxT S3SSrSSy0.05,30.01,3y 2n2y彳y1 i 1n 2F1,n 2n 2F1,n 22,n2,nSpy=0.99963F0.05 1,310.1312= 0.878F0.011,334.1212=0.959方差來(lái)源偏差平方和自由度方差F值F 0.05F 0.01顯著性回歸26522.5126522.5370

22、9.4410.1334.12*殘差21.4637.150.01,3 ,所建立的回歸方程是高度顯著的。粗大誤差判別：假設(shè)以下數(shù)據(jù)不含有系統(tǒng)誤差，試用3、肖維勒、t檢驗(yàn)、格拉布斯、狄克松準(zhǔn)則來(lái)判別該數(shù)據(jù)中是否含有粗大誤差。0.01序號(hào)12345數(shù)值186698r 7755序號(hào)678910數(shù)值935446239序號(hào)111213r 1415數(shù)值6756759126(1)28.03解:序號(hào)xvv2vv2118-39.531562.62-43.51892.252668.4771.744.520.253 19840.471637.82P 36.51332.2547719.47379.0815.5240.25

23、555-2.536.40-6.542.256 :9335.471258.12r 31.5992.25754-3.5312.46-7.556.25846-11.53132.94-15.5240.259 :2-55.533083.58-59.5-10 :39-18.53343.36-22.5506.2511679.4789.685.530.251256-1.532.34-5.530.2513 :7517.47305.2013.5182.25149133.471120.2429.5870.251526-31.47990.36-35.51260.25計(jì)算結(jié)果x 57.53x 61.5v210995.9

24、4v27695.510995.94-Vi3 , i 1,2,川,15根據(jù)3準(zhǔn)則，該數(shù)據(jù)不含粗大誤差(2)10995.941428.03 , kx 2.13kx 28.03 2.13 59.7,Vikx, i 1,2,川,15根據(jù)肖維勒準(zhǔn)則，該數(shù)據(jù)不含粗大誤差。(3) 根據(jù)t檢驗(yàn)，認(rèn)為X9可疑，剔除X9，計(jì)算X,得7695.5:1324.33，K(15,0.01)3.12K 15,0.01 =24.33 3.12 75.9 X Xl K 15,0.01 ,i 1,2,川,15根據(jù)t檢驗(yàn)準(zhǔn)則，該數(shù)據(jù)不含粗大誤差。(4) 將捲,X2，川,X15按從小到大的順序排列為:X1 2,X2 18,X3 26

25、,川,X13 91,X14 93,X1598g(15)X(15) X98 57.5328.031.4457.53228.031.98g。(15,0.01)2.70:g1 g。15,0.01 ,g 15g。15,0.01根據(jù)格拉布斯準(zhǔn)則，該數(shù)據(jù)不含粗大誤差。(5) 將九X2,|, X15按從小到大的順序排列為：X12,X218,X326,川,X1391,X1493,X15i *X(15)XgX(15)X(3)M 0.1 ,98 26d0 1X(3)X(1)X(13)X(1)26 291 20.27d0 15d0.0115 , d0 15d0.0115根據(jù)狄克松準(zhǔn)則，該數(shù)據(jù)不含粗大誤差 主效應(yīng)與交

26、互效應(yīng)：計(jì)算下列各因素的主效應(yīng)，并判斷因素間是否存在交互作用，若存在計(jì)算其交互 效應(yīng)。B1B2A12545A26022解：主效應(yīng)為：八 60 22 25 45 oA62 2廠 45 22 25 60B92 2交互效應(yīng)為：11AB A2B2 AB2A2B1 A1B1- 22 4560 252922證明題：（1）當(dāng)測(cè)量次數(shù)n 10時(shí)，即使在測(cè)量數(shù)據(jù)中含有粗大誤差，用 3準(zhǔn)則也不能 判別出來(lái)。證明：X整理上式得:對(duì)于任何一項(xiàng)vi 1 in ，必然有M代入Vin v2 得:vi &1故，當(dāng)n 10時(shí)，Vj|3因此，當(dāng)n 10時(shí)不能使用3準(zhǔn)則。(2)假設(shè)實(shí)驗(yàn)得到了 n對(duì)數(shù)據(jù)(xyj(i 1,2,

27、川,n),程? a bx ,計(jì)算可得回歸系數(shù)b的計(jì)算式為根據(jù)這些數(shù)據(jù)得到了回歸方Xi yin2XiXii 1nXi X yi yi 1n_ 2XiXi 1請(qǐng)證明以上兩式相等。 證明：XiYiXiYiXiYYiXxynnX Yi nxy nxy nxy1XiYii 1nxyi 1n1 nnXYiYiXi 1n i 1i 1nn22 2XiXXi 2XiX x1i 1nnn2X2x xi2 2 2nxXinxi 1i 1i 1n2n2XiXix yi yi 1XYi -1 nYii 1XinXiXi試驗(yàn)設(shè)計(jì)與數(shù)據(jù)處理用表肖維勒準(zhǔn)則常用的kx值n567891011kx/1.651.731.791.8

28、01.921.962.00n12131415161718kx/2.042.072.12.132.162.182.20n19202122232425kx/2.222.242.262.282.32.312.33n26272829303540kx/2.342.352.372.382.392.452.50n5060708090100150kx/2.582.642.692.732.782.812.92n185200250500100020005000kx/3.003.023.113.293.483.663.89t檢驗(yàn)系數(shù)K (n)表n顯著性水平n顯著性水平ko.o5 (n)K0.01 (n)K0.05(

29、 n)K0.01( n)44.9711.46182.183.0153.566.53192.173.0063.045.04202.162.9572.784.36212.152.9382.623.96222.142.9192.513.71232.132.90102.433.54242.122.88112.373.41252.112.86122.333.31262.102.85132.293.23272.102.84142.263.17282.092.83152.243.12292.092.82162.223.08302.082.81172.203.04格拉布斯檢驗(yàn)系數(shù)表g0(n,)nn0.050.

30、010.050.01go(n,)g0( n,)31.151.16172.482.7841.461.49182.502.8251.671.75192.532.8561.821.94202.562.8871.942.10212.582.9182.032.22222.602.9492.112.32232.622.96102.182.41242.642.99112.232.48252.663.01122.282.55302.743.10132.332.61352.813.18142.372.66402.873.24152.412.70502.963.34162.442.751003.173.59Dix

31、on檢驗(yàn)系數(shù)及d0的計(jì)算公式nd (n)do的計(jì)算公式顯著性水平X(1)可疑X(n)可疑0.010.0530.9880.941X(2)X(1)X(n)X(n 1)40.8890.76550.7800.642X(n)X(1)X(n)X(1)60.6980.56070.6370.50780.6830.554X(2)X(1)X(n)X(n 1)90.6350.512X(n 1)X(1)X(n)X(2)100.5970.477110.6790.576X(3)X(1)X(n)X(n 2)120.6420.546X(n 1)X(1)X n)X(2)130.6150.521140.6410.546X(3)X

32、(1)X(n)X(n 2)X(n)X(3)150.6160.525160.5950.507170.5770.490180.5610.475190.5470.462200.5350.450X(n 2)X(1)210.5240.440220.5140.430230.5050.421240.4970.413250.4890.406F分布表p! F(刃 I , ”2)F“( 772)i = Qa = 0.0532.b 5 5 s* 3丿«.51.U36血25黒Tl”198, 222J<».R s4 .3.1;y MU1P2w».55J- 6 i 11*5.C r 4

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