Statistical Analysis Results of Crime: ANOVA Test

Statistical Analysis Results of Crime analysis of variance TestTHE congeriesmary OF VARIANCE (analysis of variance), STUDENTS T TESTS AND MATRIX ALGEBRAATUBI, A. 0. Ph.D. cosmosThe ANOVA sometimes referred to as the F test (named after the statistician Sir Roland Fisher, the author of this test) is a hard-boiled of procedures for testing the signifi spatece of resistences among more than than deuce independent agent. This procedure determines the extremity to which there ar significant differences betwixt the means of three or more samples with a single level of significance. Because this procedure and others like it focalisation on variances, they atomic number 18 referred to as the analysis of variance or ANOVAOne Way Analysis of VarianceThe step in ANOVA can be illustrated by an exampleExample 5.1.1The payoff of crimes commit in 4 months in various parts of a township are list belowMonthArtisan quartersSlumsCDBGRAJanuaryFebruary serveApril2016321652445636364036401620 3220At 0.05 level of confidence, are crime frequencies colligate to urban morphology or month of the year?First exercise up a null hypothesis of equality of the means permit Ho be There is no significant difference between the crimes frequencies and urban morphology or month of the year.Next, find the full of the quaternary areas as listed below then their meansTotal =Artesan SlumsCBDGRA8418815288MeanNext, depend the Grand meanNext, calculate the sum of money of materials for all(prenominal) population. Note that in ANOVA the come in of samples do not accommodate to be resemblingTable 4.1.1 Means and sums of squares of crimes committedIn ANOVA, we determine differences between means by calculating their variability. Three types of variability are seedThe variation indoors each sample resultThe variation between the sample resultsThe total variation of the samples, regardless of the sample grouping to which they belong (Anyadike, 2009)Next, calculate the sums of the sum of squares for each column (i.e Variable)SSW = (Xij X)2 = 172+236+16+144 =568 figure out sum of square between = ? N (X X)2=4(21-3 2)2+4(473 2)2+4 (3832)2 + 4(2232)2= 484 +900+144+400=1928Calculate the within group sum of square. MSW = SSWn-rnWhere n=no of observationsM=no of groupsSimilarly, the mean s square between will beMSb =SSbM-lThereforeMSW =SSb=568=568n-m 16-4 12= 47.33The ratio between the variance estimate is known as the Snedecors variance ratio test or Snedecors FNow set up an ANOVA tabulateFin every(prenominal)y, test for significanceThe storys of freedom, V areV- numerator = M-1=4-1=3V- denominator = n-m= 164=12From the Table of the F-Distribution, critical value of F at 0.05 3 and 12 =3.49Calculated value is 13.18Since the calculated F of 13.58 is grander () 3.49, Ho is rejected. Therefore, There is a significant difference between crime frequencies with regards to urban morphology or month of the year.THE STUDENT T TESTThe near powerful test for the comparison of sample means is the student t test. It is a parametric test and is used to determine whether or not the differences between two sample means are sufficiently great as to justify a conclusion that the means of their populations also differ significantly. It is also used for small samplesThe student t is expressed asT = x1-x1vS12/N1 + S22/N2Where X1 and X2 are the means of the two sets of data S1 and S2 their specimen deviations and N1 and N2 the number of observations.The degree of freedom, V is expressed as followsV = N+N,-2Degree of FreedomThere is often confusion among students about the supposition of degree of freedom. Basically, if the sum of a set of chemical ingredients and the sum of all but one is of its ingredients are both(prenominal) known, then the value of the furthest element must also be known, i.e. it is not, unlike the others, free to change (Ayandike, 2009). For example, if the sum of 8 elements is 30 and the determine of 7 of the elements variously ply up to 28, then the value ofthe final (i.e. the 8) element must be (30-28) = 2, i.e. if is not free to adopt any other value. The degree of freedom in this case is thus (8-1) = 7, i.e. in the set of 8 element, 7 of them are free to take on any values to sum up to 28, leaving value inviolateELEMENTS OF MATRIX ALGEBRAIntroductionThe hyaloplasm is a rectangular array of number arranged in rows n and columns, m i.eEach of the numbers is called an elements. The position of each element is determined by its position in the row as puff up as in theThe size of the hyaloplasm is given by the number of rows (n) and number of column (m) for example.A ground substance which has the same number of rows and columns is called a square intercellular substance. In the example above, a and c are square matrices. A matrix with a single row is called a row vector, while a matrix of a single column is called a column vector. Example of a row vector is3 5 7 8example of column vector isMatrix c an be added, subtracted, multiplied and inverted but cannot be divided. However, they can merely be divided by a scalar (i.e an ordinary number). profit of MatrixMatrices to be added must be of the same size as one another(prenominal). That is they must have the same number of columns and row s. This is because each element of one matrix must be added to the same element of the other matrix e.g. supposing we are adding two matrix A BA + BI =Ci.e you add element by elementN.BThe number of columns and rows must be the same onwards it can be addedMatrix SubtractionThe same rule as addition is applied e.g to subtract BI from AA BMatrix MultiplicationThere are two aspect of matrix multiplication namelya.Multiplication of matrix by a scalerb.Multiplication of matrix by two matricesBy ScalerSupposing we are reckoning the matrix below by 4Multiplication by Two MatricesWhen two matrices are to be multiplied, the number of columns in the first matrix must be equal the number of rows in th e second matrix e.g. 23 matrix can be multiplied by 32 matrix. scarcely a 23 matrix cannot be multiplied by another 23 matrix because the number of rows there is not equal to the number of columns in the second e.g.The result will have as some(prenominal) rows as the first and as may columns as the second. Multiplication of a matrix by vector, exampleAn identity or unity matrix 1, is a matrix where the diagonal consist of ls and the put down of the elements are zero e.g.Matrix InversionIn matrix, algebra function of percentage is changed to that of inversion. The reverse of the matrix is its reciprocal i.e.Only square matrices have inverses. A matrix that cannot be inverted is called a singular matrix. several(prenominal) methods exists for finding the inverse of a matrix. They includes1.The classical methodThis is to set the matrix beside an identity matrix, and to perform all operations simultaneously in both matrices, for example, if you are to invert this matrix A, you firs t place it beside an identity matrixYou haveStep 1Subtract row 2 from 3, multiply row 1Row 2(3xrow 1)Step 32.By DeterminantsThis is the more modern one. A determining(prenominal) is a single number extracted from a square matrix by series of operations. It is represented by each det A or /A/The process of obtaining a determinant from a matrix is called evaluating the determinant. Using determinant, the inverse of matrix A becomes/A/ = adbcThe adjoint of a matrix is the transposed matrix of co -factors with the signs taken into consideration. The signs are alternating +, -, across and down the element of the matrix e.g. in a 22 matrixSo far a 22 matrix, the inverse is the adjoint of that matrix over the determinant of the matrix.The determinant of a 33 matrixThe minus for each leading element becomesUptill i.The Solution of Simultaneous equalityThe major interest in matrices (and its greatest strength) is their use in the proclamation of the unknowns in simultaneous equations (An ya dike, 2009)SIMULTANEOUS EQUATIONS WITH TWO UNKNOWNS3x+4y 102x +7y = 11 muckle in matrix form1.Using the classical method, our equation in matrix form is A x BThe matrix of the unknownx=A=BFor a 2 x 2 matrix the adjoint of it isSince our matrix of unknown isCheck with original equation3x+4y =102x +7y1lCheck3(2)+4(1)= 10 =6+4+102(2) +7 (1) =14+7=111. By determinant methodThe matrix in our example is A x BThis 13 is called the common denominator. Then we find the numerator of x, which is the determinant of the main matrix.. X2/A1.B/-/A13113 =1 =yTherefore,Y = 1X=2REFERENCESAnyadike, R.N.C (2009) Statistical methods for social and environmental Sciences. Spectrum Books Limited Ibadan.Anyadike, R.N.C (2009) Statistical methods for social and Environmental sciences. Spectrum Books limited Ibadan.Atubi, A.O. (2010d) Road Traffic Accident variations in Lagos State, Nigeria A scheme of variance Spectra. African research Review, Vol 4(2) pp. 197-218. Ethiopia.Ewhmdjakpor C, Atubi, A.O. an d Odemerho F. (2006) Statistics for social investigations. Delsu Investment Nigeria, Limited, Delta State University, Abraka.Ewhrudjakpor. C, Atubi, A.O, and Odermerho F (2006). Statistics for social Investigations. Delsu Investment Nigeria, Limited. Delta State University, Abraka.Spiegel, M.R (1972) Theory and problems of statistics. McGraw-Hill, New York.