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.
The document provides additional information on correlation analysis. It discusses various examples of correlation between variables like sugar consumption and activity level. It explains the characteristics of a relationship such as the direction, form, and degree of correlation. Correlations can be used for prediction, validity, and reliability. The document also discusses the difference between correlation and causation. It then provides examples to test the reader's understanding of correlation through multiple choice questions. Finally, it covers topics like probable error, coefficient of correlation, coefficient of determination, Spearman's rank correlation method, and concurrent deviation method for calculating correlation.
Multiple regression allows researchers to use several independent variables simultaneously to predict a continuous dependent variable. It fits a mathematical equation to the data that describes the overall relationship between the dependent variable and independent variables. The equation can be used to predict the dependent variable value based on the values of the independent variables. The technique is useful for social science research where phenomena are influenced by multiple causal factors.
The document defines correlation and regression, and describes how to calculate them. Correlation measures the strength and direction of a linear relationship between two random variables on a scale from -1 to 1. Regression finds the linear relationship between a random variable and a fixed variable to make predictions. The document provides examples of calculating correlation using Pearson's r and determining the regression line and equation from sample data.
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.
- Regression analysis is a statistical technique for modeling relationships between variables, where one variable is dependent on the others. It allows predicting the average value of the dependent variable based on the independent variables.
- The key assumptions of regression models are that the error terms are normally distributed with zero mean and constant variance, and are independent of each other.
- Linear regression specifies that the dependent variable is a linear combination of the parameters, though the independent variables need not be linearly related. In simple linear regression with one independent variable, the least squares estimates of the intercept and slope are calculated to minimize the sum of squared errors.
It is most useful for the students of BBA for the subject of "Data Analysis and Modeling"/
It has covered the content of chapter- Data regression Model
Visit for more on www.ramkumarshah.com.np/
This document discusses correlation and different types of correlation analysis. It defines correlation as a statistical analysis that measures the relationship between two variables. There are three main types of correlation: (1) simple and multiple correlation based on the number of variables, (2) linear and non-linear correlation based on the relationship between variables, and (3) positive and negative correlation based on the direction of change between variables. The degree of correlation is measured using correlation coefficients that range from -1 to +1. Common methods to study correlation include scatter diagrams and Karl Pearson's coefficient of correlation.
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.
The document provides additional information on correlation analysis. It discusses various examples of correlation between variables like sugar consumption and activity level. It explains the characteristics of a relationship such as the direction, form, and degree of correlation. Correlations can be used for prediction, validity, and reliability. The document also discusses the difference between correlation and causation. It then provides examples to test the reader's understanding of correlation through multiple choice questions. Finally, it covers topics like probable error, coefficient of correlation, coefficient of determination, Spearman's rank correlation method, and concurrent deviation method for calculating correlation.
Multiple regression allows researchers to use several independent variables simultaneously to predict a continuous dependent variable. It fits a mathematical equation to the data that describes the overall relationship between the dependent variable and independent variables. The equation can be used to predict the dependent variable value based on the values of the independent variables. The technique is useful for social science research where phenomena are influenced by multiple causal factors.
The document defines correlation and regression, and describes how to calculate them. Correlation measures the strength and direction of a linear relationship between two random variables on a scale from -1 to 1. Regression finds the linear relationship between a random variable and a fixed variable to make predictions. The document provides examples of calculating correlation using Pearson's r and determining the regression line and equation from sample data.
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.
- Regression analysis is a statistical technique for modeling relationships between variables, where one variable is dependent on the others. It allows predicting the average value of the dependent variable based on the independent variables.
- The key assumptions of regression models are that the error terms are normally distributed with zero mean and constant variance, and are independent of each other.
- Linear regression specifies that the dependent variable is a linear combination of the parameters, though the independent variables need not be linearly related. In simple linear regression with one independent variable, the least squares estimates of the intercept and slope are calculated to minimize the sum of squared errors.
It is most useful for the students of BBA for the subject of "Data Analysis and Modeling"/
It has covered the content of chapter- Data regression Model
Visit for more on www.ramkumarshah.com.np/
This document discusses correlation and different types of correlation analysis. It defines correlation as a statistical analysis that measures the relationship between two variables. There are three main types of correlation: (1) simple and multiple correlation based on the number of variables, (2) linear and non-linear correlation based on the relationship between variables, and (3) positive and negative correlation based on the direction of change between variables. The degree of correlation is measured using correlation coefficients that range from -1 to +1. Common methods to study correlation include scatter diagrams and Karl Pearson's coefficient of correlation.
Regression analysis measures the average relationship between two or more variables using their original data units. There are two main types: simple regression involving two variables, and multiple regression involving more than two variables. Regression can be linear, following a straight line, or non-linear/curvilinear. A simple linear regression model relates a dependent variable Y to an independent variable X plus an error term. Estimating the model involves calculating the slope/regression coefficient and intercept. Multiple regression relates a dependent variable to two or more independent variables using a multiple correlation coefficient.
This document discusses quantitative research methods including correlation, simple linear regression, and multiple regression. It provides examples of how to conduct simple linear regression using SPSS to analyze the relationship between two variables and predict the dependent variable based on the independent variable. It then expands the discussion to multiple linear regression, using SPSS to analyze the relationships between multiple independent variables and one dependent variable. Key steps of assessing the model such as the coefficient of determination and F-test of ANOVA are also covered.
Linear regression and correlation analysis ppt @ bec domsBabasab Patil
This document introduces linear regression and correlation analysis. It discusses calculating and interpreting the correlation coefficient and linear regression equation to determine the relationship between two variables. It covers scatter plots, the assumptions of regression analysis, and using regression to predict and describe relationships in data. Key terms introduced include the correlation coefficient, linear regression model, explained and unexplained variation, and the coefficient of determination.
Introduction to correlation and regression analysisFarzad Javidanrad
This document provides an introduction to correlation and regression analysis. It defines key concepts like variables, random variables, and probability distributions. It discusses how correlation measures the strength and direction of a linear relationship between two variables. Correlation coefficients range from -1 to 1, with values closer to these extremes indicating stronger correlation. The document also introduces determination coefficients, which measure the proportion of variance in one variable explained by the other. Regression analysis builds on correlation to study and predict the average value of one variable based on the values of other explanatory variables.
Chapter 6 simple regression and correlationRione Drevale
There is a significant positive correlation between amount of feed intake and live weight of broilers. The correlation coefficient (r) between feed intake and live weight is 0.726, which is statistically significant with p<0.017. On average, broilers gain approximately 0.5 kg of live weight for every 1 kg of feed consumed.
This document discusses multiple regression analysis and its use in predicting relationships between variables. Multiple regression allows prediction of a criterion variable from two or more predictor variables. Key aspects covered include the multiple correlation coefficient (R), squared correlation coefficient (R2), adjusted R2, regression coefficients, significance testing using t-tests and F-tests, and considerations for using multiple regression such as sample size and normality assumptions.
Regression analysis is a statistical technique used to investigate relationships between variables. It allows one to determine the strength of the relationship between a dependent variable (usually denoted by Y) and one or more independent variables (denoted by X). Multiple regression extends this to analyze the relationship between a dependent variable and multiple independent variables. The goals of regression analysis are to understand how the dependent variable changes with the independent variables and to use the independent variables to predict the value of the dependent variable. It requires the dependent variable to be continuous and the independent variables can be either continuous or categorical.
The document provides an overview of regression analysis including:
- Regression analysis is a statistical process used to estimate relationships between variables and predict unknown values.
- The document outlines different types of regression like simple, multiple, linear, and nonlinear regression.
- Key aspects of regression like scatter diagrams, regression lines, and the method of least squares are explained.
- An example problem is worked through demonstrating how to calculate the slope and y-intercept of a regression line using the least squares method.
This document contains slides from a presentation on simple linear regression and correlation. It introduces simple linear regression modeling, including estimating the regression line using the method of least squares. It discusses the assumptions of the simple linear regression model and defines key terms like the regression coefficients (intercept and slope), error variance, standard errors of the estimates, and how to perform hypothesis tests and construct confidence intervals for the regression parameters. Examples are provided to demonstrate calculating quantities like sums of squares, estimating the regression line, and evaluating the fit of the regression model.
Regression Analysis is simplified in this presentation. Starting with simple linear to multiple regression analysis, it covers all the statistics and interpretation of various diagnostic plots. Besides, how to verify regression assumptions and some advance concepts of choosing best models makes the slides more useful SAS program codes of two examples are also included.
This document discusses correlation and regression. Correlation describes the strength and direction of a linear relationship between two variables, while regression allows predicting a dependent variable from an independent variable. It provides examples of calculating the correlation coefficient r to determine the strength and direction of relationships between variables like education and self-esteem or family income and number of children. The regression equation describes the linear regression line and can be used to predict values of the dependent variable from known values of the independent variable.
- Regression analysis is a statistical technique used to measure the relationship between two quantitative variables and make causal inferences.
- A regression model graphs the relationship between a dependent variable (Y axis) and one or more independent variables (X axis). The goal is to find the linear equation that best fits the data.
- The regression equation takes the form Y = a + bX, where a is the intercept, b is the slope coefficient, and X and Y are the variables. The coefficient b indicates the strength and direction of the relationship.
Multiple regression analysis allows researchers to examine the relationship between one dependent or outcome variable and two or more independent or predictor variables. It extends simple linear regression to model more complex relationships. Stepwise regression is a technique that automates the process of building regression models by sequentially adding or removing variables based on statistical criteria. It begins with no variables in the model and adds variables one at a time based on their contribution to the model until none improve it significantly.
This document provides an overview of correlation and regression analysis concepts. It defines correlation as the strength of relationship between two variables and discusses perfect, positive, negative, and no relationships. Pearson's correlation coefficient r is described as a measure of linear correlation between -1 and +1, with values closer to these extremes indicating a stronger linear relationship. The document also explains how to calculate r using z-scores and provides examples. Finally, it introduces the concept of linear regression analysis, including the least squares regression equation and how to calculate the line of best fit, as well as the standard error of estimate.
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.
The document discusses correlation, regression analysis, and an example analysis. It defines correlation as a measure of the strength of association between two variables. Regression analysis establishes a mathematical relationship between variables to predict outcomes. The example analyzes the correlation between residents' duration of residence in a city and their attitude toward the city, finding a strong positive correlation. It then performs a bivariate regression to model this relationship mathematically.
Overviews non-parametric and parametric approaches to (bivariate) linear correlation. See also: http://en.wikiversity.org/wiki/Survey_research_and_design_in_psychology/Lectures/Correlation
The document provides an overview of regression analysis concepts including:
- Regression analysis is used to understand relationships between variables and predict the value of one variable based on another.
- A regression model has a dependent variable on the y-axis and an independent variable on the x-axis.
- Examples of how to perform regression analysis are provided including creating a scatter plot and calculating parameters like the slope and intercept.
- Key concepts for measuring the fit of a linear regression model are defined including variability, correlation coefficient, coefficient of determination, and standard error.
The document discusses regression and correlation analysis between BMI (Kg/m2) of pregnant mothers and birth weight (kg) of their newborns using data from 15 mothers. A scatter plot showed a positive linear relationship between BMI and birth weight. Linear regression was used to calculate the regression line as y=1.775351+0.0330817x, which can be used to predict birth weight based on a mother's BMI. The correlation coefficient (R) between BMI and birth weight was 0.94, indicating a strong positive correlation.
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 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.
Regression analysis measures the average relationship between two or more variables using their original data units. There are two main types: simple regression involving two variables, and multiple regression involving more than two variables. Regression can be linear, following a straight line, or non-linear/curvilinear. A simple linear regression model relates a dependent variable Y to an independent variable X plus an error term. Estimating the model involves calculating the slope/regression coefficient and intercept. Multiple regression relates a dependent variable to two or more independent variables using a multiple correlation coefficient.
This document discusses quantitative research methods including correlation, simple linear regression, and multiple regression. It provides examples of how to conduct simple linear regression using SPSS to analyze the relationship between two variables and predict the dependent variable based on the independent variable. It then expands the discussion to multiple linear regression, using SPSS to analyze the relationships between multiple independent variables and one dependent variable. Key steps of assessing the model such as the coefficient of determination and F-test of ANOVA are also covered.
Linear regression and correlation analysis ppt @ bec domsBabasab Patil
This document introduces linear regression and correlation analysis. It discusses calculating and interpreting the correlation coefficient and linear regression equation to determine the relationship between two variables. It covers scatter plots, the assumptions of regression analysis, and using regression to predict and describe relationships in data. Key terms introduced include the correlation coefficient, linear regression model, explained and unexplained variation, and the coefficient of determination.
Introduction to correlation and regression analysisFarzad Javidanrad
This document provides an introduction to correlation and regression analysis. It defines key concepts like variables, random variables, and probability distributions. It discusses how correlation measures the strength and direction of a linear relationship between two variables. Correlation coefficients range from -1 to 1, with values closer to these extremes indicating stronger correlation. The document also introduces determination coefficients, which measure the proportion of variance in one variable explained by the other. Regression analysis builds on correlation to study and predict the average value of one variable based on the values of other explanatory variables.
Chapter 6 simple regression and correlationRione Drevale
There is a significant positive correlation between amount of feed intake and live weight of broilers. The correlation coefficient (r) between feed intake and live weight is 0.726, which is statistically significant with p<0.017. On average, broilers gain approximately 0.5 kg of live weight for every 1 kg of feed consumed.
This document discusses multiple regression analysis and its use in predicting relationships between variables. Multiple regression allows prediction of a criterion variable from two or more predictor variables. Key aspects covered include the multiple correlation coefficient (R), squared correlation coefficient (R2), adjusted R2, regression coefficients, significance testing using t-tests and F-tests, and considerations for using multiple regression such as sample size and normality assumptions.
Regression analysis is a statistical technique used to investigate relationships between variables. It allows one to determine the strength of the relationship between a dependent variable (usually denoted by Y) and one or more independent variables (denoted by X). Multiple regression extends this to analyze the relationship between a dependent variable and multiple independent variables. The goals of regression analysis are to understand how the dependent variable changes with the independent variables and to use the independent variables to predict the value of the dependent variable. It requires the dependent variable to be continuous and the independent variables can be either continuous or categorical.
The document provides an overview of regression analysis including:
- Regression analysis is a statistical process used to estimate relationships between variables and predict unknown values.
- The document outlines different types of regression like simple, multiple, linear, and nonlinear regression.
- Key aspects of regression like scatter diagrams, regression lines, and the method of least squares are explained.
- An example problem is worked through demonstrating how to calculate the slope and y-intercept of a regression line using the least squares method.
This document contains slides from a presentation on simple linear regression and correlation. It introduces simple linear regression modeling, including estimating the regression line using the method of least squares. It discusses the assumptions of the simple linear regression model and defines key terms like the regression coefficients (intercept and slope), error variance, standard errors of the estimates, and how to perform hypothesis tests and construct confidence intervals for the regression parameters. Examples are provided to demonstrate calculating quantities like sums of squares, estimating the regression line, and evaluating the fit of the regression model.
Regression Analysis is simplified in this presentation. Starting with simple linear to multiple regression analysis, it covers all the statistics and interpretation of various diagnostic plots. Besides, how to verify regression assumptions and some advance concepts of choosing best models makes the slides more useful SAS program codes of two examples are also included.
This document discusses correlation and regression. Correlation describes the strength and direction of a linear relationship between two variables, while regression allows predicting a dependent variable from an independent variable. It provides examples of calculating the correlation coefficient r to determine the strength and direction of relationships between variables like education and self-esteem or family income and number of children. The regression equation describes the linear regression line and can be used to predict values of the dependent variable from known values of the independent variable.
- Regression analysis is a statistical technique used to measure the relationship between two quantitative variables and make causal inferences.
- A regression model graphs the relationship between a dependent variable (Y axis) and one or more independent variables (X axis). The goal is to find the linear equation that best fits the data.
- The regression equation takes the form Y = a + bX, where a is the intercept, b is the slope coefficient, and X and Y are the variables. The coefficient b indicates the strength and direction of the relationship.
Multiple regression analysis allows researchers to examine the relationship between one dependent or outcome variable and two or more independent or predictor variables. It extends simple linear regression to model more complex relationships. Stepwise regression is a technique that automates the process of building regression models by sequentially adding or removing variables based on statistical criteria. It begins with no variables in the model and adds variables one at a time based on their contribution to the model until none improve it significantly.
This document provides an overview of correlation and regression analysis concepts. It defines correlation as the strength of relationship between two variables and discusses perfect, positive, negative, and no relationships. Pearson's correlation coefficient r is described as a measure of linear correlation between -1 and +1, with values closer to these extremes indicating a stronger linear relationship. The document also explains how to calculate r using z-scores and provides examples. Finally, it introduces the concept of linear regression analysis, including the least squares regression equation and how to calculate the line of best fit, as well as the standard error of estimate.
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.
The document discusses correlation, regression analysis, and an example analysis. It defines correlation as a measure of the strength of association between two variables. Regression analysis establishes a mathematical relationship between variables to predict outcomes. The example analyzes the correlation between residents' duration of residence in a city and their attitude toward the city, finding a strong positive correlation. It then performs a bivariate regression to model this relationship mathematically.
Overviews non-parametric and parametric approaches to (bivariate) linear correlation. See also: http://en.wikiversity.org/wiki/Survey_research_and_design_in_psychology/Lectures/Correlation
The document provides an overview of regression analysis concepts including:
- Regression analysis is used to understand relationships between variables and predict the value of one variable based on another.
- A regression model has a dependent variable on the y-axis and an independent variable on the x-axis.
- Examples of how to perform regression analysis are provided including creating a scatter plot and calculating parameters like the slope and intercept.
- Key concepts for measuring the fit of a linear regression model are defined including variability, correlation coefficient, coefficient of determination, and standard error.
The document discusses regression and correlation analysis between BMI (Kg/m2) of pregnant mothers and birth weight (kg) of their newborns using data from 15 mothers. A scatter plot showed a positive linear relationship between BMI and birth weight. Linear regression was used to calculate the regression line as y=1.775351+0.0330817x, which can be used to predict birth weight based on a mother's BMI. The correlation coefficient (R) between BMI and birth weight was 0.94, indicating a strong positive correlation.
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 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.
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.
2016 SH2 H2 Maths Correlation & Regression E-Learning ModuleJason Lee
This document provides an overview of correlation and regression topics including: bivariate data shown on scatter diagrams, identifying independent and dependent variables, linear regression analysis, and the product moment correlation coefficient. It includes an example that illustrates sketching a scatter diagram from data values, describing the relationship between the variables, and determining whether one variable directly causes the other.
The document outlines the key components of findings, presentation, and discussion sections of a research study report. It notes that findings should present objective results through descriptive and inferential statistics. These findings are then discussed in a narrative, using tables and figures as needed, to explain their meaning and significance. The discussion also compares results to previous studies and addresses reliability and validity. The conclusions summarize what was learned and if research goals were met, while implications and recommendations suggest applications and directions for future work.
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 scatter diagrams and their use in showing the relationship between two variables. Scatter diagrams involve plotting paired data points for each variable on a graph, allowing one to see if there is a correlation between the variables. Examples are provided of scatter diagrams plotting ice cream sales against hours of sunshine and city temperature against latitude. Scatter diagrams can reveal trends and relationships in data.
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.
The document discusses several topics related to contemporary policies around violence against women and security issues. It provides statistics showing that nearly 1 in 3 women experience domestic violence, 1 in 4 experience sexual assault, and 1 in 10 have been stalked. It also notes that the Forced Marriage Unit assisted over 1,600 cases in 2009, most involving females. Additionally, it outlines several policies and acts related to football banning orders, counterterrorism, CCTV surveillance, neighborhood watches, crime prevention programs, and tools to address anti-social behavior.
This document discusses different measures of variability in data, including range, interquartile range, standard deviation, and variance. It provides examples calculating each measure using students' quiz scores. Range is the distance between the highest and lowest values, while interquartile range describes the middle 50% of scores. Standard deviation and variance measure how dispersed all values are from the mean by taking the average of squared deviations from the mean. Standard deviation is the square root of variance and is commonly used in inferential statistics.
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.
This document discusses official statistics as a source of data for sociological research. It notes both the advantages and disadvantages of official statistics from different theoretical perspectives. The advantages include being cheap and available sources of large-scale, representative quantitative data. However, critics argue they may lack validity, be socially constructed, and serve political interests rather than objective representation. Interpretivists see them as simple counts that do not capture meanings or motives.
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.
The document discusses different statistical methods for organizing and summarizing data, including frequency tables, stem-and-leaf plots, histograms, and scatter plots. It provides examples of each method and explains how to interpret the results, such as looking for relationships between variables in scatter plots. Key terms defined include correlation, variables, and linear regression lines.
Scatter plots are used to analyze the relationship between two sets of data by plotting points on a graph without connecting them. Points that form a positive sloping pattern from bottom left to top right indicate a direct relationship, while an inverse pattern shows an indirect relationship, and no pattern means no relationship exists between the variables. The stat key in a graphing calculator can be used to choose the lists of data for the x and y axes and determine the window ranges to plot scatter plot graphs for analysis.
This document outlines the schedule and topics for an advanced econometrics and Stata training course taking place in Beijing from November 17-26, 2019. The course will cover topics including introduction to econometrics and Stata, single and multiple regression, hypothesis testing, time series models, panel data models, and frontier analysis. Sessions are planned each morning and evening, with exercises and practice sessions interspersed.
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Chapter 10: Correlation and Regression
10.1: Correlation
This document discusses correlation and regression analysis. It defines correlation as a statistical measure of how strongly two variables are related. A correlation coefficient between -1 and 1 indicates the strength and direction of the linear relationship between variables. Regression analysis allows us to predict the value of a dependent variable based on the value of one or more independent variables. Simple linear regression involves one independent variable, while multiple regression involves two or more independent variables to predict the dependent variable. The document provides examples and formulas for calculating correlation, regression lines, explained and unexplained variance, and the coefficient of determination.
This document discusses correlation and regression analysis. It defines correlation as a statistical measure of how related two variables are. A correlation coefficient between -1 and 1 indicates the strength and direction of the relationship. Scatterplots visually depict the relationship between variables. Regression analysis predicts the value of a dependent variable based on the value of one or more independent variables. The regression equation represents the line of best fit through the data points that minimizes the residuals.
The document discusses correlation, which is a statistic that measures the strength and direction of the relationship between two variables. It provides an example of calculating the correlation between height and self-esteem using made-up data from 20 individuals. The correlation is found to be 0.73, indicating a strong positive relationship. The significance of the correlation is then tested to determine if it is likely due to chance.
This document discusses correlation and regression analysis. It defines correlation as a statistical measure of how two variables are related. A correlation coefficient between -1 and 1 indicates the strength and direction of the linear relationship between variables. A scatterplot can show this graphically. Regression analysis involves using one variable to predict scores on another variable. Simple linear regression uses one independent variable to predict a dependent variable, while multiple regression uses two or more independent variables. The goal is to identify the regression line that best fits the data with the least error. The coefficient of determination, R2, indicates how much variance in the dependent variable is explained by the independent variables.
FSE 200AdkinsPage 1 of 10Simple Linear Regression Corr.docxbudbarber38650
FSE 200
Adkins Page 1 of 10
Simple Linear Regression
Correlation only measures the strength and direction of the linear relationship between two quantitative variables. If the relationship is linear, then we would like to try to model that relationship with the equation of a line. We will use a regression line to describe the relationship between an explanatory variable and a response variable.
A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. We often use a regression line to predict the value of y for a given value of x.
Ex. It has been suggested that there is a relationship between sleep deprivation of employees and the ability to complete simple tasks. To evaluate this hypothesis, 12 people were asked to solve simple tasks after having been without sleep for 15, 18, 21, and 24 hours. The sample data are shown below.
Subject
Hours without sleep, x
Tasks completed, y
1
15
13
2
15
9
3
15
15
4
18
8
5
18
12
6
18
10
7
21
5
8
21
8
9
21
7
10
24
3
11
24
5
12
24
4
Draw a scatterplot and describe the relationship. Lay a straight-edge on top of the plot and move it around until you find what you think might be a “line of best fit.” Then try to predict the number of tasks completed for someone having been without sleep 16 hours.
Was your line the same as that of the classmate sitting next to you? Probably not. We need a method that we can use to find the “best” regression line to use for prediction. The method we will use is called least-squares. No line will pass exactly through all the points in the scatterplot. When we use the line to predict a y for a given x value, if there is a data point with that same x value, we can compute the error (residual):
Our goal is going to be to make the vertical distances from the line as small as possible. The most commonly used method for doing this is the least-squares method.
The least-squares regression line of y on x is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible.
Equation of the Least-Squares Regression Line
· Least-Squares Regression Line:
· Slope of the Regression Line:
· Intercept of the Regression Line:
Generally, regression is performed using statistical software. Clearly, given the appropriate information, the above formulas are simple to use.
Once we have the regression line, how do we interpret it, and what can we do with it?
The slope of a regression line is the rate of change, that amount of change in when x increases by 1.
The intercept of the regression line is the value of when x = 0. It is statistically meaningful only when x can take on values that are close to zero.
To make a prediction, just substitute an x-value into the equation and find .
To plot the line on a scatterplot, just find a couple of points on the regression line, one near each end of the range of x in the data. Plot the points and connect them with a line. .
This document provides an overview of correlation and linear regression analysis. It defines correlation as a statistical measure of the relationship between two variables. Pearson's correlation coefficient (r) ranges from -1 to 1, with values farther from 0 indicating a stronger linear relationship. Positive values indicate an increasing relationship, while negative values indicate a decreasing relationship. The coefficient of determination (r2) represents the proportion of shared variance between variables. While correlation indicates linear association, it does not imply causation. Multiple regression allows predicting a continuous dependent variable from two or more independent variables.
Linear regression is a statistical method used to model the relationship between variables. It finds the line of best fit for the data and uses this to predict the value of the dependent variable based on the independent variable. Simple linear regression involves one independent variable, while multiple linear regression can have multiple independent variables. In Python, the Scikit-learn library can be used to perform linear regression on data and evaluate the model performance using metrics like R-squared and root mean square error.
This document provides an overview of simple linear regression models. It defines regression analysis as concerned with studying the dependence of a dependent variable on one or more independent variables. The major objectives of regression are to estimate mean dependent variable values based on independent variable values, test hypotheses about the relationship, and predict/forecast dependent variable values. It describes estimating regression coefficients using the ordinary least squares method and the properties of least squares estimators established by the Gauss-Markov theorem. Examples are provided to demonstrate estimating and interpreting a simple linear regression model.
This document discusses key statistical concepts including random variables, probability distributions, expected value, variance, and correlation. It defines discrete and continuous random variables and explains how probability distributions assign probabilities to the possible values of a random variable. It also defines important metrics like expected value and variance, and how they are calculated for discrete and continuous random variables. The document concludes by explaining correlation, how the correlation coefficient measures the strength and direction of linear association between two variables, and how it is calculated.
This document discusses standards for graphing data from scientific experiments. It explains that the independent variable is plotted horizontally and the dependent variable vertically. All graphs must be clearly labeled with units and titles. Data points should be circled to indicate possible error. A line of best fit can be drawn to represent the overall trend, ignoring individual error. The area under a velocity-time graph represents displacement. Graphing data allows relationships between variables to be identified and communicated.
This document provides an overview of probability, statistics, and their applications in engineering. It defines key probability and statistics concepts like trials, outcomes, random experiments, and frequency distributions. It explains how engineers use statistics and probability to analyze data from tests and experiments to better understand product quality and failure rates. Examples are given of measures of central tendency like mean and median, measures of variation like standard deviation and variance, and the normal distribution curve. Engineering applications include using these analytical techniques to assess results from a class and compare two data histograms.
- The document discusses computing correlations between variables in R and interpreting the results.
- It provides an example of calculating the correlation between happiness and other life factors like friends and salary.
- The document uses real data from the World Happiness Report to explore correlations between variables like freedom to make life choices and confidence in national government. It finds a positive correlation between these two variables.
Scatter plots are a quality tool used to show the relationship between two variables. They graph pairs of numerical data with one variable on each axis to look for correlation. If the variables are correlated, the data points will fall along a line or curve, indicating a relationship. Scatter plots are useful for determining potential causes of problems by identifying which process elements are related and how strongly. They involve collecting paired data, plotting the independent variable on the x-axis and dependent variable on the y-axis, and examining the shape and slope of the resulting cluster of points.
This document discusses various aspects of data distributions including their shape, modality, symmetry, and skewness. It provides definitions and examples of key terms such as:
- Modality, which refers to the number of peaks in a distribution. Unimodal distributions have one peak while multimodal distributions have two or more.
- Symmetry, which means a distribution could be split down the middle to form mirror images. Asymmetric or skewed distributions have an off-center peak with a tail on one side.
- Skewness, which is assessed using measures like Pearson's coefficient and Fisher's measure that quantify the degree of asymmetry. Positive skewness indicates a right tail while negative indicates a left tail
This document provides an overview of simple linear regression and correlation analysis. It defines regression as estimating the relationship between two variables and correlation as measuring the strength and direction of that relationship. The key points covered include:
- Regression finds an estimating equation to relate known and unknown variables. Correlation determines how well that equation fits the data.
- Pearson's correlation coefficient r measures the linear relationship between two variables on a scale from -1 to 1.
- The coefficient of determination r2 indicates what percentage of variation in the dependent variable is explained by the independent variable.
- Statistical tests can evaluate whether a correlation is statistically significant or could be due to chance.
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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.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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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.
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In an education system, it is understood that assessment is only for the students, but on the other hand, the Assessment of teachers is also an important aspect of the education system that ensures teachers are providing high-quality instruction to students. The assessment process can be used to provide feedback and support for professional development, to inform decisions about teacher retention or promotion, or to evaluate teacher effectiveness for accountability purposes.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
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Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
1. Introduction toIntroduction to
Correlation and RegressionCorrelation and Regression
Ginger Holmes Rowell, Ph. D.Ginger Holmes Rowell, Ph. D.
Associate Professor of MathematicsAssociate Professor of Mathematics
Middle Tennessee State UniversityMiddle Tennessee State University
2. OutlineOutline
IntroductionIntroduction
Linear CorrelationLinear Correlation
RegressionRegression
Simple LinearSimple Linear
RegressionRegression
Using the TI-83Using the TI-83
Model/FormulasModel/Formulas
3. Outline continuedOutline continued
ApplicationsApplications
Real-life ApplicationsReal-life Applications
Practice ProblemsPractice Problems
Internet ResourcesInternet Resources
AppletsApplets
Data SourcesData Sources
4. CorrelationCorrelation
CorrelationCorrelation
A measure of association betweenA measure of association between
two numerical variables.two numerical variables.
Example (positive correlation)Example (positive correlation)
Typically, in the summer as theTypically, in the summer as the
temperature increases people aretemperature increases people are
thirstier.thirstier.
5. Specific ExampleSpecific Example
For sevenFor seven
random summerrandom summer
days, a persondays, a person
recorded therecorded the
temperaturetemperature andand
theirtheir waterwater
consumptionconsumption,,
during a three-hourduring a three-hour
period spentperiod spent
outside.outside.
Temperature
(F)
Water
Consumption
(ounces)
75 16
83 20
85 25
85 27
92 32
97 48
99 48
6. How would you describe the graph?How would you describe the graph?
7. How “strong” is the linear relationship?How “strong” is the linear relationship?
8. Measuring the RelationshipMeasuring the Relationship
Pearson’s SamplePearson’s Sample
Correlation Coefficient,Correlation Coefficient, rr
measures themeasures the directiondirection and theand the
strengthstrength of the linear associationof the linear association
between two numerical pairedbetween two numerical paired
variables.variables.
10. Strength of Linear AssociationStrength of Linear Association
r
value
Interpretation
1
perfect positive linear
relationship
0 no linear relationship
-1
perfect negative linear
relationship
12. Other Strengths of AssociationOther Strengths of Association
r value Interpretation
0.9 strong association
0.5 moderate association
0.25 weak association
14. FormulaFormula
= the sum
n = number of paired
items
xi
= input variable yi
= output variable
x = x-bar = mean of
x’s
y = y-bar = mean of
y’s
sx
= standard deviation
of x’s
sy
= standard
deviation of y’s
sum
15. RegressionRegression
RegressionRegression
Specific statistical methodsSpecific statistical methods forfor
finding the “line of best fit” for onefinding the “line of best fit” for one
response (dependent) numericalresponse (dependent) numerical
variable based on one or morevariable based on one or more
explanatory (independent)explanatory (independent)
variables.variables.
16. Curve Fitting vs. RegressionCurve Fitting vs. Regression
RegressionRegression
Includes using statistical methodsIncludes using statistical methods
to assess the "goodness of fit" ofto assess the "goodness of fit" of
the model. (ex. Correlationthe model. (ex. Correlation
Coefficient)Coefficient)
17. Regression: 3 Main PurposesRegression: 3 Main Purposes
To describeTo describe (or model)(or model)
To predictTo predict ((or estimate)or estimate)
To controlTo control (or administer)(or administer)
18. Simple Linear RegressionSimple Linear Regression
Statistical method for findingStatistical method for finding
the “line of best fit”the “line of best fit”
for one response (dependent)for one response (dependent)
numerical variablenumerical variable
based on one explanatorybased on one explanatory
(independent) variable.(independent) variable.
19. Least Squares RegressionLeast Squares Regression
GOALGOAL --
minimize theminimize the
sum of thesum of the
square ofsquare of
the errors ofthe errors of
the datathe data
points.points.
This minimizes theThis minimizes the Mean Square ErrorMean Square Error
20. ExampleExample
Plan an outdoor party.Plan an outdoor party.
EstimateEstimate number of soft drinks tonumber of soft drinks to
buy per person, based on how hotbuy per person, based on how hot
the weather is.the weather is.
Use Temperature/Water data andUse Temperature/Water data and
regressionregression..
21. Steps to Reaching a SolutionSteps to Reaching a Solution
Draw a scatterplot of the data.Draw a scatterplot of the data.
22. Steps to Reaching a SolutionSteps to Reaching a Solution
Draw a scatterplot of the data.Draw a scatterplot of the data.
Visually, consider the strength of theVisually, consider the strength of the
linear relationship.linear relationship.
23. Steps to Reaching a SolutionSteps to Reaching a Solution
Draw a scatterplot of the data.Draw a scatterplot of the data.
Visually, consider the strength of theVisually, consider the strength of the
linear relationship.linear relationship.
If the relationship appears relativelyIf the relationship appears relatively
strong, find the correlation coefficientstrong, find the correlation coefficient
as a numerical verification.as a numerical verification.
24. Steps to Reaching a SolutionSteps to Reaching a Solution
Draw a scatterplot of the data.Draw a scatterplot of the data.
Visually, consider the strength of theVisually, consider the strength of the
linear relationship.linear relationship.
If the relationship appears relativelyIf the relationship appears relatively
strong, find the correlation coefficientstrong, find the correlation coefficient
as a numerical verification.as a numerical verification.
If the correlation is still relativelyIf the correlation is still relatively
strong, then find the simple linearstrong, then find the simple linear
regression line.regression line.
25. Our Next StepsOur Next Steps
Learn to Use the TI-83 forLearn to Use the TI-83 for
Correlation and Regression.Correlation and Regression.
Interpret the Results (in theInterpret the Results (in the
Context of the Problem).Context of the Problem).
26. Finding the Solution: TI-83Finding the Solution: TI-83
Using the TI- 83 graphing calculatorUsing the TI- 83 graphing calculator
Turn on the calculator diagnostics.Turn on the calculator diagnostics.
Enter the data.Enter the data.
Graph a scatterplot of the data.Graph a scatterplot of the data.
Find the equation of the regression lineFind the equation of the regression line
and the correlation coefficient.and the correlation coefficient.
Graph the regression line on a graphGraph the regression line on a graph
with the scatterplot.with the scatterplot.
27. Preliminary StepPreliminary Step
Turn the Diagnostics On.Turn the Diagnostics On.
PressPress 2nd 02nd 0 (for Catalog).(for Catalog).
Scroll down toScroll down to DiagnosticOnDiagnosticOn. The. The
marker points to the right of themarker points to the right of the
words.words.
PressPress ENTERENTER. Press. Press ENTERENTER
again.again.
The wordThe word DoneDone should appear onshould appear on
the right hand side of the screen.the right hand side of the screen.
29. 1. Enter the Data into Lists1. Enter the Data into Lists
PressPress STATSTAT..
UnderUnder EDITEDIT, select, select 1: Edit1: Edit..
Enter x-values (input) intoEnter x-values (input) into L1L1
Enter y-values (output) intoEnter y-values (output) into L2L2..
After data is entered in the lists, goAfter data is entered in the lists, go
toto 2nd MODE2nd MODE to quit and return toto quit and return to
the home screen.the home screen.
Note:Note: If you need to clear out a list,If you need to clear out a list,
for example list 1, place the cursor onfor example list 1, place the cursor on
L1 then hit CLEAR and ENTER .L1 then hit CLEAR and ENTER .
30. 2. Set up the Scatterplot.2. Set up the Scatterplot.
PressPress 2nd Y=2nd Y= (STAT PLOTS).(STAT PLOTS).
SelectSelect 1: PLOT 11: PLOT 1 and hitand hit ENTERENTER..
Use the arrow keys to move theUse the arrow keys to move the
cursor down tocursor down to OnOn and hitand hit ENTERENTER..
Arrow down toArrow down to Type:Type: and select theand select the
first graphfirst graph under Type.under Type.
UnderUnder Xlist:Xlist: EnterEnter L1L1..
UnderUnder Ylist:Ylist: EnterEnter L2L2..
UnderUnder Mark:Mark: select any of these.select any of these.
31. 3. View the Scatterplot3. View the Scatterplot
PressPress 2nd MODE2nd MODE to quit andto quit and
return to the home screen.return to the home screen.
To plot the points, pressTo plot the points, press ZOOMZOOM
and selectand select 9: ZoomStat9: ZoomStat..
The scatterplot will then beThe scatterplot will then be
graphed.graphed.
32. 4. Find the regression line.4. Find the regression line.
PressPress STATSTAT..
PressPress CALCCALC..
SelectSelect 4: LinReg(ax + b)4: LinReg(ax + b)..
PressPress 2nd 12nd 1 (for List 1)(for List 1)
Press thePress the comma keycomma key,,
PressPress 2nd 22nd 2 (for List 2)(for List 2)
PressPress ENTERENTER..
33. 5. Interpreting and Visualizing5. Interpreting and Visualizing
Interpreting the result:Interpreting the result:
y = ax + by = ax + b
The valueThe value ofof aa is theis the slopeslope
The value ofThe value of bb is theis the y-intercepty-intercept
rr is theis the correlation coefficientcorrelation coefficient
rr22
is theis the coefficient of determinationcoefficient of determination
34. 5. Interpreting and Visualizing5. Interpreting and Visualizing
Write down the equation of theWrite down the equation of the
line in slope intercept form.line in slope intercept form.
PressPress Y=Y= and enter the equationand enter the equation
under Y1. (Clear all otherunder Y1. (Clear all other
equations.)equations.)
PressPress GRAPHGRAPH and the line willand the line will
be graphed through the databe graphed through the data
points.points.
36. Interpretation in ContextInterpretation in Context
Regression Equation:Regression Equation:
y=1.5*x - 96.9y=1.5*x - 96.9
Water Consumption =Water Consumption =
1.5*Temperature - 96.91.5*Temperature - 96.9
37. Interpretation in ContextInterpretation in Context
Slope = 1.5 (ounces)/(degrees F)Slope = 1.5 (ounces)/(degrees F)
for each 1 degree F increase infor each 1 degree F increase in
temperature, you expect an increasetemperature, you expect an increase
of 1.5 ounces of water drank.of 1.5 ounces of water drank.
38. Interpretation in ContextInterpretation in Context
y-intercept = -96.9y-intercept = -96.9
For this example,For this example,
when the temperature is 0 degrees F,when the temperature is 0 degrees F,
then a person would drink about -97then a person would drink about -97
ounces of water.ounces of water.
That does not make any sense!That does not make any sense!
Our model is not applicable for x=0.Our model is not applicable for x=0.
39. Prediction ExamplePrediction Example
PredictPredict the amount ofthe amount of
water a person would drink when thewater a person would drink when the
temperature istemperature is 95 degrees F.95 degrees F.
Solution:Solution: Substitute the value of x=95Substitute the value of x=95
(degrees F) into the regression equation(degrees F) into the regression equation
and solve for y (water consumption).and solve for y (water consumption).
If x=95, y=1.5*95 - 96.9 =If x=95, y=1.5*95 - 96.9 = 45.6 ounces.45.6 ounces.
40. Strength of the Association:Strength of the Association: rr22
Coefficient of Determination –Coefficient of Determination – rr22
General Interpretation:General Interpretation: TheThe
coefficient of determination tells thecoefficient of determination tells the
percent of the variationpercent of the variation in thein the
response variable that isresponse variable that is explainedexplained
(determined) by the model and the(determined) by the model and the
explanatory variable.explanatory variable.
41. Interpretation ofInterpretation of rr22
Example:Example: rr22
=92.7%.=92.7%.
Interpretation:Interpretation:
Almost 93% of the variability in theAlmost 93% of the variability in the
amount of water consumed isamount of water consumed is
explained by outside temperatureexplained by outside temperature
using this model.using this model.
Note: Therefore 7% of theNote: Therefore 7% of the
variation in the amount of watervariation in the amount of water
consumed is not explained by thisconsumed is not explained by this
model using temperature.model using temperature.
43. Simple Linear Regression ModelSimple Linear Regression Model
The model forThe model for
simple linear regression issimple linear regression is
There are mathematical assumptions behindThere are mathematical assumptions behind
the concepts thatthe concepts that
we are covering today.we are covering today.
45. Real Life ApplicationsReal Life Applications
Cost Estimating for Future SpaceCost Estimating for Future Space
Flight Vehicles (MultipleFlight Vehicles (Multiple
Regression)Regression)
46. Nonlinear ApplicationNonlinear Application
Predicting when Solar Maximum WillPredicting when Solar Maximum Will
OccurOccur
http://science.msfc.nasa.gov/ssl/pad/http://science.msfc.nasa.gov/ssl/pad/
solar/predict.htmsolar/predict.htm
47. Real Life ApplicationsReal Life Applications
Estimating Seasonal Sales forEstimating Seasonal Sales for
Department Stores (Periodic)Department Stores (Periodic)
48. Real Life ApplicationsReal Life Applications
Predicting Student Grades BasedPredicting Student Grades Based
on Time Spent Studyingon Time Spent Studying
49. Real Life ApplicationsReal Life Applications
. . .. . .
What ideas can you think of?What ideas can you think of?
What ideas can you think of thatWhat ideas can you think of that
your students will relate to?your students will relate to?
50. Practice ProblemsPractice Problems
Measure Height vs. Arm SpanMeasure Height vs. Arm Span
Find line of best fit for height.Find line of best fit for height.
Predict height forPredict height for
one student not inone student not in
data set. Checkdata set. Check
predictability of model.predictability of model.
51. Practice ProblemsPractice Problems
Is there any correlation betweenIs there any correlation between
shoe size and height?shoe size and height?
Does gender make a differenceDoes gender make a difference
in this analysis?in this analysis?
52. Practice ProblemsPractice Problems
Can the number of pointsCan the number of points
scored in a basketball game bescored in a basketball game be
predicted bypredicted by
The time a player plays inThe time a player plays in
the game?the game?
By the player’s height?By the player’s height?
Idea modified from Steven King, Aiken,Idea modified from Steven King, Aiken,
SC. NCTM presentation 1997.)SC. NCTM presentation 1997.)
53. ResourcesResources
Data Analysis and StatisticsData Analysis and Statistics..
Curriculum and EvaluationCurriculum and Evaluation
Standards for School Mathematics.Standards for School Mathematics.
Addenda Series, Grades 9-12.Addenda Series, Grades 9-12.
NCTM. 1992.NCTM. 1992.
Data and Story LibraryData and Story Library. Internet. Internet
Website.Website.
http://lib.stat.cmu.edu/DASL/http://lib.stat.cmu.edu/DASL/
2001.2001.
54. Internet ResourcesInternet Resources
CorrelationCorrelation
Guessing CorrelationsGuessing Correlations - An- An
interactive site that allows you tointeractive site that allows you to
try to match correlation coefficientstry to match correlation coefficients
to scatterplots. University of Illinois,to scatterplots. University of Illinois,
Urbanna Champaign StatisticsUrbanna Champaign Statistics
Program.Program.
http://www.stat.uiuc.edu/~stat100/java/guhttp://www.stat.uiuc.edu/~stat100/java/gu
55. Internet ResourcesInternet Resources
RegressionRegression
Effects of adding anEffects of adding an
OutlierOutlier..
W. West, University of SouthW. West, University of South
Carolina.Carolina.
http://www.stat.sc.edu/~west/javahtml/Rhttp://www.stat.sc.edu/~west/javahtml/R
56. Internet ResourcesInternet Resources
RegressionRegression
Estimate the Regression LineEstimate the Regression Line..
Compare the mean square errorCompare the mean square error
from different regression lines. Canfrom different regression lines. Can
you find the minimum mean squareyou find the minimum mean square
error? Rice University Virtualerror? Rice University Virtual
Statistics Lab.Statistics Lab.
http://www.ruf.rice.edu/~lane/stat_sim/reghttp://www.ruf.rice.edu/~lane/stat_sim/reg
57. Internet Resources: Data SetsInternet Resources: Data Sets
Data and Story Library.Data and Story Library.
Excellent source for small data sets.Excellent source for small data sets.
Search for specific statistical methodsSearch for specific statistical methods
(e.g. boxplots, regression) or for data(e.g. boxplots, regression) or for data
concerning a specific field of interestconcerning a specific field of interest
(e.g. health, environment, sports).(e.g. health, environment, sports).
http://lib.stat.cmu.edu/DASL/http://lib.stat.cmu.edu/DASL/
58. Internet Resources: Data SetsInternet Resources: Data Sets
FEDSTATS.FEDSTATS. "The gateway to"The gateway to
statistics from over 100 U.S. Federalstatistics from over 100 U.S. Federal
agencies"agencies" http://www.fedstats.gov/http://www.fedstats.gov/
"Kid's Pages.""Kid's Pages." (not all related to(not all related to
statistics)statistics)
http://www.fedstats.gov/kids.htmlhttp://www.fedstats.gov/kids.html
59. Internet ResourcesInternet Resources
OtherOther
Statistics Applets. Using WebStatistics Applets. Using Web
Applets to Assist in StatisticsApplets to Assist in Statistics
Instruction. Robin Lock, St.Instruction. Robin Lock, St.
Lawrence University.Lawrence University.
http://it.stlawu.edu/~rlock/maa99/http://it.stlawu.edu/~rlock/maa99/
60. Internet ResourcesInternet Resources
OtherOther
Ten Websites Every StatisticsTen Websites Every Statistics
Instructor Should Bookmark.Instructor Should Bookmark.
Robin Lock, St. LawrenceRobin Lock, St. Lawrence
University.University.
http://it.stlawu.edu/~rlock/10sites.htmhttp://it.stlawu.edu/~rlock/10sites.htm
61. For More Information…For More Information…
On-line version of this presentationOn-line version of this presentation
http://www.mtsu.edu/~statshttp://www.mtsu.edu/~stats
/corregpres/index.html/corregpres/index.html
More information about regressionMore information about regression
VisitVisit STATS @ MTSUSTATS @ MTSU web siteweb site
http://www.mtsu.edu/~statshttp://www.mtsu.edu/~stats