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International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 8These result force the research to the following conclusion on each of the 4 type of pipe used in the analysis. M eaningthere was no need to obtain a linear combination of the variables but conduct hypothesis test using Hotellings T 2 and estimateRoy-Bose Confidence interval for each characteristics of each particular type of pipe.(Hotelling, 1947)(1) From matrix or Table 3.11,3.12,3.13 and 3.14, (see appendix) all the correlation matrix shows a very low relationshipbetween the variables i.e diameter, circumference and length which indicate that there is no further need to conductprincipal component analysis.(2) Also, matrix or table 3.11,3.12,3.13 and 3.14,shows that in each of the type of pipe produced, the diameter contributehigher variation to the total variation in the production process. This is clearly d emonstrated by the eigen values, andfrom the graphs m, n, o and p prove that by the sharp drop from the first eigen value in each graph.3.2 MULTIVARIATE TES TSA multivariate statistical tests specifies conditions about the parameters of the population from which sample wasdrawn. This means that multivariate model considers that the population distribution should h ave a particular form (e.g anormal distribution) and also involves hypotheses about population parameters.( Morrison,2005) The reliability of the resultsof multivariate tests depends on the validity of these assumptions:It is paramount to note that, the most powerful tests are those having more extensive assumptions. The T 2 test, for itto be applied the underlying conditions must be at least 70% satisfied, (Chatfield & Collins, 1980),1. Observations must be independent (uncorrelated)2. Observations must be drawn from normally distributed populations3. Population must have variance-covariance matrix and mean vector.4. Variables must have been measured in at least interval or ratio scale.If these conditions are satisfied in the data under analysis, one can choose T 2 or wilks lambda test. When these set ofassumptions are met, the tests are most likely to reject H O when it is false.3.3 HOTELLING T 2 DIS TRIBUTIONHarold Hotelling (1931), propose a multivariate generalization of the student t -distribution and T 2 is used inmultivariate hypothesis testing.If x, x, ---,x n ~N ( µ, 2 ) with µ and unknown, we can test the hypothesis; Ho: µ=µ o usingt = ( X - )//nSo that t 2 = (x- µ o ) 2 /( 2 /n)t 2 = n (x- µ o )¹ ( 2 ) - ¹ (x- µ o ).the generalization for x 1 , x 2 --, x n ,~ N p (µ,Σ) with P≥1, is the Hotelling T 2 statisticT 2 = n(x- µ o )¹ s - ¹(x- µ o )Where mean X = 1/n Σx, and s - ¹ is the inverse of the variance covariance matrix. n is the sample size. The diagonalelements of S are the variances of the x i and the off diagonal are the covariance for p variables.The multivariate techniques should posses‟ three important properties:1. They produce a single answer to the question: is the process in control?2. Has the specified type I error been maintained?.3. These techniques must take in to account the relationship between the variables.3.3.1 HOTELLING’S T 2 TES TBelow is the covariance result of the 3`` round pipe data.diam circum lengthdiam 0.00119900 0.00025500 -0.00004317circum 0.00025500 0.01726667 -0.02476333length -0.00004317 -0.02476333 0.41491233Inverse of covariance matrix843.11 -12.89 0.00-12.89 63.17 2.580.00 2.58 2.58T 2 = 47.3684.T =14.47.F(0.05)(3,22)=3.05.||Issn 2250-3005(online)|| ||December||2012|| Page 213
International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 8From the result, the null hypothesis is rejected at α= 0.05, that the means are not equal i.e. the mean of diameter,circumference and length for the individual 3``round pipe differs. And that the following intervals should be considered for theacceptance of 3`` round pipes produced.3.4 CONFIDENCE INTERVAL (Roy – Bose confidence interval) for 3`` round pipe data.C.I= X Ka/2,(n-1) S/Ka/2 = [p(n-1)/(n-p) Fα, , ( p, n-p )]C.I(diameter)= (8.99,9.03).C.I(circumference)= (30.09,30.25).C.I(length)= (599.22,600.04).nnAlso we can use, C.I= X ta/2,(n-1) S/C.I(diameter)= (9,9.02).C.I(circumference)= (30.12,30.22).C.I(length)= (599.36,599.9).Note that this test was done for all 3 types of pipe with interpretation of each before the (ROY- BOS E C.I)4.0 SUMMARYIn this research Quality Control using multivariate techniques tests for independence and equality of means wasconducted using multivariate data. From the result of each tests using principal component analysis, the variables are found tobe reasonably uncorrelated that the degree of relationship(correlation) is not strong enough. The hypothesis conducted usingHotelling‟s T 2 , each tests rejected the null hypothesis(H o : means are the the same).4.1 CONCLUS IONThe multivariate tools used, principal component analysis and Hotelling T 2 are both statistical method of inference.Multivariate methods usually depend upon the assumption of a specific distribution form, for example an approximatemultivariate normal distribution. And data for these methods will be in interval or ratio scale . The analysis of the data frompipe Production Company using the multivariate techniques produced a fairly reasonable result. From the analysis usingprincipal component analysis to have a “feel” for the set of data and understanding of their correlation structure, the data setfor diameter shows higher variation contribution compared to that for circumference and length. And the c orrelation that existsfrom each product variable is “substantial”. Meaning nearly all the correlation coefficients are “small”, there is probably n otmuch point to further conduct complete principal component analysis. The Hotelling T 2 analysis result for the differentproduct rejected the null hypothesis in favor of the alternative hypothesis ( H o ; the means are different ). The Roy-Boseconfidence interval techniques tend to estimate interval for the production department to consider for the acceptance of a pipethat would be produced when the process is in statistical control with a confidence that items produced reached 95% standardquality. Based on the findings, it is therefore concluded that the process that generates the data is in statistical cont rol.Numerous quality control related problems are multivariate in nature, and using univariate statistical process control charts isnot effective when dealing with multivariate data exhibiting correlated behavior. Therefore, multivariate statistical processcontrol procedure provides reliable result.APPENDIX3.2 PRINCIPAL COMPONENT ANALYS ISTable 3.11 shows the result of the 3`` round pipe data using PCA.Table 3.11 :————— 10/10/2012 9:08:34 AM ——————————————————Welcome to Minitab, press F1 for help.CORRELATIOND C LD 1 0.056 -0.002C 0.056 1 -0.269L -0.002 -0.269 1Covariances: diameter(cm), circumf(cm), length(cm)diam circu m lengthdiam 0.00119900 0.00025500 -0.00004317circum 0.00025500 0.01726667 -0.02476333length -0.00004317 -0.02476333 0.41491233||Issn 2250-3005(online)|| ||December||2012|| Page 214
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