The document provides solutions to calculating various statistical measures - arithmetic mean, median, mode, harmonic mean, and geometric mean - for 5 sets of data. For each data set, the document calculates the measures using the relevant formulas. The statistical measures included arithmetic mean, median, mode, harmonic mean, and geometric mean. Formulas are provided for calculating each measure.
The work deals finite frequency H∞ control design for continuous time nonlinear systems, we provide sufficient conditions, ensuring that the closed-loop model is stable. Simulations will be gifted to show level of attenuation that a H∞ lower can be by our method obtained developed where further comparison.
Vector mechanics for engineers statics 7th chapter 5 Nahla Hazem
This problem involves locating the centroid of a plane area shown in multiple problems. The solution provides the area (A) of each section, the x and y coordinates, the moment of area about the x-axis (xA), and the moment of area about the y-axis (yA). It then calculates the x and y coordinates of the centroid by taking the sum of the xA and yA values, respectively, and dividing each by the total area.
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.
This document contains solutions to problems from Chapter 5 of an engineering textbook. Problem 5-3 calculates the torque and allowable twist in a torsion bar made of two springs in parallel. Problem 5-12 calculates the maximum deflection and stress in a beam loaded by two point loads. Problem 5-19 involves selecting the appropriate cross-sectional dimensions to achieve a required stiffness for a beam of given length.
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.
This document provides equations and calculations for determining the mean cycles to failure (x-bar) and standard deviation of cycles to failure (s_x) for a sample of fatigue test data. The sample data consists of the number of cycles to failure (x) and applied force (f) for 10 tests. The mean x-bar is calculated as the sum of the product of f and x divided by the sum of f, which equals 122.9 kcycles. The standard deviation s_x is calculated using the variance formula, which equals 30.3 kcycles.
This document provides calculations for determining the specifications of compression springs. It analyzes music wire, phosphor bronze, and stainless steel springs given various dimensional parameters. Equations are used to calculate properties like spring rate, shear stress, yield point, and critical buckling length. The summaries indicate some designs are not solid-safe due to exceeding the shear yield strength, and suggest adjusting the free length to achieve a solid-safe design.
The work deals finite frequency H∞ control design for continuous time nonlinear systems, we provide sufficient conditions, ensuring that the closed-loop model is stable. Simulations will be gifted to show level of attenuation that a H∞ lower can be by our method obtained developed where further comparison.
Vector mechanics for engineers statics 7th chapter 5 Nahla Hazem
This problem involves locating the centroid of a plane area shown in multiple problems. The solution provides the area (A) of each section, the x and y coordinates, the moment of area about the x-axis (xA), and the moment of area about the y-axis (yA). It then calculates the x and y coordinates of the centroid by taking the sum of the xA and yA values, respectively, and dividing each by the total area.
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.
This document contains solutions to problems from Chapter 5 of an engineering textbook. Problem 5-3 calculates the torque and allowable twist in a torsion bar made of two springs in parallel. Problem 5-12 calculates the maximum deflection and stress in a beam loaded by two point loads. Problem 5-19 involves selecting the appropriate cross-sectional dimensions to achieve a required stiffness for a beam of given length.
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.
This document provides equations and calculations for determining the mean cycles to failure (x-bar) and standard deviation of cycles to failure (s_x) for a sample of fatigue test data. The sample data consists of the number of cycles to failure (x) and applied force (f) for 10 tests. The mean x-bar is calculated as the sum of the product of f and x divided by the sum of f, which equals 122.9 kcycles. The standard deviation s_x is calculated using the variance formula, which equals 30.3 kcycles.
This document provides calculations for determining the specifications of compression springs. It analyzes music wire, phosphor bronze, and stainless steel springs given various dimensional parameters. Equations are used to calculate properties like spring rate, shear stress, yield point, and critical buckling length. The summaries indicate some designs are not solid-safe due to exceeding the shear yield strength, and suggest adjusting the free length to achieve a solid-safe design.
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 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.
There are several numerical methods described for approximating derivatives and integrals:
1) Forward difference formula approximates the derivative as the slope of the secant line through two nearby points.
2) Three-point formulas approximate the derivative using the slopes of secant lines through three evenly spaced points, reducing the error term to O(h^2).
3) Trapezoidal rule approximates the integral of a function as the area of trapezoids formed between the function graph and the x-axis at evenly spaced points, with error term O(h^2).
This document discusses numerical integration methods for calculating ship geometrical properties. It introduces the Trapezoidal rule, Simpson's 1st rule, and Simpson's 2nd rule for numerical integration when the ship's shape cannot be represented by a mathematical equation. It then provides examples of applying Simpson's 1st rule to calculate properties like waterplane area, sectional area, submerged volume, and the longitudinal center of floatation (LCF). The document explains the calculation steps and provides generalized Simpson's equations for these examples.
(1) The document presents several bending problems involving the determination of stress at various points on beams subjected to bending couples.
(2) Solutions are provided that calculate the stress based on the couple magnitude, beam geometry, and bending axis.
(3) Stresses are determined at points A, B, C, D, and E on different beams and range from -136 MPa to 91.7 MPa depending on the couple magnitude and distance from the beam's neutral axis.
The document presents data on the math and English scores of 5 students. It asks to compute the Spearman Rank Correlation Coefficient and Pearson's Correlation Coefficient between the scores. The Spearman Rank Correlation Coefficient is -0.96, indicating a very strong negative correlation between ranks. Pearson's Correlation Coefficient is -0.99, also indicating a very strong negative correlation. The document also provides price and age data for 8 cars and asks to find the regression line and make predictions based on the line.
This document discusses methods for identifying the source node of information spread in networks based on the observed spread over time. It begins by introducing epidemic models like SIS and SI for modeling information spread over networks. It then discusses maximum likelihood methods for identifying the source node on regular tree networks based on the observed subgraph. The accuracy of these methods increases with network size and degree. Extensions to other network structures and SIR models are also proposed. Overall, the document reviews mathematical models and algorithms for source identification in networks from limited observations of information spread.
- Hiroaki Shiokawa's research interests include graph mining, network analysis, and efficient algorithms. He was previously employed at NTT from 2011 to 2015.
- His current research focuses on developing clustering algorithms for large-scale networks and evaluating their performance on real-world network datasets.
- He has published highly cited papers in top data mining and network science conferences such as KDD, CIKM, and WSDM.
Hand book of Howard Anton calculus exercises 8th editionPriSim
The document contains the table of contents for a calculus textbook. It lists 17 chapters covering topics such as functions, limits, derivatives, integrals, vector calculus, and applications of calculus. It also includes 6 appendices reviewing concepts in real numbers, trigonometry, coordinate planes, and polynomial equations.
The document contains solutions to multiple problems involving calculating stresses in beams subjected to bending moments. For problem 4.1, it is determined that the stress at point A is 61.14 MPa (compressive) and at point B is 91.7 MPa (tensile). For problem 4.2, the stresses at points A and B are calculated to be -5.31 GPa and 3.365 GPa, respectively. Problem 4.3 involves calculating the largest bending moment that can be applied to an aluminum beam before yielding, which is determined to be 5.283 KN.m.
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 presents an overview of optimization algorithms on Riemannian manifolds. It begins by introducing concepts such as vector transport and retraction mappings that are used to generalize algorithms from Euclidean spaces to manifolds. It then summarizes several classical optimization methods including gradient descent, conjugate gradient, and variants of quasi-Newton methods adapted to the Riemannian setting using these geometric concepts. The convergence of the Fletcher-Reeves method is analyzed under standard assumptions on the objective function. Overall, the document provides a conceptual and mathematical foundation for optimization on manifolds.
Complete presentation On Regression Analysis.
Proved By Three methods, Least Square Method, Deviation method by assumed mean, Deviation method By Arithmetic mean.
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.
Kushal should draw the following conclusion from the sample data:
The coefficient of correlation (R) between size (x) and price (y) is 0.6725. The coefficient of determination (R2) is 0.452. This indicates that the relationship between size and price is moderately correlated.
1) Four positive charges are located at the corners of a square in the xy-plane. A fifth positive charge is located 8cm from the others. The total force on the fifth charge is calculated to be 4.0x10-4 N directed along the z-axis.
2) Two charges of Q1 coulombs are located at z=±1. For a third charge Q2 to produce zero total electric field at (0,1,0), Q2 must lie along the y-axis at y=1±21/4|Q2|/Q1, where the sign depends on the sign of Q2.
3) The total force on a 50nC charge
This document discusses numerical methods for solving engineering problems. It begins by introducing numerical methods and their use in solving problems that cannot be solved exactly or are intractable. It then provides an example of using numerical methods to solve a problem involving a bascule bridge where the trunnion got stuck in the hub after cooling. The document walks through modeling the problem, estimating the contraction using integration and the thermal expansion coefficient, and proposes immersing the trunnion in liquid nitrogen as a solution to achieve greater contraction.
Este conto de fadas conta a história de uma princesa que foi expulsa de casa por seu pai por dizer que o amava tanto quanto a comida ama o sal. Ela se torna cozinheira em outro reino e acaba se casando com o príncipe. Seu pai é convidado para o casamento e só come quando reconhece sua filha, entendendo o significado de suas palavras originais.
Los retenedores ayudan a mantener los resultados del tratamiento ortodóncico al retener los movimientos de los dientes. Existen retenedores extraíbles hechos de plástico o metal que deben quitarse para comer y cepillarse, y retenedores fijos cementados a los dientes. El proceso de fabricar un retenedor extraíble implica tomar una impresión dental, hacer un modelo de trabajo, aislarlo y moldear acrílico sobre este. Se recomienda usar retenedores de por vida para mantener una correct
Este documento describe los diferentes tipos de juicios, que son la forma en que se establecen relaciones entre conceptos. Explica que hay 14 combinaciones de formas simples de juicios, incluyendo juicios singulares, particulares y universales. Detalla cada tipo de juicio, como la prófasis, discordancia, implicación e incompatibilidad.
Ahmed Fakhrany is an Egyptian national who currently works as an IT Network Administrator at EUN. He has a BSC in Electronic Engineering and holds several Cisco certifications including CCNA, CCNP, and CCNA voice CME. He has over 8 years of experience installing, repairing, and managing networks for governmental Egyptian universities.
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 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.
There are several numerical methods described for approximating derivatives and integrals:
1) Forward difference formula approximates the derivative as the slope of the secant line through two nearby points.
2) Three-point formulas approximate the derivative using the slopes of secant lines through three evenly spaced points, reducing the error term to O(h^2).
3) Trapezoidal rule approximates the integral of a function as the area of trapezoids formed between the function graph and the x-axis at evenly spaced points, with error term O(h^2).
This document discusses numerical integration methods for calculating ship geometrical properties. It introduces the Trapezoidal rule, Simpson's 1st rule, and Simpson's 2nd rule for numerical integration when the ship's shape cannot be represented by a mathematical equation. It then provides examples of applying Simpson's 1st rule to calculate properties like waterplane area, sectional area, submerged volume, and the longitudinal center of floatation (LCF). The document explains the calculation steps and provides generalized Simpson's equations for these examples.
(1) The document presents several bending problems involving the determination of stress at various points on beams subjected to bending couples.
(2) Solutions are provided that calculate the stress based on the couple magnitude, beam geometry, and bending axis.
(3) Stresses are determined at points A, B, C, D, and E on different beams and range from -136 MPa to 91.7 MPa depending on the couple magnitude and distance from the beam's neutral axis.
The document presents data on the math and English scores of 5 students. It asks to compute the Spearman Rank Correlation Coefficient and Pearson's Correlation Coefficient between the scores. The Spearman Rank Correlation Coefficient is -0.96, indicating a very strong negative correlation between ranks. Pearson's Correlation Coefficient is -0.99, also indicating a very strong negative correlation. The document also provides price and age data for 8 cars and asks to find the regression line and make predictions based on the line.
This document discusses methods for identifying the source node of information spread in networks based on the observed spread over time. It begins by introducing epidemic models like SIS and SI for modeling information spread over networks. It then discusses maximum likelihood methods for identifying the source node on regular tree networks based on the observed subgraph. The accuracy of these methods increases with network size and degree. Extensions to other network structures and SIR models are also proposed. Overall, the document reviews mathematical models and algorithms for source identification in networks from limited observations of information spread.
- Hiroaki Shiokawa's research interests include graph mining, network analysis, and efficient algorithms. He was previously employed at NTT from 2011 to 2015.
- His current research focuses on developing clustering algorithms for large-scale networks and evaluating their performance on real-world network datasets.
- He has published highly cited papers in top data mining and network science conferences such as KDD, CIKM, and WSDM.
Hand book of Howard Anton calculus exercises 8th editionPriSim
The document contains the table of contents for a calculus textbook. It lists 17 chapters covering topics such as functions, limits, derivatives, integrals, vector calculus, and applications of calculus. It also includes 6 appendices reviewing concepts in real numbers, trigonometry, coordinate planes, and polynomial equations.
The document contains solutions to multiple problems involving calculating stresses in beams subjected to bending moments. For problem 4.1, it is determined that the stress at point A is 61.14 MPa (compressive) and at point B is 91.7 MPa (tensile). For problem 4.2, the stresses at points A and B are calculated to be -5.31 GPa and 3.365 GPa, respectively. Problem 4.3 involves calculating the largest bending moment that can be applied to an aluminum beam before yielding, which is determined to be 5.283 KN.m.
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 presents an overview of optimization algorithms on Riemannian manifolds. It begins by introducing concepts such as vector transport and retraction mappings that are used to generalize algorithms from Euclidean spaces to manifolds. It then summarizes several classical optimization methods including gradient descent, conjugate gradient, and variants of quasi-Newton methods adapted to the Riemannian setting using these geometric concepts. The convergence of the Fletcher-Reeves method is analyzed under standard assumptions on the objective function. Overall, the document provides a conceptual and mathematical foundation for optimization on manifolds.
Complete presentation On Regression Analysis.
Proved By Three methods, Least Square Method, Deviation method by assumed mean, Deviation method By Arithmetic mean.
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.
Kushal should draw the following conclusion from the sample data:
The coefficient of correlation (R) between size (x) and price (y) is 0.6725. The coefficient of determination (R2) is 0.452. This indicates that the relationship between size and price is moderately correlated.
1) Four positive charges are located at the corners of a square in the xy-plane. A fifth positive charge is located 8cm from the others. The total force on the fifth charge is calculated to be 4.0x10-4 N directed along the z-axis.
2) Two charges of Q1 coulombs are located at z=±1. For a third charge Q2 to produce zero total electric field at (0,1,0), Q2 must lie along the y-axis at y=1±21/4|Q2|/Q1, where the sign depends on the sign of Q2.
3) The total force on a 50nC charge
This document discusses numerical methods for solving engineering problems. It begins by introducing numerical methods and their use in solving problems that cannot be solved exactly or are intractable. It then provides an example of using numerical methods to solve a problem involving a bascule bridge where the trunnion got stuck in the hub after cooling. The document walks through modeling the problem, estimating the contraction using integration and the thermal expansion coefficient, and proposes immersing the trunnion in liquid nitrogen as a solution to achieve greater contraction.
Este conto de fadas conta a história de uma princesa que foi expulsa de casa por seu pai por dizer que o amava tanto quanto a comida ama o sal. Ela se torna cozinheira em outro reino e acaba se casando com o príncipe. Seu pai é convidado para o casamento e só come quando reconhece sua filha, entendendo o significado de suas palavras originais.
Los retenedores ayudan a mantener los resultados del tratamiento ortodóncico al retener los movimientos de los dientes. Existen retenedores extraíbles hechos de plástico o metal que deben quitarse para comer y cepillarse, y retenedores fijos cementados a los dientes. El proceso de fabricar un retenedor extraíble implica tomar una impresión dental, hacer un modelo de trabajo, aislarlo y moldear acrílico sobre este. Se recomienda usar retenedores de por vida para mantener una correct
Este documento describe los diferentes tipos de juicios, que son la forma en que se establecen relaciones entre conceptos. Explica que hay 14 combinaciones de formas simples de juicios, incluyendo juicios singulares, particulares y universales. Detalla cada tipo de juicio, como la prófasis, discordancia, implicación e incompatibilidad.
Ahmed Fakhrany is an Egyptian national who currently works as an IT Network Administrator at EUN. He has a BSC in Electronic Engineering and holds several Cisco certifications including CCNA, CCNP, and CCNA voice CME. He has over 8 years of experience installing, repairing, and managing networks for governmental Egyptian universities.
Este documento presenta información sobre cómo elaborar un protocolo de investigación. Explica los pasos para plantear un problema de investigación, incluyendo la descripción del problema, los elementos del problema, y la formulación del problema. También cubre cómo definir los objetivos de la investigación, tanto el objetivo general como los objetivos específicos. El propósito es guiar a los estudiantes en el proceso de desarrollar un protocolo de investigación válido y bien estructurado.
Este documento presenta una introducción general sobre la importancia de integrar los contenidos de metodología y estadística en las carreras universitarias, ya que tradicionalmente se imparten de forma independiente, lo que dificulta que los estudiantes adquieran una visión completa sobre cómo realizar investigaciones cuantitativas. También señala que esto ha provocado enormes dificultades para los estudiantes cuando deben formular y conducir proyectos de investigación.
1. El documento ofrece instrucciones para construir el perfil de investigación, incluyendo cómo identificar y definir las variables del problema, establecer el tema, describir el problema y delimitar el alcance, objetivos y factibilidad.
2. Se especifica que el perfil debe incluir una introducción, el problema, la justificación y el alcance, objetivos, ámbitos geográfico e institucional, y factibilidad teórica y práctica.
3. Los objetivos deben expresarse claramente y ser consistentes, incluyendo un objetivo general y
FEF Congratulates President Elect Rodrigo DuterteFEF Philippines
The Foundation for Economic Freedom (FEF) congratulates President-elect Rodrigo Duterte for his unmistakable mandate to lead the Philippines for the next six years. FEF supports Duterte's proposed reforms to remove foreign ownership restrictions, improve competition, investment climate, peace and order, reduce bureaucratic red tape, reform the tax system, and improve infrastructure. FEF also supports Duterte's program to make growth inclusive by focusing on education, rural development, and basic services for the poor, and urges him to end labor contractualization only in the context of total labor market reform.
Route optimization algorithm are the mathematical formula that solve routing problems..
Some types of routing:
1) Vehicle Routing Problem (VRP)
2) Traveling Salesman Problem (TSP)
3) Ant Colony Optimization (ACO)
This presentation covers the topic of access control in software. Access control is an essential part of every software application that manages data of any value. However, access control is also complex and hard to get right, both from a development and management point of view.
In this presentation, we first explore the concept and goals of access control in general. We then discuss the different models that exist in practice and in literature to reason about access control. We then investigate different approaches of how to enforce access control in an application. Overall, this sessions aims to provide deeper insights into access control in order to better reason about it and implement it correctly and efficiently.
The document discusses the geometric mean and how to calculate it from data. It provides examples of calculating the geometric mean from individual observations, discrete series, and continuous series. For individual observations, the geometric mean is calculated as the nth root of the product of the values. For series with frequencies, the geometric mean is calculated as the antilog of the sum of the logarithms of the values times their frequencies divided by the total number of values.
The document contains data arranged in tables with columns for variables x, y, f, x^2, etc. It discusses calculating means, standard deviations, and fitting distributions such as normal and lognormal to the data. It also contains examples of using the method of least squares to fit linear and quadratic regression models to data.
Numerical Methods: Solution of system of equationsNikolai Priezjev
This document provides information about solving systems of equations using LU decomposition. It begins with an example system of equations and shows the steps to decompose the coefficient matrix [A] into lower [L] and upper [U] triangular matrices. It then explains that solving the original system involves first solving [L][Z]=[B] for [Z], then [U][X]=[Z] for [X]. The document provides an example using a 4x4 matrix to decompose it into [L] and [U], then uses the matrices to solve the system.
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 provides material properties data from tables for various steels and metals. It includes yield strengths, ultimate tensile strengths, ductility values, and stiffness for different materials. Equations are also provided to calculate properties like specific strength and Poisson's ratio from the data. Graphs are plotted showing stress-strain curves and the relationship between yield strength and strain for one material.
This document contains data and calculations related to linear regression analysis. It includes regression equations, calculations of mean and standard deviation, and use of Cramer's rule to determine regression coefficients from sample data. Regression lines are fitted to several data sets to determine the relationships between variables.
This document provides examples and explanations of laws of indices. It includes expressing numbers in index form, writing numbers in index notation, evaluating expressions using laws of indices, and simplifying combinations of indices. Examples range from single term expressions to more complex expressions combining multiple laws of indices. The document aims to teach readers how to manipulate expressions involving indices and apply the laws of indices.
The document discusses numerical integration methods such as Newton-Cotes formulas, the trapezoidal rule, and Simpson's rules. The trapezoidal rule approximates the integral of a function f(x) between bounds a and b by taking the average of f(a) and f(b) and multiplying by the width b-a. Simpson's rules use higher order polynomials to connect function values for a more accurate approximation of the integral. Gauss quadrature implements strategic positioning of points to define straight lines that balance positive and negative errors, improving the integral estimate.
Diseno en ingenieria mecanica de Shigley - 8th ---HDes
descarga el contenido completo de aqui http://paralafakyoumecanismos.blogspot.com.ar/2014/08/libro-para-mecanismos-y-elementos-de.html
The document discusses calculating the geometric mean of discrete and continuous data series. For the discrete series, the logarithms of the products of the frequencies and values are summed and divided by the total frequency to calculate the geometric mean. For the continuous series, midpoints are used rather than discrete values, and the geometric mean formula is applied to calculate the overall mean as 17.89. An example is provided for the reader to calculate the geometric mean from given discrete data.
The document provides information about calculating mean, variance, and standard deviation from a data set. It includes a table of values for number of cycles (x) and failure cycles (f) for a sample of bearings. It then shows the calculations to find:
1) The mean number of cycles is 122.9 thousand cycles.
2) The variance is 912.9 thousand cycles squared.
3) The standard deviation is 30.3 thousand cycles.
The document provides information about calculating mean, variance, and standard deviation from a data set. It includes a table of values for number of cycles (x) and failure cycles (f) for a sample. It then shows the calculations to find:
1) The mean number of cycles is 122.9 thousand cycles
2) The variance is 912.9 thousand cycles squared
3) The standard deviation is 30.3 thousand cycles
The intent is to demonstrate calculating statistics from a data set to characterize the distribution and variability. The example uses cycle life data from a fatigue test to find the central tendency and spread.
The document provides information about normal probability distributions and how to solve problems using normal distributions. It defines the normal distribution and standard normal distribution. It gives the equation for a normal distribution and how to standardize a normal variable. Examples are provided on finding probabilities and areas under the normal curve. The document also discusses using normal approximations to the binomial and Poisson distributions and provides continuity correction rules for such approximations.
The file includes solved numerical problems on standard deviation with direct and indirect methods, in individual, discrete and continuous series. The file also has some unsolved problems for practice and self assessment after learning standard deviation.
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.
The document defines and provides examples for calculating the coefficient of variation, which is a measure used to compare the dispersion of data sets. It gives the formula for coefficient of variation as the standard deviation divided by the mean, expressed as a percentage. Two examples are shown comparing the stability of prices between two cities and production between two manufacturing plants, with the data set having the lower coefficient of variation considered more consistent or stable.
This document discusses multicollinearity, beginning with definitions and the case of perfect multicollinearity. It then examines the case of near or imperfect multicollinearity using data on the demand for widgets. There is high multicollinearity between the price and income variables, resulting in unstable coefficient estimates with large standard errors and insignificant t-statistics. The document outlines methods to detect multicollinearity such as high R-squared but insignificant variables, high pairwise correlations, auxiliary regressions, and variance inflation factors. It provides an example using data on chicken demand.
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 defines key terms and concepts related to standard deviation and variance. It provides formulas for calculating range, deviation, variance, and standard deviation for both ungrouped and grouped data. Examples are given to demonstrate calculating these metrics from raw data sets and grouped data tables. Interpreting skewness is also discussed.
Similar to Central tedancy & correlation project - 2 (20)
Current Affair 2020 Pakistan for those Candidates who are appearing for any type of competitive exam would swear by the importance of studying the current affairs books. In a lot of exams, including PPSC, FPSC, KPSC, SPSC, NTS, PTS, OTS, & BTS. Current Affairs is one of the most important pillars.
Title: Pakistan Current Affairs
Authors: Faizan Akhtar
Publisher: Easynokri.com
Edition: 2nd Latest 2020
Pages: 74
If you have any Question then Contact Us: 03336973219
1. The document discusses the author's experiences with change management and lessons learned from studying the subject. It provides insights from the author's personal and professional experiences undergoing changes without awareness of change management principles.
2. The author learned that effective communication is key to managing change and helping employees understand why changes are happening. Resistance to change is common and can be overcome through two-way communication that addresses employees' questions and issues.
3. Case studies of companies like Qantas, Woolworths, and Samsung demonstrate how adopting change management frameworks like the McKinsey 7S model helped implement successful transformations through improved communication. Failure to change and innovate led to the downfall of companies like Kodak and
This document compares and contrasts money markets and capital markets. It defines money markets as markets for short-term debt instruments with maturities of 1 year or less, like treasury bills and commercial paper, while capital markets deal in longer term securities like stocks, bonds and debentures. Key differences include money markets focusing on liquidity and short-term borrowing needs, while capital markets help raise long term financing for businesses and infrastructure. Risk is also generally lower in money markets due to shorter durations, while returns are higher in capital markets. Both play important roles in channeling funds between lenders and borrowers in an economy.
This document provides information about a training program on health and safety conducted by Haleeb Foods Ltd. It includes an executive summary describing the training process, from conducting a needs assessment of employees to evaluating the training. The training was delivered over four days in January and February in response to a previous incident at the organization. It aimed to educate laborers on safety and its importance through lectures, videos and practical demonstrations. Overall, the training was found to be a positive experience for employees and the company hopes it will help prevent future incidents.
The document discusses the Chartered Accountants Ordinance 1961 and the Institute of Chartered Accountants of Pakistan (ICAP). It summarizes that ICAP is the statutory body that regulates the accountancy profession in Pakistan according to the Chartered Accountants Ordinance 1961. It establishes ICAP's headquarters in Karachi and regional offices in other major cities. The document then outlines ICAP's vision, management structure, requirements to become a chartered accountant, salaries, job description, and amendments made to the ordinance over time.
This document is a project report on the impact of macroeconomic determinants like inflation, unemployment, foreign direct investment, and poverty on India's GDP growth rate. It provides an economic profile of India, including key statistics on GDP composition and growth. It then analyzes various macroeconomic factors in depth, discussing inflation rates and causes, types and measurement of unemployment, trends in foreign investment, and the challenges of poverty. The report concludes by emphasizing the importance of investment and effective policymaking for India's continued economic development.
This document contains a final term project report submitted by three students to their professor. The report summarizes statistical techniques including correlation, regression, and measures of central tendency. For correlation, the document defines correlation, describes the correlation coefficient and different types of correlation. It also discusses the history and uses of correlation. For regression, it defines regression, describes the regression coefficient and line, and discusses the history and uses of regression. Finally, it defines different measures of central tendency including mean, median, mode, and discusses their advantages and disadvantages. The report is presented in a table of contents and contains examples, formulas and multiple choice questions.
1. The document discusses various types of companies under corporate law in Pakistan including listed vs unlisted companies, public vs private companies, limited by share vs limited by guarantee, public vs single member companies, public companies vs joint ventures, and limited by guarantee vs joint ventures.
2. The key differences between each type of company are explained such as membership requirements, ability to invite public subscription, filing of accounts and reports, and liability of members.
3. Examples are provided for each type of company to illustrate the concepts discussed.
This document provides a marketing channel project report for Olper's Milk, a brand of Engro Foods Ltd. in Pakistan. It includes an introduction to Engro Foods and Olper's Milk, an analysis of the current state of Olper's milk marketing channels, and recommendations. The key points are:
- Olper's Milk uses a two-level distribution channel consisting of distributors who supply retailers, who then supply customers. It has an intensive distribution network across Pakistan.
- A gap analysis found differences between existing and desired states, such as opportunities for bulk breaking and new product development.
- Recommendations include focusing on customer demands through research, using advanced technology, value addition, cost
This document discusses the importance of setting goals and provides guidance on how to effectively set and achieve goals. It notes that goals provide direction and purpose, and increase motivation. It then outlines a seven step process for setting SMART goals, including writing goals down, identifying obstacles and support systems, using visualization, and setting deadlines. Regularly reviewing and updating goals is important to stay on track to achievement. The overall message is that clearly defining and working towards specific, measurable goals improves the chances of success.
This document provides information about demand, supply, and market equilibrium for Coca-Cola. It discusses the history and invention of Coca-Cola, the basic concepts of demand and supply, factors that affect demand and supply, and the relationship between price, demand, and supply. Specifically, it explains that demand for Coca-Cola depends on factors like price, income, tastes, and policies while supply depends on price, technology, number of consumers, and input prices. It also illustrates the laws of demand and supply through demand and supply curves for Coca-Cola.
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
How Barcodes Can Be Leveraged Within Odoo 17Celine George
In this presentation, we will explore how barcodes can be leveraged within Odoo 17 to streamline our manufacturing processes. We will cover the configuration steps, how to utilize barcodes in different manufacturing scenarios, and the overall benefits of implementing this technology.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...
Central tedancy & correlation project - 2
1. 1 | P a g e
Q#No.1
For the Data Give below calculates: (1) Arithmetic mean (2) Median (3) Mode (4) Harmonic
mean (5) Geometric mean.
Solution:
(1) Arithmetic means
fX
X
f
4265
61
X
69.918X
(2) Median
2
h n
Median L c
f
10
60 30.5 16
15
Median
69.67Median
(3) Mode
1
1 2
m
m m
f f
Mode L h
f f f f
15 12
60 10
15 12 15 14
Mode
67.5Mode
C-B f x fx f (
𝟏
𝒙
) C.f F log x
30-40 1 35 35 0.0285 1 1.5441
40-50 3 45 135 0.6666 4 4.9596
50-60 12 55 660 0.218 16 20.8844
60-70 15 65 975 0.230 31 27.1957
70-80 14 75 1050 0.1866 45 26.2508
80-90 11 85 935 0.129 56 21.2237
90-100 5 95 475 0.052 61 9.8886
61f 4265fx 1
( ) 0.9113f
x
log 111.9449f x
2. 2 | P a g e
(4)Harmonic mean:
.
1
.
f
H M
f
x
61
.
0.9113
H M . 66.93H M
(5) Geometric mean:
(f logX)
. logG M anti
f
41.9449
. log
61
G M anti . 68.4227G M
Q#NO.2
Find the (1) Arithmetic mean (2) Median (3) Mode (4) Harmonic mean (5) Geometric mean.
Solution:
(1) Arithmetic mean:
FX
X
F
18765
164
X 162.44X
C-B f x fx Flogx f (
𝟏
𝒙
) C.f
0-40 6 20 120 7.8062 0.3 6
40-80 15 60 900 26.6723 0.25 21
80-120 22 100 2200 44 0.22 43
120-160 30 140 4200 64.3839 0.2143 73
160-200 45 180 8100 101.4873 0.25 118
200-140 27 220 5940 63.2455 0.1228 145
240-180 13 260 3380 31.3947 0.05 15.8
280-320 6 300 1800 14.8628 0.02 164
164f 26640fx log 353.8527f x 1
( ) 1.427f
x
3. 3 | P a g e
(2) Median
2
h n
Median L c
f
40
160 82 73
45
Median 168Median
(3) Mode
1
1 2
m
m m
f f
Mode L h
f f f f
45 30
160 40
45 30 45 27
Mode
45 30
160 40
45 30 45 27
Mode
15
160 40
33
Mode
178.18Mode
(4)Harmonic mean:
.
1
.
f
H M
f
x
164
.
1.4271
H M
. 114.92H M
(5) Geometric mean:
(f logX)
. logG M anti
f
353.8527
. log
164
G M anti
. log 2.1577G M anti . 143.789G M
4. 4 | P a g e
Q#No.3
Calculate the: (1) Arithmetic mean (2) Median (3) Mode (4) Harmonic mean (5) Geometric
mean.
Solution:
(1) Arithmetic mean:
fX
X
f
3372.5
55
X 61.318X
(2) Median
2
h n
Median L c
f
55
60 27.5 2
18
Median 61.18Median
(3) Mode
1
1 2
m
m m
f f
Mode L h
f f f f
18 12
60 5
18 12 18 13
Mode
30Mode
(4)Harmonic mean:
.
1
.
f
H M
f
x
55
.
0.90587
H M . 60.715H M
C-B f X fx Flogx f (
𝟏
𝒙
) C.f
45-50 2 47.5 94 3.3533 0.0421 2
50-55 7 52.5 367.5 12.0411 0.1333 9
55-60 12 57.5 690 21.1160 0.2086 21
60-65 18 62.5 1125 32.3258 0.288 39
65-70 13 67.5 877.5 23.7809 0.1925 52
70-75 3 72.5 217.5 5.5810 0.04137 54
55f 3372.5fx log 98.1981f x 1
( ) 0.90587f
x
5. 5 | P a g e
(5) Geometric mean:
(f logX)
. logG M anti
f
98.1981
. log
55
G M anti
. log 1.78542G M anti . 61.012G M
Q#No.4
For the Data Give below calculates: (1) Arithmetic mean (2) Median (3) Mode (4) Harmonic
mean (5) Geometric mean.
Solution:
(1) Arithmetic mean:
fX
X
f
46350
850
X 54.526X
(2) Median
2
h n
Median L c
f
10
60 425 420
190
Median 60.263Median
(3) Mode
1
1 2
m
m m
f f
Mode L h
f f f f
190 125
60 10
190 125 190 240
Mode
65
60 10
65 50
Mode
8.333Mode
C-B f x fx Flogx f (
𝟏
𝒙
) C.f
0-10 25 5 125 17.47 5 25
10-20 40 15 600 47.043 2.667 65
20-30 60 25 1500 83.876 2.4 125
30-40 75 35 2625 115.80 2.142 200
40-50 95 45 4275 157.05 2.111 295
50-60 125 55 6875 217.54 2.272 420
60-70 190 65 12350 344.45 2.923 610
70-80 240 75 18000 445.75 3.2 850
850f 46350fx log 1428.98f x 1
( ) 22.75f
x
6. 6 | P a g e
(4) Harmonic mean:
.
1
.
f
H M
f
x
850
.
22.75
H M . 37.36H M
(5) Geometric mean:
(f logX)
. logG M anti
f
1428.98
. log
850
G M anti . 47.98G M
Q#No.5
For the Data Give below calculates: (1) Arithmetic mean (2) Median (3) Mode (4) Harmonic
mean (5) Geometric mean.
Solution:
(1) Arithmetic mean:
fX
X
f
2005
72
X 27.847X
(2) Median
2
h n
Median L c
f
5
22.5 36 24
19
Median 25.66Median
C-B f X fx C.f F log x 1
f
x
12.5-17.5 2 15 30 2 2.3522 0.4
17.5-22.5 22 20 440 24 28.6227 1.1
22.5-27.5 19 25 475 43 26.5609 0.76
27.5-32.5 14 30 420 57 20.6797 0.4667
32.5-37.5 3 35 105 60 4.6322 0.857
37.5-42.5 4 40 160 64 6.4082 0.1
42.5-47.5 6 45 270 70 9.9193 0.1333
47.5-52.5 1 50 50 71 1.6989 0.02
52.5-57.5 1 55 55 72 1.7407 0.0181
∑f=72 ∑fx=2005 ∑flogx=102.6145 1
f
x
7. 7 | P a g e
(3) Mode:
1
1 2
m
m m
f f
Mode L h
f f f f
22 2
17.5 5
22 2 22 19
Mode
20
17.5 5
20 3
Mode
21.85Mode
(4) Harmonic mean:
.
1
.
f
H M
f
x
72
.
3.0837
H M . 23.35H M
(5) Geometric mean:
(f logX)
. logG M anti
f
102.6145
. log
72
G M anti . 26.62G M
8. 8 | P a g e
Q#No.1
Calculate the correlation co-efficient between percentage of marks scored by 12 students in
statistics and economics.
Solution:
(1). Find the co-efficient of correlation:
2 2 2 2
( ) ( )
n xy x y
r
n x x n y y
2 2
12(23136) (750)(365)
12(47470) (750) 12(11385) (365)
r
3882
7140 3395
r
710
45054900
r
3882
4923.443917
r 0.7884r
(2). Regression line:
Find the regression line of following data:
Y on X:
Y=a+bx
2 2
( )
yx
n xy x y
b
n x x
2
12 23136 750 365
12 47470 (750)
yxb
277632 273750
569640 562500
yxb
3882
7140
yxb 0.543yxb
X y xy 𝒙 𝟐
𝒚 𝟐
50 22 1100 2500 484
54 25 1350 2916 625
56 34 1904 3136 1156
59 25 1652 3481 784
60 26 1560 3600 678
61 30 1830 3721 900
62 32 1984 3844 1024
65 30 1950 4225 900
67 28 1876 4489 784
71 34 2414 5041 1156
71 36 2556 5041 1296
74 40 2960 5476 1600
∑x=750 ∑y=365 ∑xy=23136 ∑𝒙 𝟐=47470 ∑𝒚 𝟐=11385
9. 9 | P a g e
yx
yx
y b x
a
n
365 0.543 750
12
yxa
3.52yxa
The estimated regression line is as follows.
ˆ 3.52 0.543Y X
(3). Regression line:
X on Y:
X=a+by
2 2
( )
xy
n xy x y
b
n y y
2
12 23136 750 365
12 11385 (365)
xyb
277632 27375
136620 133225
xyb
3882
3395
xyb 1.143xyb
.
xy
x bxy y
a
n
750 1.143 (365)
12
xya
27.73xya
The estimated regression line is as follows.
ˆ 27.73 1.143x Y
10. 10 | P a g e
Q#No.2
Calculate the Co-efficientof correlationfromthe followingdataandalsocompute RegressionlineY onX
Solution:
(1).Find the co-efficient of correlation:
2 2 2 2
( ) ( )
n xy x y
r
n x x n y y
2 2
(11)(1336) (110)(125)
(11)(1210) (110) (11)(153) (125)
r
14696 13750
(13310 12100)(16841 15625)
r
946
(1210)(1216)
r
946
1471360
r
946
1212.99
r 0.779r
(2). Regression line:
Y on X
Y a bx
2 2
( )
yx
n xy x y
b
n x x
2
(11)(1336) (110)(125)
(11)(1210) (110)
yxb
(14696) (13750)
(13310) (12100)
yxb
946
1210
yxb 0.781yxb
X Y XY 𝒙 𝟐
𝒚 𝟐
5 9 45 25 81
6 6 36 36 36
7 10 70 49 100
8 8 64 64 64
9 13 117 81 169
10 11 110 100 121
11 14 154 121 196
12 10 120 144 100
13 14 182 169 196
14 12 168 196 144
15 18 270 225 324
∑X=110 ∑Y=125 ∑XY=1336 ∑𝑿 𝟐
=1240 ∑𝒀 𝟐
=1531
11. 11 | P a g e
yx
yx
y b x
a
n
(125) (0.781)(110)
11
yxa
125 85.91
11
yxa
39.09
11
yxa 3.553yxa
The estimated regression line is as follows.
ˆ 3.553 0.781Y x
Q#NO.3
Calculate the Co-efficient of correlation from the following data and also compute Regression
line Y on X and X on Y.
Solution:
(1). Find the co-efficient of correlation:
2 2 2 2
( ) ( )
n xy x y
r
n x x n y y
2 2
(10)(33535) (655)(500)
(10)(46059) (655) (10)(25464) (500)
r
(335350) (327500)
(460590 429025)(254640 250000)
r
(7850)
(31565)(4640)
r
X Y XY 𝒙 𝟐
𝒚 𝟐
16 40 640 256 1600
72 52 3744 5184 2704
73 43 3139 5329 1849
63 49 3087 3969 2401
83 61 5063 6889 3721
80 58 4640 6400 3364
66 44 2904 4359 1936
66 58 3828 4356 3364
74 50 3700 5476 2500
62 45 2790 3844 2025
∑X=655 ∑Y=500 ∑XY=33535 ∑𝑿 𝟐
=46059 ∑𝒀 𝟐
=25464
12. 12 | P a g e
(7850)
12102.13204
r 0.6486r
(2)Regression line Y on X
Y a bx
2 2
( )
yx
n xy x y
b
n x x
2
(10)(33535) (655)(500)
(10)(46059) (655)
yxb
(335350) (327500)
(460590) (429025)
yxb
7850
31565
yxb 0.249yxb
yx
yx
y b x
a
n
(500) (0.249)(655)
10
yxa
(500) (163.095)
10
yxa
336.905
10
yxa 33.6905yxa
The estimated regression line is as follows.
ˆ 33.6905 0.249Y x
(3)Regression line X on Y:
X a by
2 2
( )
xy
n xy x y
b
n y y
2
(10)(33535) (655)(500)
(10)(25464) (500)
xyb
(335350) (327500)
(254640) (250000)
xyb
7850
4640
xyb 1.6918103xyb
xy
xy
x b y
a
n
(655) (1.6918)(500)
10
xya
(655) (845.9)
10
xya
190.9
10
xya
19.09xya
13. 13 | P a g e
The estimated regression line is as follows.
ˆ 19.09 1.6918X x
Q#NO.4
Find the co-efficient of correlation and fit the regression lines of the given data and also discuss
its result.
Solution:
(1). Find the co-efficient of correlation:
2 2 2 2
( ) ( )
n xy x y
r
n x x n y y
2 2
8(11245) (595)(150)
8(47375) (595) 8(3038) (150)
r
710
24975 1804
r
710
45054900
r 710
6712.29469
r
0.1058r
(2). Regression line Y on X
Y a bx
2 2
( )
yx
n xy x y
b
n x x
2
8 11245 595 150
8 47375 (595)
yxb
X Y XY 𝒙 𝟐
𝒚 𝟐
40 17 680 1600 289
55 19 1045 3025 361
60 23 1380 3600 529
75 15 1125 5625 225
80 18 1440 6400 324
90 17 1530 8100 289
95 11 1045 9025 121
100 30 3000 10000 900
∑x=595 ∑y=150 ∑xy=11245 ∑𝒙 𝟐
=47375 ∑𝒚 𝟐
=3038
14. 14 | P a g e
2
89960 89250
8 47375 (595)
yxb
710
37900 354025
yxb
710
24975
yxb 0.02843yxb
yx
yx
y b x
a
n
150 0.024843 595
8
yxa
16.635yxa
The estimated regression line is as follows.
ˆ 16.635 0.02843Y X
(3) Regression line X on Y:
X a by
2 2
( )
xy
n xy x y
b
n y y
2
8 11245 595 150
8 3038 (150)
xyb
89960 89250
379000 22500
xyb
710
1804
xyb 0.39356xyb
.
xy
x bxy y
a
n
595 0.39356 (150)
8
xya
67.99xya
The estimated regression line is as follows.
ˆ 67.99 0.39356x
15. 15 | P a g e
Q#No.5: calculate the co-efficient of correlation from given data. And also find out regression
line Y on X and X on Y.
Solution:
x y xy 𝒙 𝟐
𝒚 𝟐
10 7 70 1000 49
15 8 120 225 64
20 3 60 400 9
25 7 175 625 49
30 18 540 900 324
35 6 210 1225 36
40 17 680 1600 289
∑x=175 ∑y=66 ∑xy=1855 ∑𝒙 𝟐
∑𝒚 𝟐
= 𝟖𝟐𝟎
(1). Find the co-efficient of correlation:
2 2 2 2
( ) ( )
n xy x y
r
n x x n y y
2 2
7 1855 175 66
7 5075 175 7 820 (66)
r
12985 11550
4900 1384
r
1435
6781600
r
1435
2604.1505
r 0.551043r
(2) . Regression line Y on X
Y a bx
2 2
( )
yx
n xy x y
b
n x x
1435
4900
yxb 0.2928yxb
yx
y byx x
a
n
66 0.2928 175
7
yxa
66 51.24
7
yxa
2.1085yxa
The estimated regression line is as follows.
ˆ 2.1085 0.2928x x
16. 16 | P a g e
(3)Regression line X on Y:
X a by
2 2
( )
xy
n xy x y
b
n y y
2
7 1855 175 66
7 820 (66)
xyb
1435
1384
xyb 1.0368xyb
xy
x bxy y
a
n
175 1.036 66
7
xya
106.63
7
xya 15.2328xya
The estimated regression line is as follows.
ˆ 15.2328 1.0368x y