This document discusses correlation and regression analysis. It defines correlation as dealing with the association between two or more variables, and identifies different types including positive/negative, simple/multiple, and linear/non-linear. Regression analysis predicts the value of a dependent variable based on an independent variable. Key aspects covered include Karl Pearson's coefficient of correlation, Spearman's rank correlation coefficient, regression lines, coefficients, and estimating values from the regression equation.
This document discusses correlation and regression analysis. It defines correlation as dealing with the association between two or more variables. Regression analysis develops a statistical model to predict a dependent variable from an independent variable. The key methods covered are Karl Pearson's coefficient of correlation, Spearman's rank correlation coefficient, and linear regression lines and coefficients. Formulas are provided for calculating these statistical measures.
Kishan Kasundra presented on Cramer's rule for solving systems of equations. Cramer's rule uses determinants of coefficient matrices to find variables. It can be applied to systems with 2 or 3 equations. For 2 equations, the determinant of the original coefficient matrix and matrices with the coefficient columns replaced by the constants are calculated. The ratios of these determinants over the original give the variables. For 3 equations, a similar process is followed using a 3x3 coefficient matrix. An example showed applying Cramer's rule to solve a 2x2 system.
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
Matrices Slide For B.Sc Students As Well For F.Sc StudentsAbu Bakar Soomro
The document provides information about matrices, including:
- Matrices are two dimensional arrays that can store ordered data.
- The order or size of a matrix is defined by the number of rows and columns, represented as RxC.
- Elements in a matrix are accessed by their row and column position, represented as aij which refers to the element in the ith row and jth column.
- Operations on matrices include addition, subtraction, multiplication, and determining the determinant. The determinant is a number associated with square matrices.
This document contains exercises and solutions for line integrals from a chapter on the topic. It includes 6 exercises evaluating line integrals over various curves defined parametrically or through equations. It also contains exercises using Green's Theorem and evaluating line integrals for conservative vector fields. The solutions provide the parametrizations needed to set up and evaluate the line integrals.
Here we have included details about relaxation method and some examples .
Contribution - Parinda Rajapakha, Hashan Wanniarachchi, Sameera Horawalawithana, Thilina Gamalath, Samudra Herath and Pavithri Fernando.
This document provides an overview of matrix methods for solving systems of linear equations. It begins with an example from structural engineering of setting up a system of 6 equations with 6 unknowns to model the forces and reactions in a statically determinant truss. The equations are represented in matrix notation as [A]{x}={c}. The document then reviews key matrix concepts and operations used to solve systems of linear equations, such as matrix addition, multiplication, transposes, inverses, and types of matrices. It aims to help readers understand how to set up and solve systems of linear equations using matrices.
This document discusses correlation and regression analysis. It defines correlation as dealing with the association between two or more variables. Regression analysis develops a statistical model to predict a dependent variable from an independent variable. The key methods covered are Karl Pearson's coefficient of correlation, Spearman's rank correlation coefficient, and linear regression lines and coefficients. Formulas are provided for calculating these statistical measures.
Kishan Kasundra presented on Cramer's rule for solving systems of equations. Cramer's rule uses determinants of coefficient matrices to find variables. It can be applied to systems with 2 or 3 equations. For 2 equations, the determinant of the original coefficient matrix and matrices with the coefficient columns replaced by the constants are calculated. The ratios of these determinants over the original give the variables. For 3 equations, a similar process is followed using a 3x3 coefficient matrix. An example showed applying Cramer's rule to solve a 2x2 system.
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
Matrices Slide For B.Sc Students As Well For F.Sc StudentsAbu Bakar Soomro
The document provides information about matrices, including:
- Matrices are two dimensional arrays that can store ordered data.
- The order or size of a matrix is defined by the number of rows and columns, represented as RxC.
- Elements in a matrix are accessed by their row and column position, represented as aij which refers to the element in the ith row and jth column.
- Operations on matrices include addition, subtraction, multiplication, and determining the determinant. The determinant is a number associated with square matrices.
This document contains exercises and solutions for line integrals from a chapter on the topic. It includes 6 exercises evaluating line integrals over various curves defined parametrically or through equations. It also contains exercises using Green's Theorem and evaluating line integrals for conservative vector fields. The solutions provide the parametrizations needed to set up and evaluate the line integrals.
Here we have included details about relaxation method and some examples .
Contribution - Parinda Rajapakha, Hashan Wanniarachchi, Sameera Horawalawithana, Thilina Gamalath, Samudra Herath and Pavithri Fernando.
This document provides an overview of matrix methods for solving systems of linear equations. It begins with an example from structural engineering of setting up a system of 6 equations with 6 unknowns to model the forces and reactions in a statically determinant truss. The equations are represented in matrix notation as [A]{x}={c}. The document then reviews key matrix concepts and operations used to solve systems of linear equations, such as matrix addition, multiplication, transposes, inverses, and types of matrices. It aims to help readers understand how to set up and solve systems of linear equations using matrices.
This document discusses numerical methods for solving linear systems of equations. It begins by introducing linear systems in general and matrix forms, and classifying them as homogeneous or non-homogeneous. It then discusses checking for consistency. The main methods covered for obtaining solutions are: Gauss elimination, Gauss-Jordan elimination, using the inverse matrix if it exists, and iterative techniques. Specific examples are provided to demonstrate how to apply Gauss elimination, Gauss-Jordan elimination and using the inverse matrix to solve sample systems.
B.tech ii unit-3 material multiple integrationRai University
1. The document discusses multiple integrals and double integrals. It defines double integrals and provides two methods for evaluating them: integrating first with respect to one variable and then the other, or vice versa.
2. Examples are given of evaluating double integrals using these methods over different regions of integration in the xy-plane, including integrals over a circle and a hyperbolic region.
3. The document also discusses calculating double integrals over a region when the limits of integration are not explicitly given, but the region is described geometrically.
This document discusses numerical differentiation and integration using Newton's forward and backward difference formulas. It provides examples of using these formulas to calculate derivatives from tables of ordered data pairs. Specifically, it shows how to calculate derivatives at interior points using central difference formulas, and at endpoints using forward or backward formulas depending on if the point is near the start or end of the data range. Formulas are derived for calculating the first and second derivatives, and examples are worked through to find acceleration and rates of cooling from given temperature-time tables.
This document discusses methods for solving systems of linear equations. It describes direct methods like Gauss elimination and LU decomposition that obtain solutions in a finite number of steps. It also describes iterative methods like Jacobi's method and Gauss-Seidel method that obtain solutions through successive approximations that converge to the required solution. Pseudocode and MATLAB implementations are provided for various algorithms.
Numerical solution of a system of linear equations by
1) LU FACTORIZATION METHOD.
2) GAUSS ELIMINATION METHOD.
3) MATRIX INVERSION BY GAUSS ELIMINATION METHOD.
The document discusses several numerical methods for solving systems of linear equations, including Jacobi, Gauss-Seidel, LU decomposition, and Cholesky decomposition. It provides the algorithms and formulas for each method. As an example, it applies the Jacobi and Gauss-Seidel methods to solve a system of 3 equations with 3 unknowns, showing how Gauss-Seidel converges faster by immediately using updated values at each step.
This document provides an overview of vector differentiation, including gradient, divergence, curl, and related concepts. It begins with definitions of scalar and vector point functions. It then defines the vector differential operator Del and explores using it to calculate the gradient of a scalar function, directional derivatives, and normal derivatives. The document also covers divergence and curl, providing their definitions and formulas. Examples are given for calculating gradient, divergence, curl, and directional derivatives. The document concludes with exercises and references for further reading.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIRai University
This document provides an overview of Unit II - Complex Integration from the Engineering Mathematics-IV course at RAI University, Ahmedabad. It covers key topics such as:
1) Complex line integrals and Cauchy's integral theorem which states that the integral of an analytic function around a closed curve is zero.
2) Cauchy's integral formula which can be used to evaluate integrals and find derivatives of analytic functions.
3) Taylor and Laurent series expansions of functions, including their regions of convergence.
4) The residue theorem which can be used to evaluate real integrals involving trigonometric or rational functions.
This document discusses matrix representations of linear transformations and changes of basis in linear algebra. It defines the matrix associated with a linear transformation with respect to two bases and introduces the change of basis matrices. It provides examples of finding the matrix associated with a linear transformation, the change of basis matrices between two bases, and using the change of basis matrices to transform component representations of a vector between bases.
The chapter discusses numerical methods for solving the 1D and 2D heat equation. Four methods are described for the 1D equation: Schmidt, Crank-Nicolson, iterative (Jacobi and Gauss-Seidel), and Du Fort-Frankel. The Schmidt method is explicit but conditionally stable, while Crank-Nicolson is implicit and unconditionally stable. Examples are solved using each method and compared to analytical solutions. The alternating direction explicit (ADE) method is described for the 2D equation.
This document discusses vector differentiation and provides examples and exercises. Some key topics covered include:
- The definition of scalar and vector point functions
- Vector differential operator Del (∇)
- Gradient of a scalar function and its properties
- Normal and directional derivatives
- Divergence and curl of vector functions
- Examples calculating gradients, directional derivatives, and verifying identities
- Exercises involving finding normals, gradients, directional derivatives, and divergence
The document discusses the composition of linear transformations. It defines the composition of two linear transformations T1 and T2 as the transformation T2 o T1, which maps an element X in the domain of T1 to the result of applying T2 to the output of T1. The key points made are:
1) The composition T2 o T1 is a linear transformation.
2) The matrix associated with the composition T2 o T1 is the product of the matrices associated with T1 and T2.
3) Examples are provided to illustrate finding the matrix of a composition and applying a composition to vectors and subspaces.
Comparison Results of Trapezoidal, Simpson’s 13 rule, Simpson’s 38 rule, and ...BRNSS Publication Hub
Numerical integration plays very important role in mathematics. In this paper, overviews on the most common one, namely, trapezoidal Simpson’s 13 rule, Simpson’s 38 rule and Weddle’s rule. Different procedures compared and tried to evaluate the value of some definite integrals. A combined approach of different integral rules has been proposed for a definite integral to get more accurate value for all case
This document discusses methods for solving systems of linear equations, including Gauss elimination and iterative Gauss methods like the Gauss-Jacobi and Gauss-Seidel methods. It provides explanations of row-echelon form and reduced row-echelon form. Examples are given to demonstrate using elementary row operations to solve systems of linear equations and the Gauss-Jacobi iterative method.
LU decomposition is a method to solve systems of linear equations by decomposing the coefficient matrix A into the product of a lower triangular matrix L and an upper triangular matrix U. It involves (1) decomposing A into L and U, (2) solving LZ = C for Z, and (3) solving UX = Z for X to find the solution vector X. The document provides an example using LU decomposition to solve a system of 3 linear equations.
The document discusses rules for simplifying radical expressions. It states the square root and multiplication rules, which are that the square root of a perfect square is the number itself, and that the square root of a product is the product of the individual square roots. Examples are provided to demonstrate simplifying radical expressions by extracting square roots from the radicand in steps using these rules.
1) The document discusses differentiation techniques such as differentiating polynomials, composite functions, and logarithmic and exponential functions.
2) It also covers applications of differentiation like finding maxima and minima, rates of change, and using differentiation to approximate small changes.
3) The key concepts covered are the derivative, differentiation rules, the relationship between the derivative and tangents/normals to curves, and using the derivative to solve optimization problems.
The document discusses numerical methods for solving linear algebraic equations. It begins by introducing the general form of a system of n linear equations with n unknowns.
It then describes two classes of solution methods: direct and iterative. Gaussian elimination is presented as a direct method that transforms the system of equations into an upper triangular system which can then be solved using back substitution. Gauss-Seidel iteration is introduced as an iterative method that uses successive approximations to obtain solutions. The document provides examples to illustrate how to apply both Gaussian elimination and Gauss-Seidel iteration to solve systems of linear equations.
This document discusses correlation and regression analysis. It defines correlation as dealing with the association between two or more variables. There are different types of correlation including positive, negative, simple, and multiple. Methods for measuring correlation include scatter diagrams, graphs, and Karl Pearson's coefficient of correlation. Regression analysis develops a statistical model to predict a dependent variable from an independent variable. Regression coefficients and the correlation coefficient can be used to describe the relationship between variables.
Regression and correlation analysis allow researchers to assess relationships between variables. Regression fits a line to two variables that minimizes the sum of squared errors, representing how well the independent variable predicts the dependent variable. Correlation assesses the strength and direction of association, ranging from -1 to 1. R-squared indicates the proportion of variance in the dependent variable explained by the independent variable.
This document introduces correlation and regression. It defines correlation as a measure of the association between two numerical variables and discusses how the Pearson correlation coefficient measures both the direction and strength of the linear relationship between two variables. Regression is introduced as a statistical method for finding the line of best fit for one variable based on another. Simple linear regression finds the line of best fit for one dependent variable based on one independent variable. The document provides steps for performing simple linear regression using a TI-83 graphing calculator.
This document discusses numerical methods for solving linear systems of equations. It begins by introducing linear systems in general and matrix forms, and classifying them as homogeneous or non-homogeneous. It then discusses checking for consistency. The main methods covered for obtaining solutions are: Gauss elimination, Gauss-Jordan elimination, using the inverse matrix if it exists, and iterative techniques. Specific examples are provided to demonstrate how to apply Gauss elimination, Gauss-Jordan elimination and using the inverse matrix to solve sample systems.
B.tech ii unit-3 material multiple integrationRai University
1. The document discusses multiple integrals and double integrals. It defines double integrals and provides two methods for evaluating them: integrating first with respect to one variable and then the other, or vice versa.
2. Examples are given of evaluating double integrals using these methods over different regions of integration in the xy-plane, including integrals over a circle and a hyperbolic region.
3. The document also discusses calculating double integrals over a region when the limits of integration are not explicitly given, but the region is described geometrically.
This document discusses numerical differentiation and integration using Newton's forward and backward difference formulas. It provides examples of using these formulas to calculate derivatives from tables of ordered data pairs. Specifically, it shows how to calculate derivatives at interior points using central difference formulas, and at endpoints using forward or backward formulas depending on if the point is near the start or end of the data range. Formulas are derived for calculating the first and second derivatives, and examples are worked through to find acceleration and rates of cooling from given temperature-time tables.
This document discusses methods for solving systems of linear equations. It describes direct methods like Gauss elimination and LU decomposition that obtain solutions in a finite number of steps. It also describes iterative methods like Jacobi's method and Gauss-Seidel method that obtain solutions through successive approximations that converge to the required solution. Pseudocode and MATLAB implementations are provided for various algorithms.
Numerical solution of a system of linear equations by
1) LU FACTORIZATION METHOD.
2) GAUSS ELIMINATION METHOD.
3) MATRIX INVERSION BY GAUSS ELIMINATION METHOD.
The document discusses several numerical methods for solving systems of linear equations, including Jacobi, Gauss-Seidel, LU decomposition, and Cholesky decomposition. It provides the algorithms and formulas for each method. As an example, it applies the Jacobi and Gauss-Seidel methods to solve a system of 3 equations with 3 unknowns, showing how Gauss-Seidel converges faster by immediately using updated values at each step.
This document provides an overview of vector differentiation, including gradient, divergence, curl, and related concepts. It begins with definitions of scalar and vector point functions. It then defines the vector differential operator Del and explores using it to calculate the gradient of a scalar function, directional derivatives, and normal derivatives. The document also covers divergence and curl, providing their definitions and formulas. Examples are given for calculating gradient, divergence, curl, and directional derivatives. The document concludes with exercises and references for further reading.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIRai University
This document provides an overview of Unit II - Complex Integration from the Engineering Mathematics-IV course at RAI University, Ahmedabad. It covers key topics such as:
1) Complex line integrals and Cauchy's integral theorem which states that the integral of an analytic function around a closed curve is zero.
2) Cauchy's integral formula which can be used to evaluate integrals and find derivatives of analytic functions.
3) Taylor and Laurent series expansions of functions, including their regions of convergence.
4) The residue theorem which can be used to evaluate real integrals involving trigonometric or rational functions.
This document discusses matrix representations of linear transformations and changes of basis in linear algebra. It defines the matrix associated with a linear transformation with respect to two bases and introduces the change of basis matrices. It provides examples of finding the matrix associated with a linear transformation, the change of basis matrices between two bases, and using the change of basis matrices to transform component representations of a vector between bases.
The chapter discusses numerical methods for solving the 1D and 2D heat equation. Four methods are described for the 1D equation: Schmidt, Crank-Nicolson, iterative (Jacobi and Gauss-Seidel), and Du Fort-Frankel. The Schmidt method is explicit but conditionally stable, while Crank-Nicolson is implicit and unconditionally stable. Examples are solved using each method and compared to analytical solutions. The alternating direction explicit (ADE) method is described for the 2D equation.
This document discusses vector differentiation and provides examples and exercises. Some key topics covered include:
- The definition of scalar and vector point functions
- Vector differential operator Del (∇)
- Gradient of a scalar function and its properties
- Normal and directional derivatives
- Divergence and curl of vector functions
- Examples calculating gradients, directional derivatives, and verifying identities
- Exercises involving finding normals, gradients, directional derivatives, and divergence
The document discusses the composition of linear transformations. It defines the composition of two linear transformations T1 and T2 as the transformation T2 o T1, which maps an element X in the domain of T1 to the result of applying T2 to the output of T1. The key points made are:
1) The composition T2 o T1 is a linear transformation.
2) The matrix associated with the composition T2 o T1 is the product of the matrices associated with T1 and T2.
3) Examples are provided to illustrate finding the matrix of a composition and applying a composition to vectors and subspaces.
Comparison Results of Trapezoidal, Simpson’s 13 rule, Simpson’s 38 rule, and ...BRNSS Publication Hub
Numerical integration plays very important role in mathematics. In this paper, overviews on the most common one, namely, trapezoidal Simpson’s 13 rule, Simpson’s 38 rule and Weddle’s rule. Different procedures compared and tried to evaluate the value of some definite integrals. A combined approach of different integral rules has been proposed for a definite integral to get more accurate value for all case
This document discusses methods for solving systems of linear equations, including Gauss elimination and iterative Gauss methods like the Gauss-Jacobi and Gauss-Seidel methods. It provides explanations of row-echelon form and reduced row-echelon form. Examples are given to demonstrate using elementary row operations to solve systems of linear equations and the Gauss-Jacobi iterative method.
LU decomposition is a method to solve systems of linear equations by decomposing the coefficient matrix A into the product of a lower triangular matrix L and an upper triangular matrix U. It involves (1) decomposing A into L and U, (2) solving LZ = C for Z, and (3) solving UX = Z for X to find the solution vector X. The document provides an example using LU decomposition to solve a system of 3 linear equations.
The document discusses rules for simplifying radical expressions. It states the square root and multiplication rules, which are that the square root of a perfect square is the number itself, and that the square root of a product is the product of the individual square roots. Examples are provided to demonstrate simplifying radical expressions by extracting square roots from the radicand in steps using these rules.
1) The document discusses differentiation techniques such as differentiating polynomials, composite functions, and logarithmic and exponential functions.
2) It also covers applications of differentiation like finding maxima and minima, rates of change, and using differentiation to approximate small changes.
3) The key concepts covered are the derivative, differentiation rules, the relationship between the derivative and tangents/normals to curves, and using the derivative to solve optimization problems.
The document discusses numerical methods for solving linear algebraic equations. It begins by introducing the general form of a system of n linear equations with n unknowns.
It then describes two classes of solution methods: direct and iterative. Gaussian elimination is presented as a direct method that transforms the system of equations into an upper triangular system which can then be solved using back substitution. Gauss-Seidel iteration is introduced as an iterative method that uses successive approximations to obtain solutions. The document provides examples to illustrate how to apply both Gaussian elimination and Gauss-Seidel iteration to solve systems of linear equations.
This document discusses correlation and regression analysis. It defines correlation as dealing with the association between two or more variables. There are different types of correlation including positive, negative, simple, and multiple. Methods for measuring correlation include scatter diagrams, graphs, and Karl Pearson's coefficient of correlation. Regression analysis develops a statistical model to predict a dependent variable from an independent variable. Regression coefficients and the correlation coefficient can be used to describe the relationship between variables.
Regression and correlation analysis allow researchers to assess relationships between variables. Regression fits a line to two variables that minimizes the sum of squared errors, representing how well the independent variable predicts the dependent variable. Correlation assesses the strength and direction of association, ranging from -1 to 1. R-squared indicates the proportion of variance in the dependent variable explained by the independent variable.
This document introduces correlation and regression. It defines correlation as a measure of the association between two numerical variables and discusses how the Pearson correlation coefficient measures both the direction and strength of the linear relationship between two variables. Regression is introduced as a statistical method for finding the line of best fit for one variable based on another. Simple linear regression finds the line of best fit for one dependent variable based on one independent variable. The document provides steps for performing simple linear regression using a TI-83 graphing calculator.
This document provides an introduction to correlation and regression analysis. It defines correlation as a measure of the association between two variables and regression as using one variable to predict another. The key aspects covered are:
- Calculating correlation using Pearson's correlation coefficient r to measure the strength and direction of association between variables.
- Performing simple linear regression to find the "line of best fit" to predict a dependent variable from an independent variable.
- Using a TI-83 calculator to graphically display scatter plots of data and calculate the regression equation and correlation coefficient.
Correlation by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
The regression coefficients are 0.8 and 0.2.
The coefficient of correlation r is the geometric mean of the regression coefficients, which is:
√(0.8 × 0.2) = 0.4
Therefore, the value of the coefficient of correlation is 0.4.
The document discusses simple linear regression analysis. It provides definitions and formulas for simple linear regression, including that the regression equation is y = a + bx. An example is shown of using the stepwise method to determine if there is a significant relationship between number of absences (x) and grades (y) for students. The analysis finds a significant negative relationship, meaning more absences correlated with lower grades. Formulas are provided for calculating the slope, intercept, and testing significance of the regression model.
Establishing Construct Validity using a Correlation Matrix with Survey DataKen Plummer
This document discusses using a correlation matrix to provide construct-related evidence of validity. It shows high inter-item correlations between items targeting reading efficacy and items targeting math efficacy, demonstrating convergent validity. It also shows low inter-scale correlations between the reading efficacy items and math efficacy items, demonstrating divergent validity by showing the scales measure different constructs as intended.
The document discusses correlation analysis and correlation coefficients. It explains that correlation analysis finds how well data points fit a line of best fit, with perfect correlation being 1. The correlation coefficient r is a measure of the linear relationship between two variables, ranging from -1 to 1. A scattergram plots the variables to show the strength, shape, direction, and outliers of their relationship. The interpretation of r values is provided, with higher absolute values indicating stronger correlation. Steps for correlation analysis and an example are also presented.
This document provides an overview of key concepts in business statistics including correlation, scatter plots, the correlation coefficient, linear regression, and calculating the regression line. It defines correlation as a linear association between two variables and explains how scatter plots can show positive, negative, or no relationship. The correlation coefficient r measures the strength and direction of a relationship between -1 and 1. Linear regression finds the linear relationship between a dependent and independent variable to predict future values using the regression line equation y=a+bx.
This was a presentation I gave to my firm's internal CPE in December 2012. It related to correlation and simple regression models and how we can utilize these statistics in both income and market approaches.
1) The document discusses risk-return analysis and the efficient frontier. It introduces the Capital Market Line (CML), which shows superior portfolio combinations when investing in both risky and risk-free assets.
2) The CML is tangent to the efficient frontier at the market portfolio, which offers the highest Sharpe Ratio. The Sharpe Ratio represents excess return per unit of risk.
3) With access to risk-free borrowing and lending, investors are no longer confined to the efficient frontier, but can choose portfolios along the CML based on their individual risk preferences.
Correlational research examines relationships between two or more variables without manipulating them. It investigates whether changes in one variable are associated with changes in another. Correlational studies describe relationships using a correlation coefficient and can be used to predict scores on one variable based on scores on another. Common correlational techniques include scatterplots, regression analysis, and factor analysis. Threats to internal validity like subject characteristics, mortality, history, and instrumentation must be controlled.
- Regression analysis determines the relationship between two quantitative variables and derives an equation to describe their relationship.
- A scatter plot is used to display the relationship between the independent and dependent variables and determine if it is linear or nonlinear.
- The method of least squares is used to fit a linear regression line that minimizes the sum of the squared residuals between observed and predicted values of the dependent variable.
- The regression equation can be used to predict values of the dependent variable for given values of the independent variable.
The document discusses simple linear regression analysis. It provides definitions and formulas for simple linear regression, including that the regression equation is y = a + bx. An example is shown of using the stepwise method to determine if there is a significant relationship between number of absences (x) and grades (y) for students. The analysis finds a significant negative relationship, meaning more absences correlated with lower grades. The document also discusses using the regression equation to predict outcomes and the significance test for the slope of the regression line.
1) The document provides an overview of key concepts in probability and statistics, including random variables, probability distributions, and characteristics of distributions such as expected value and variance.
2) It defines key probability terms such as population, sample, mutually exclusive events, independent events, and exhaustive events. It also covers how to calculate the probability of single and multiple events.
3) The document distinguishes between discrete and continuous random variables and probability distributions. It explains how probability distributions associate probabilities with individual outcomes for discrete variables but use probability density functions to provide probabilities over intervals for continuous variables.
This document provides an overview of the banking system in Nepal. It begins by explaining the purpose of banks and then outlines the different types of banks in Nepal, including central banks, commercial banks, development banks, finance companies, and microcredit development banks. A total of 30 commercial banks, 82 development banks, 48 finance companies, and 37 microcredit development banks currently operate in Nepal. The document also includes organizational charts of the banking hierarchy and describes some of the key roles and services provided by banks, such as accepting deposits, lending money, remittances, safe deposit services, and capital market activities.
The document discusses different types of two-sample hypothesis tests, including tests comparing two population means of independent samples, two population proportions, and paired or dependent samples. It provides examples and step-by-step explanations of how to conduct two-sample t-tests, z-tests, and tests of proportions. Key points covered include determining the appropriate test statistic based on sample size and characteristics, stating the null and alternative hypotheses, test criteria, and decisions rules.
1. The document discusses sampling methods and the central limit theorem. It describes various probability sampling methods like simple random sampling, systematic random sampling, and stratified random sampling.
2. It defines the sampling distribution of the sample mean and explains that according to the central limit theorem, the sampling distribution will follow a normal distribution as long as the sample size is large.
3. The mean of the sampling distribution is equal to the population mean, and its variance is equal to the population variance divided by the sample size. This allows probabilities to be determined about a sample mean falling within a certain range.
The document discusses correlation and regression analysis. It describes correlation as dealing with the association between two or more variables. There are different types of correlation including positive, negative, simple, and multiple correlations. Methods for measuring correlation include graphical, diagrammatic, and algebraic methods such as Karl Pearson's coefficient of correlation and Spearman's rank correlation coefficient. Karl Pearson's coefficient of correlation is defined using formulas involving the summation of deviations between variables. The assumptions, properties, merits, and limitations of Karl Pearson's coefficient are also outlined. Examples are provided to demonstrate calculating correlation coefficients.
The document discusses correlation and regression analysis. It describes correlation as dealing with the association between two or more variables. There are different types of correlation including positive, negative, simple, and multiple correlations. Methods for measuring correlation include graphical, diagrammatic, and algebraic methods such as Karl Pearson's coefficient of correlation and Spearman's rank correlation coefficient. Karl Pearson's coefficient formula is provided and its assumptions and properties are described.
The document is a maths project report for class 12th student Tabrez Khan on the topic of determinants. It contains definitions and properties of determinants of order 1, 2 and 3 matrices. It discusses minors, cofactors and applications of determinants like solving systems of linear equations using Cramer's rule. It also contains examples of evaluating determinants and applying properties of determinants to simplify expressions.
This document discusses different methods of calculating correlation coefficients and regression equations between two variables X and Y, including:
- Direct and short cut methods to calculate the Pearson correlation coefficient r
- Spearman's rank correlation coefficient rs
- Regression equations to predict X from Y and Y from X using the least squares method
- The relationship between the correlation coefficient r and the regression coefficients
- The document discusses determinants of square matrices, including how to calculate the determinant of matrices of various orders, properties of determinants, and some applications of determinants.
- Key concepts covered include minors, cofactors, expanding determinants in terms of minors and cofactors, properties such as how determinants change with row/column operations, and using determinants to solve systems of linear equations.
- Examples are provided to demonstrate calculating determinants and using properties to simplify or prove identities about determinants.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
1. Indices involve rules for exponents like xa+b = xaxb and (xa)b = xab. Solving exponential equations uses these rules.
2. Graph transformations include translations, stretches, reflections, and asymptotes. Translations replace x with (x-a) and y with (y-b).
3. Sequences are functions with successive terms defined by a rule. Geometric sequences multiply successive terms by a constant ratio while arithmetic sequences add a constant.
This document discusses various measures of dispersion including range, interquartile range, quartile deviation, mean deviation, and standard deviation. It provides formulas and procedures for calculating each measure along with examples of calculating dispersion measures from data sets. Key measures discussed include range, interquartile range, quartile deviation, mean deviation, and standard deviation. Procedures are outlined for determining class intervals and cumulative frequencies needed to calculate certain dispersion measures.
The document discusses regression analysis, including definitions, uses, calculating regression equations from data, graphing regression lines, the standard error of estimate, and limitations. Regression analysis is a statistical technique used to understand the relationship between variables and allow for predictions. The document provides examples of calculating regression equations from various data sets and determining the standard error of estimate.
Here are the steps to solve this problem:
1) The given equation is: y = x2 + 2x
2) To complete the table, we need to calculate the value of y when x = -3 and when x = 1
3) When x = -3:
y = (-3)2 + 2(-3)
y = 9 - 6
y = 3
4) When x = 1:
y = 12 + 2(1)
y = 1 + 2
y = 3
So the completed table is:
Table 1
x -3 1
y 3 3
(b) Sketch the graph of y = x2 + 2
Gauss Jorden and Gauss Elimination method.pptxAHSANMEHBOOB12
Gauss elimination and Gauss-Jordan elimination are methods for solving systems of linear equations. Gauss elimination puts the matrix of coefficients into row-echelon form using elementary row operations, while Gauss-Jordan elimination further reduces the matrix to reduced row-echelon form. These methods can also be used to find the rank of a matrix, calculate the determinant, and compute the inverse of an invertible square matrix. Examples demonstrate applying the methods to solve systems of 3 equations with 3 unknowns.
Here are the key steps to solve this problem:
1. Calculate the coefficient of variation for each factory:
Factory A:
Coefficient of variation = Standard deviation/Average
= 6.5/19.7 = 33%
Factory B:
Coefficient of variation = Standard deviation/Average
= 8.64/21 = 41%
2. The factory with the lower coefficient of variation has more consistent profits.
Factory A has a coefficient of variation of 33%, lower than Factory B's 41%. Therefore, the profits of Factory A are more consistent.
The coefficient of variation allows us to compare the extent of variability in relation to the average of the data set. A lower coefficient
This document discusses various measures of dispersion used to quantify the spread or variability in data. It defines absolute and relative measures of dispersion and describes key measures such as range, interquartile range, mean deviation, standard deviation, and coefficient of variation. Examples are provided to demonstrate calculating these measures from data sets. The standard deviation is identified as the most common measure of dispersion and its properties are outlined.
This document provides information on mathematical concepts and formulas relevant to economics, including:
- Exponential functions such as y=ex and their graphs showing exponential growth and decay
- Quadratic functions of the form y=ax2+bx+c and total cost functions
- Differentiation rules for common functions like exponentials, logarithms, and the product, quotient and chain rules
- Integration basics and formulas for integrating common functions
- Concepts like inverse functions, the mean, variance and standard deviation in statistics
- Information is also provided on fractions, ratios, percentages, and algebraic rules involving exponents, logarithms and sigma notation.
The document describes the Jacobi iterative method for solving systems of linear equations. It begins with an initial estimate for the solution variables, inserts them into the equations to get updated estimates, and repeats this process iteratively until the estimates converge to the desired solution. As an example, it applies the method to a set of 3 equations in 3 unknowns, showing the estimates after each iteration getting progressively closer to the exact solution obtained using Gaussian elimination. A Fortran program implementing the Jacobi method is also presented.
Determinants provide a scalar quantity associated with square matrices. There are several properties of determinants, including that the determinant of a matrix does not change if rows or columns are interchanged. The determinant can be expressed as the sum of the products of each element and its corresponding cofactor. Examples show how to evaluate determinants by expanding along rows or columns and applying properties such as identical rows resulting in a determinant of zero.
Determinants provide a scalar quantity associated with square matrices. There are several properties of determinants, including that the determinant of a matrix does not change if rows or columns are interchanged. The determinant can be expressed as the sum of the products of each element and its corresponding cofactor. Examples show how to evaluate determinants by expanding along rows or columns and applying properties such as identical rows resulting in a determinant of zero.
Unitedworld School of Business has campuses located in Ahmadabad and Kolkata, India. The Ahmadabad campus is located at A/907, Uvarsad, Gandhinagar 382422. The Ahmedabad corporate office is located at 407, 4th Floor, Zodiac Square, Opp. Gurudwara, Sarkhej-Gandhinagar Highway, Bodakdev, Ahmedabad-380054, Gujarat. The Kolkata campus is located at Infinity Benchmark Tower, 10th Floor, Plot – G1, Block – EP& GP, Sec -V, Salt Lake, Kolkata 700091.
The programme is designed to render the students with a holistic education and deeper understanding of business tactics of global magnitude. We stress on conducting interactive study sessions which give birth to rational ideas and develop innovative thinking, live cases, e-learning and positive influence of our renowned guest speakers facilitates students’ abilities and aspirations. http://www.unitedworld.in/school-of-business/
The programme is designed to render the students with a holistic education and deeper understanding of business tactics of global magnitude. We stress on conducting interactive study sessions which give birth to rational ideas and develop innovative thinking, live cases, e-learning and positive influence of our renowned guest speakers facilitates students’ abilities and aspirations. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
UWSB offers Post Graduate courses at its Ahmedabad and Kolkata campuses. Ahmedabad campus, approved by AICTE, offers Post Graduate Diploma in Management (PGDM). Kolkata campus offers Post Graduate Programme in Management along with an option of AICTE approved Post Graduate Diploma in Management (PGDM) and/or MBA. Our programmes aim to create value-instilled potential leaders by incorporating higher-management functions. http://www.unitedworld.in/school-of-business/
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
3. It deals with association between two or
more variables
Correlation analysis deals with
covariation between two or more
variables
Types
1. Positive or negative
Simple or multiple
Linear or non-linear
4. Methods of Measuring correlation
1. Graphic Method
2. Diagramatic Method- Scatter Diagram
3. Algebraic method
a. Karl Pearson’s Coefficient of correlation
b. Spearman’s Rank Co-efficient Correlation
c. Coefficient of Concurrent deviations
d. Least Squares Method
5. Karl Pearson’s Coefficient of Correlation
Σ dx dy
γ ( Gamma) = -------------------------
√ Σ dx2
Σ dy2
Σ dx dy
= -------------------------
N σxσy
dx = x-xbar
dy = y- ybar
dx dy = sum of products of deviations from respective
arithmetic means of both series
6. Karl Pearson’s Coefficient of Correlation
After calculating assumed or working mean Ax & Ay
Σ dx dy – (Σ dx) x( Σ dy)
γ ( Gamma) = --------------------------------
√ [ NΣ dx2
- (Σ dx)2
x [Σ Ndy2
- (Σ dy)2
]
Σ dx dy = total of products of deviation from assumed
means of x and y series
Σ dx = total of deviations of x series
Σ dy = total of deviations of y series
Σ dx2
= total of squared deviations of x series
Σ dy2
= total of squared deviations of y series
N= No. of items ( no. of paired items
7. Karl Pearson’s Coefficient of Correlation
After calculating assumed or working mean Ax &
Ay
Σ dx x Σ dy
Σ dx dy - ----------------
N
γ ( Gamma) = -------------------------
(Σ dx)2
(Σ dy)2
√ [ Σ dx2
- --------- ] x [ Σ dy2
- ------------]
N N
8. Assumptions of Karl Pearson’s Coefficient of Correlation
1. Linear relationship exists between the variables
Properties of Karl Pearson’s Coefficient of Correlation
1.value lies between +1 & - 1
2.Zero means no correlation
3.γ ( Gamma) = √ bxy X byx
Where bxy X byx are two regression coefficicent
Merit
Convenient for accurate interpretation as it gives degree &
direction of relationship between two variables
9. Limitations
1. Assumes linear relationship , even though it
may not be
2. Method & process of calculation is difficult &
time consuming
3. Affected by extreme values in distribution
10. Probable Error of Karl Pearson’s Coefficient of
Correlation
1- γ2
Probable Error of γ ( Gamma) = 0.6745 --------
√ N
11. Q7.Calculate coefficient of correlation for following data
X
65 63 67 64 68 62 70 66 68 67 69 71
Y 68 66 68 65 69 66 68 65 71 67 68 70
Ans Σ dx dy
γ ( Gamma) = -------------------------
√ Σ dx2
Σ dy2
Σ dx dy
= -------------------
N σxσy
15. Rank Correlation : some times variable are not
quantitative in nature but can be arranged in
serial order.
Specially while eading with attributes like –
honesty , beauty , character , morality etc
To deal with such situations , Charles Edward
Spearman , in 1904 developed a formula for
obtaining correlation coefficient between ranks
of n individuals in two attributes under study , or
ranks given by two or three judges
16. Rank coefficient of correlation
6Σ d2
ρ (rho) = 1 - -------------------
N3
-N
6Σ d2
ρ (rho) = 1 - -------------------
N(N2
-1)
Σ d2
= total of squared difference
N = number of items
17. Q9. ten competitors in a cooking competition are ranked
by three judges in the following way .by using rank
coorelation method find out which pair of judges have
nearest approach
P Q R
1 1 3 6
2 6 5 4
3 5 8 9
4 10 4 8
5 3 7 1
6 2 10 2
7 4 2 3
8 9 1 10
9 7 6 5
10 8 9 7
19. Regression Analysis is the process of
developing a statistical model which is used
to predict the value of a dependant variable
by an independent variable
Application
Advertising v/s sales revenue
First used by Sir Francis Gatton in 1877 for
study of height of sons w.r.t height of fathers
20. Regression Analysis – going back or to revert to
the former condition or return
Refers to functional relationship between x & y
and estimates of value of depebdent variable y
for given values of independeny variable x
Relationship between income of employees and
savings
Regression coefficients can be used to calculate ,
correlation coeffecient.γ ( Gamma) = √ bxy X
byx
21. Types of Regression
1. Simple & Multiple Regression
2. Total or Partial
3. Linear / Non-linear
Methods of Regression Analysis
1. Scatter Diagram
2. Regression Equations
3. Regression Lines
22. Line of Regression of y on x y= a + bx
Coefficient b is slope of line of regression of y on x.
It represents the increment in the value of the dependent
variable y for a unit change in the value of independent
variable x i.e. rate of change of y w.r.t. x. It is written as byx
Regression coefficients/ coefficient of regression of y on x
Σ( x- x-
) (y- y-
) σdx dy
byx= ------------------= ----------
Σ (x- x-
)2
Σ dx2
i.e. Equation of Line of Regression of x on y
y-y-
= byx (x-x-
)
23. Line of Regression of x on y x= a + by
Coefficient b is slope of line of regression of x on y.
It represents the increment in the value of the dependent
variable x for a unit change in the value of independent
variabley i.e. rate of change of x w.r.t. y. It is written as bxy
Regression coefficients/ coefficient of regression of x on y
Σ( x- x-
) (y- y-
) σdx dy
bxy= ------------------= ----------
Σ (y- y-
)2
Σ dy2
i.e. Equation of Line of Regression of x on y
y-y-
= bxy (x-x-
)
24. Q2.From the data given below find
two regression coefficients
two regression equations
coefficient of correlation between marks in
Economics & statistics
most likely marks in statistics when marks in
Economics are 30
let marks in Economics be x and that in statistics
be y
Marks in Eco 25 28 35 32 31 36 29 38 34 32
Marks in Stat 43 46 49 41 36 32 31 30 33 39
25. Marks in
Eco
25 28 35 32 31 36 29 38 34 32 Σx 320 x-
32
Marks in
Stat
43 46 49 41 36 32 31 30 33 39 Σy 380 y-
38
28. Regression coefficients / coefficient of regression of y on
x =
Σ( x- x-
) (y- y-
) Σdx dy -93
byx= ------------------= ---------- = --------= -0.6643
Σ (x- x
-
)2
Σ dx2
140
regression of y on x
y-y-
= byx (x-x-
)
y-38 = -0.6643(x-32)
y -38= -0.6643x+0.6643*32
y = -0.6643x+38+0.6643*32
y = -0.6643x+38+21.2576
y = -0.6643x+59.2576
29. coefficient of regression of x on y
Σ( x- x-
) (y- y-
) Σdx dy -93
bxy= ------------------= ------- = ------ = -0.2337
Σ (y- y-
)2
Σ dy2
398
Equation of regression of x on y
x-x-
= bxy (y-y-
)
x-32 = -0.2337(y-38)
= - 0.2337 y +0.2337 *38
= -0.2337y + 8.8806
x = -0.2337y +32 + 8.8806
x = -0.2337y +40.8806
30. Correlation Coefficient = √ bxy *byx
= √ -0.2337 *-0.6643 = √ 0.1552 = -0.394
Since byx & bxy are both negative
31. In order to estimate most likely marks in statistics
(y) when Economics (x) are 30 , we shall use the
line regression of y x viz
The required estimate is given by
y = -0.6643* 30+59.2576= -19.929+59.2576 =
=39.3286
32. Sum of Squares- x&y
(Σx )*(Σy)
SSxy=Σ( x-x-
)(y-y-
)= Σdxdy = Σxy - --------------
n
Sum of Squares xx
(Σx )
SSxx = Σ ( x-x-
)2
= Σdx
2
=Σx2
- -------------
n
34. Sum of Squares- x&y
(Σx )*(Σy)
SSxy=Σ( x-x-
)(y-y-
)= Σdxdy = Σxy - --------------
n
Sum of Squares xx
(Σx )
SSxx = Σ ( x-x-
)2
= Σdx
2
=Σx2
- -------------
n
35. SSxy Σdxdy
b = ------------=---------
SSxx Σdx
2
y=a+bx
Σ y= Σ a+b Σ x
Σ y= n* a+b Σ x
n* a = b Σ x - Σ y
Σ y - bΣ x Σ y bΣ x
a = ----------- = ------- - -------
n n n
39. SSxy 6565
b = ------------- = ----------------= 19.0704
SSxx 344.25
y=a+bx
Σ y= Σ a+b Σ x
Σ y= n* a+b Σ x
n* a = b Σ x - Σ y
Σ y - bΣ x Σ y bΣ x 13060 19.0704*1221
a = ----------- = ------- - ------- = ---------- - --------------
n n n 12 12
= - 852.08
40. equation for simple regression line
y= a+bx
y= -852.08+ 19.0704 x
for regression of y on x
41. For testing the Fit
yi = yi- value of y –recorded value in the given data
y-
= Mean ( Average )of y
y^ = Predicted Values from regression line
deviation = (yi- y-
) = difference in actual value of y from
mean
Residuals = (yi- y^)= gap ( error , difference ) between
actual value of y & predicted value calculated from
regression line
Deviation of predicted value from mean = (y^- y-
)
a = intercept on y -axis
b= slope of regression line
42. total sum of squares = SST = Σ (yi-y-
)2
regression sum of squares = SSR = Σ (y^- y-
)2
Error sum of squares = SSE = Σ (yi-y^)2
SSR
coefficient of determination = γ2= -------
SST
43. SSE
Standard Error of Estimate =Syx= √----------------
n-2
In order to to determine whether a significant
linear relationship exists between independent
variable x and dependent variable y we perform
whether population slope is zero
b - β
t= ----------
Sb
Syx
Sb = Standard error of b= -----------
√ SSxx
44. H0:Slope of thr regression line is zero
H1-Slope of the regression line is not zero
45. SSE
Syx= Standard Error of Estimate =√--------
n-2
Σ (yi-y^)2 13769.21
=√ -------- = √------------ = √1376.92 = 37.1068
n-2 10-2
(Σx )2 (1221)2
SSxx = Σx2 - -------- = 124581 - -------= 344.25
n 12
Syx
Sb = Standard error of b= -----------
√ SSxx
46. Syx
Sb = Standard error of b= -----------
√ SSxx
b- β 19.07-0
t= ---------- = ------------------------------- = 9.53
Sb 37.1068/( √344.25)
As calculated value of t is more than table
value of t for 12-2 = 10 degrees of freedom
Null hypothesis is rejected
47. Coefficient of Determination Definition
The Coefficient of Determination, also known as R
Squared, is interpreted as the goodness of fit of a
regression.
The higher the coefficient of determination, the
better the variance that the dependent variable is
explained by the independent variable.
The coefficient of determination is the overall
measure of the usefulness of a regression.
For example,r2
is given at 0.95. This means that the
variation in the regression is 95% explained by the
independent variable. That is a good regression.
48. The Coefficient of Determination can be
calculated as the Regression sum of squares,
SSR, divided by the total sum of squares, SST
SSR
Coefficient of Determination γ2
= ---------- SST
49. Campus Overview
907/A Uvarshad,
Gandhinagar
Highway, Ahmedabad –
382422.
Ahmedabad Kolkata
Infinity Benchmark,
10th
Floor, Plot G1,
Block EP & GP,
Sector V, Salt-Lake,
Kolkata – 700091.
Mumbai
Goldline Business Centre
Linkway Estate,
Next to Chincholi Fire
Brigade, Malad (West),
Mumbai – 400 064.