This document summarizes bivariate data and linear regression analysis. It introduces scatterplots and the Pearson correlation coefficient as ways to examine relationships between two variables. A positive correlation indicates that as one variable increases, so does the other, while a negative correlation means one variable increases as the other decreases. The least squares line provides the best fit linear relationship between two variables by minimizing the sum of squared residuals. Calculating the slope and y-intercept of this line allows predicting y-values from x-values. Examples using bus fare and distance data demonstrate these concepts.
The document discusses the t-test, which is a statistical method used to determine if there is a significant difference between the means of two groups. It can be used to compare the means of two independent groups, related groups, or a group's mean to a hypothesized population mean. There are assumptions that must be met for a t-test, including independent observations, normal distribution of data, and homogeneity of variances. The t-test calculates a t-score or t-value which is compared to a critical value to determine if the null hypothesis can be rejected.
This document presents an overview of statistical methods for comparing two populations. It discusses paired sample comparisons and independent sample comparisons. For paired samples, it covers the paired t-test and constructing confidence intervals. For independent samples, it explains how to test whether population means are equal using a z-test or t-test. Several examples are provided to demonstrate these techniques. The document also briefly discusses testing differences in population proportions and variances.
Spearman's rank correlation coefficient is used to measure the strength of association between two ranked variables. It involves ranking the data values and calculating the differences between the ranks to determine if there is a monotonic relationship between the variables. The coefficient value ranges from +1 to -1, where +1 is a perfect increasing monotonic relationship and -1 is a perfect decreasing relationship. The example calculates the Spearman's rank correlation coefficient between the distance of convenience stores from a museum and the price of water bottles sold. It finds the ranks of the distances and prices, takes the differences of the ranks, sums the squared differences, and plugs the values into the Spearman's rank correlation formula to determine the coefficient value.
This document discusses the Spearman rank order correlation coefficient, which is used to examine the association between two ordinally scaled variables when the relationship is not linear. It provides an example of calculating Spearman's rs using rankings of actors by two judges. The rs value of 0.83 is greater than the critical value of 0.738 for the sample size, so the null hypothesis of no difference is rejected.
Standard deviation is a measure of how spread out numbers are in a data set from the mean. It is calculated by taking the difference of each value from the mean, squaring the differences, summing them, and dividing by the number of values minus one, then taking the square root. The higher the standard deviation, the more varied the data.
Binary logistic regression analysis is used to predict a dichotomous dependent variable from continuous and/or categorical independent variables. SPSS is used to conduct binary logistic regression by entering the dependent variable as 1/0 and independent variables as predictors, and the output provides coefficients, odds ratios, classification tables, and goodness of fit tests. Factors like multicollinearity between predictors and sample size need to be considered to develop the best fitting and most predictive logistic regression model.
Correlation and Regression Analysis using SPSS and Microsoft ExcelSetia Pramana
This document discusses correlation and linear regression analysis. It covers correlation coefficients, linear relationships between variables, assumptions of linear regression, and using SPSS and Excel to conduct correlation and regression analyses. Pearson and Spearman correlation coefficients are introduced as measures of the linear association between two continuous variables. Simple and multiple linear regression models are explained as tools to predict an outcome variable from one or more predictor variables.
Mathematics, Statistics, Probability, Randomness, General Probability Rules, General Addition Rules, Conditional Probability, General Multiplication Rules, Bayes’s Rule, Independence
The document discusses the t-test, which is a statistical method used to determine if there is a significant difference between the means of two groups. It can be used to compare the means of two independent groups, related groups, or a group's mean to a hypothesized population mean. There are assumptions that must be met for a t-test, including independent observations, normal distribution of data, and homogeneity of variances. The t-test calculates a t-score or t-value which is compared to a critical value to determine if the null hypothesis can be rejected.
This document presents an overview of statistical methods for comparing two populations. It discusses paired sample comparisons and independent sample comparisons. For paired samples, it covers the paired t-test and constructing confidence intervals. For independent samples, it explains how to test whether population means are equal using a z-test or t-test. Several examples are provided to demonstrate these techniques. The document also briefly discusses testing differences in population proportions and variances.
Spearman's rank correlation coefficient is used to measure the strength of association between two ranked variables. It involves ranking the data values and calculating the differences between the ranks to determine if there is a monotonic relationship between the variables. The coefficient value ranges from +1 to -1, where +1 is a perfect increasing monotonic relationship and -1 is a perfect decreasing relationship. The example calculates the Spearman's rank correlation coefficient between the distance of convenience stores from a museum and the price of water bottles sold. It finds the ranks of the distances and prices, takes the differences of the ranks, sums the squared differences, and plugs the values into the Spearman's rank correlation formula to determine the coefficient value.
This document discusses the Spearman rank order correlation coefficient, which is used to examine the association between two ordinally scaled variables when the relationship is not linear. It provides an example of calculating Spearman's rs using rankings of actors by two judges. The rs value of 0.83 is greater than the critical value of 0.738 for the sample size, so the null hypothesis of no difference is rejected.
Standard deviation is a measure of how spread out numbers are in a data set from the mean. It is calculated by taking the difference of each value from the mean, squaring the differences, summing them, and dividing by the number of values minus one, then taking the square root. The higher the standard deviation, the more varied the data.
Binary logistic regression analysis is used to predict a dichotomous dependent variable from continuous and/or categorical independent variables. SPSS is used to conduct binary logistic regression by entering the dependent variable as 1/0 and independent variables as predictors, and the output provides coefficients, odds ratios, classification tables, and goodness of fit tests. Factors like multicollinearity between predictors and sample size need to be considered to develop the best fitting and most predictive logistic regression model.
Correlation and Regression Analysis using SPSS and Microsoft ExcelSetia Pramana
This document discusses correlation and linear regression analysis. It covers correlation coefficients, linear relationships between variables, assumptions of linear regression, and using SPSS and Excel to conduct correlation and regression analyses. Pearson and Spearman correlation coefficients are introduced as measures of the linear association between two continuous variables. Simple and multiple linear regression models are explained as tools to predict an outcome variable from one or more predictor variables.
Mathematics, Statistics, Probability, Randomness, General Probability Rules, General Addition Rules, Conditional Probability, General Multiplication Rules, Bayes’s Rule, Independence
Correlation and regression analysis are statistical methods used to determine if a relationship exists between variables and describe the nature of that relationship. A scatter plot graphs the independent and dependent variables and allows visualization of any trends in the data. The correlation coefficient measures the strength and direction of the linear relationship between variables, ranging from -1 to 1. Regression finds the linear "best fit" line that minimizes the residuals and can be used to predict dependent variable values.
This presentation introduces regression analysis. It discusses key concepts such as dependent and independent variables, simple and multiple regression, and linear and nonlinear regression models. It also covers different types of regression including simple linear regression, cross-sectional vs time series data, and methods for building regression models like stepwise regression and forward/backward selection. Examples are provided to demonstrate calculating regression equations using the least squares method and computing deviations from mean values.
John Napier, a Scottish mathematician and astronomer, discovered logarithms in the late 16th century as a way to simplify calculations. He introduced the concept of logarithms to ease complex mathematical computations. Napier was also an astrologer and believer in black magic who would travel with a spider and black rooster he claimed were his familiars.
The Chi-Square test of independence is used to determine if two categorical variables are independent or dependent. It examines if understanding one variable depends on the other. The test calculates an observed versus expected frequency for each cell. If the Chi-Square value exceeds the critical value, the null hypothesis of independence is rejected, indicating a dependent relationship. The document provides an example comparing education level and news source, finding the variables are dependent based on a significant Chi-Square value.
The document defines exponential functions as functions of the form f(x) = bx, where b is a positive constant other than 1. It discusses how the graph of an exponential function depends on whether b is greater than or less than 1. Specifically, if b > 1 the graph increases to the right, and if 0 < b < 1 the graph decreases to the right. The document also covers transformations of exponential functions, including vertical and horizontal shifting, reflecting, and stretching/shrinking. It introduces the special number e, defines it as the limit of (1 + 1/n)n as n approaches infinity, and discusses its role in compound interest formulas.
This document discusses statistical tests for comparing groups on continuous and categorical outcomes. For binary outcomes, it describes chi-square tests, logistic regression, McNemar's tests, and conditional logistic regression for independent and correlated groups. For continuous outcomes, it discusses t-tests, ANOVA, linear regression, paired t-tests, repeated measures ANOVA, mixed models, and non-parametric alternatives. It also provides examples of calculating odds ratios, standard errors, and performing hypothesis tests like the two-sample t-test.
Difference between grouped and ungrouped dataAtiq Rehman
This document discusses the topics of ungrouped versus grouped data, frequency distributions, and data range. Ungrouped data refers to raw data that has not been organized, while grouped data has been organized into a frequency distribution by classifying the data into intervals. An example shows ages of managers organized into a frequency distribution with class intervals of 10 years each. The data range is calculated by finding the difference between the largest and smallest values in the set of data.
The document discusses exponential functions and exponential equations. Exponential functions have the form f(x) = bx, where b is the base and x is the exponent. These functions are important in modeling real-world phenomena like population growth. Exponential equations set the exponents of the same base equal to solve for the variable. They can be solved by rewriting all terms to have the same base, setting the exponents equal, and solving the resulting equation.
This document provides an overview of rational functions, equations, and inequalities. It discusses rational expressions, rational equations, rational inequalities, and rational functions. Students are assigned to answer assessment questions on pages 23-25 to test their understanding of these concepts.
- MAP testing will take place this week, with detailed information available in announcements
- Next week, students will begin working on their end-of-year projects
- This document provides information about bivariate data, scatter plots, and lines of best fit for a statistics and probability lesson
The document provides examples for calculating the Pearson Product Moment Correlation Coefficient (r) from bivariate data. It defines r as a measure of the strength of the linear relationship between two variables. Several fully worked examples are shown calculating r from tables of paired data and interpreting the resulting r value based on established thresholds for strength of correlation. Formulas and steps for calculating r are demonstrated throughout.
The document defines and provides information about correlation coefficients. It discusses how correlation coefficients measure the strength and direction of linear relationships between two variables. The range of correlation coefficients is from -1 to 1, where values closer to -1 or 1 indicate stronger linear relationships and a value of 0 indicates no linear relationship. It also provides the formula to calculate correlation coefficients and an example of calculating the correlation coefficient for age and blood pressure data.
How to Find the Slope of a Tangent Line? The slope of a tangent line at a point is its derivative at that point. If a tangent line is drawn for a curve y = f(x) at a point (x0, y0), then its slope (m) is obtained by simply substituting the point in the derivative of the function. i.e., m = (f '(x))(x0, y0).
Probability Distributions for Discrete Variablesgetyourcheaton
This document discusses probability distributions for discrete variables. It begins by defining a probability distribution as a relative frequency distribution of all possible outcomes of an experiment. It provides examples of probability distributions for discrete variables like the binomial distribution. It discusses key aspects of probability distributions like the mean, standard deviation, and different types of distributions like binomial. It provides examples of calculating probabilities, means, and standard deviations for binomial distributions. It discusses the basic characteristics of the binomial distribution and provides an example of constructing a binomial distribution and calculating related probabilities.
This document discusses hypothesis testing, which involves testing whether evidence supports an alternative hypothesis against a null hypothesis. It defines key terms like null hypothesis, alternative hypothesis, type 1 and type 2 errors. The null hypothesis generally assumes no difference, while the alternative hypothesis states a difference. Formulas are provided to calculate sample sizes needed when testing differences between means or proportions. The power, effect size, significance level, and variability estimates all influence the required sample size, with larger power, effect sizes and variability requiring larger samples.
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 document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.
Percentiles are positional measures used to indicate an individual's position within a group. They divide a data set into 100 equal parts, with percentiles (denoted Px) indicating what percent of values are less than a specified value. Common percentiles include the median (P50), quartiles (P25, P50, P75), and deciles. Percentiles are calculated using a formula that determines the position number based on the total number of data points and percentile value. This position is then used to find the corresponding value within ordered data.
This document provides information on estimating population characteristics from sample data, including:
- Point estimates are single numbers based on sample data that represent plausible values of population characteristics.
- Confidence intervals provide a range of plausible values for population characteristics with a specified degree of confidence.
- Formulas are given for constructing confidence intervals for population proportions and means using large sample approximations or t-distributions.
- Guidelines for determining necessary sample sizes to estimate population values within a specified margin of error are also outlined.
The document describes multiple regression models and their applications. It begins by defining a general multiple regression model that relates a dependent variable to multiple predictor variables. It then discusses key aspects of multiple regression models like regression coefficients, the regression function, polynomial regression models, and qualitative predictor variables. The document provides examples of applying multiple regression to model lung capacity based on variables like height, age, gender, and activity level. It describes building different regression models and evaluating their fit and significance.
Correlation and regression analysis are statistical methods used to determine if a relationship exists between variables and describe the nature of that relationship. A scatter plot graphs the independent and dependent variables and allows visualization of any trends in the data. The correlation coefficient measures the strength and direction of the linear relationship between variables, ranging from -1 to 1. Regression finds the linear "best fit" line that minimizes the residuals and can be used to predict dependent variable values.
This presentation introduces regression analysis. It discusses key concepts such as dependent and independent variables, simple and multiple regression, and linear and nonlinear regression models. It also covers different types of regression including simple linear regression, cross-sectional vs time series data, and methods for building regression models like stepwise regression and forward/backward selection. Examples are provided to demonstrate calculating regression equations using the least squares method and computing deviations from mean values.
John Napier, a Scottish mathematician and astronomer, discovered logarithms in the late 16th century as a way to simplify calculations. He introduced the concept of logarithms to ease complex mathematical computations. Napier was also an astrologer and believer in black magic who would travel with a spider and black rooster he claimed were his familiars.
The Chi-Square test of independence is used to determine if two categorical variables are independent or dependent. It examines if understanding one variable depends on the other. The test calculates an observed versus expected frequency for each cell. If the Chi-Square value exceeds the critical value, the null hypothesis of independence is rejected, indicating a dependent relationship. The document provides an example comparing education level and news source, finding the variables are dependent based on a significant Chi-Square value.
The document defines exponential functions as functions of the form f(x) = bx, where b is a positive constant other than 1. It discusses how the graph of an exponential function depends on whether b is greater than or less than 1. Specifically, if b > 1 the graph increases to the right, and if 0 < b < 1 the graph decreases to the right. The document also covers transformations of exponential functions, including vertical and horizontal shifting, reflecting, and stretching/shrinking. It introduces the special number e, defines it as the limit of (1 + 1/n)n as n approaches infinity, and discusses its role in compound interest formulas.
This document discusses statistical tests for comparing groups on continuous and categorical outcomes. For binary outcomes, it describes chi-square tests, logistic regression, McNemar's tests, and conditional logistic regression for independent and correlated groups. For continuous outcomes, it discusses t-tests, ANOVA, linear regression, paired t-tests, repeated measures ANOVA, mixed models, and non-parametric alternatives. It also provides examples of calculating odds ratios, standard errors, and performing hypothesis tests like the two-sample t-test.
Difference between grouped and ungrouped dataAtiq Rehman
This document discusses the topics of ungrouped versus grouped data, frequency distributions, and data range. Ungrouped data refers to raw data that has not been organized, while grouped data has been organized into a frequency distribution by classifying the data into intervals. An example shows ages of managers organized into a frequency distribution with class intervals of 10 years each. The data range is calculated by finding the difference between the largest and smallest values in the set of data.
The document discusses exponential functions and exponential equations. Exponential functions have the form f(x) = bx, where b is the base and x is the exponent. These functions are important in modeling real-world phenomena like population growth. Exponential equations set the exponents of the same base equal to solve for the variable. They can be solved by rewriting all terms to have the same base, setting the exponents equal, and solving the resulting equation.
This document provides an overview of rational functions, equations, and inequalities. It discusses rational expressions, rational equations, rational inequalities, and rational functions. Students are assigned to answer assessment questions on pages 23-25 to test their understanding of these concepts.
- MAP testing will take place this week, with detailed information available in announcements
- Next week, students will begin working on their end-of-year projects
- This document provides information about bivariate data, scatter plots, and lines of best fit for a statistics and probability lesson
The document provides examples for calculating the Pearson Product Moment Correlation Coefficient (r) from bivariate data. It defines r as a measure of the strength of the linear relationship between two variables. Several fully worked examples are shown calculating r from tables of paired data and interpreting the resulting r value based on established thresholds for strength of correlation. Formulas and steps for calculating r are demonstrated throughout.
The document defines and provides information about correlation coefficients. It discusses how correlation coefficients measure the strength and direction of linear relationships between two variables. The range of correlation coefficients is from -1 to 1, where values closer to -1 or 1 indicate stronger linear relationships and a value of 0 indicates no linear relationship. It also provides the formula to calculate correlation coefficients and an example of calculating the correlation coefficient for age and blood pressure data.
How to Find the Slope of a Tangent Line? The slope of a tangent line at a point is its derivative at that point. If a tangent line is drawn for a curve y = f(x) at a point (x0, y0), then its slope (m) is obtained by simply substituting the point in the derivative of the function. i.e., m = (f '(x))(x0, y0).
Probability Distributions for Discrete Variablesgetyourcheaton
This document discusses probability distributions for discrete variables. It begins by defining a probability distribution as a relative frequency distribution of all possible outcomes of an experiment. It provides examples of probability distributions for discrete variables like the binomial distribution. It discusses key aspects of probability distributions like the mean, standard deviation, and different types of distributions like binomial. It provides examples of calculating probabilities, means, and standard deviations for binomial distributions. It discusses the basic characteristics of the binomial distribution and provides an example of constructing a binomial distribution and calculating related probabilities.
This document discusses hypothesis testing, which involves testing whether evidence supports an alternative hypothesis against a null hypothesis. It defines key terms like null hypothesis, alternative hypothesis, type 1 and type 2 errors. The null hypothesis generally assumes no difference, while the alternative hypothesis states a difference. Formulas are provided to calculate sample sizes needed when testing differences between means or proportions. The power, effect size, significance level, and variability estimates all influence the required sample size, with larger power, effect sizes and variability requiring larger samples.
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 document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.
Percentiles are positional measures used to indicate an individual's position within a group. They divide a data set into 100 equal parts, with percentiles (denoted Px) indicating what percent of values are less than a specified value. Common percentiles include the median (P50), quartiles (P25, P50, P75), and deciles. Percentiles are calculated using a formula that determines the position number based on the total number of data points and percentile value. This position is then used to find the corresponding value within ordered data.
This document provides information on estimating population characteristics from sample data, including:
- Point estimates are single numbers based on sample data that represent plausible values of population characteristics.
- Confidence intervals provide a range of plausible values for population characteristics with a specified degree of confidence.
- Formulas are given for constructing confidence intervals for population proportions and means using large sample approximations or t-distributions.
- Guidelines for determining necessary sample sizes to estimate population values within a specified margin of error are also outlined.
The document describes multiple regression models and their applications. It begins by defining a general multiple regression model that relates a dependent variable to multiple predictor variables. It then discusses key aspects of multiple regression models like regression coefficients, the regression function, polynomial regression models, and qualitative predictor variables. The document provides examples of applying multiple regression to model lung capacity based on variables like height, age, gender, and activity level. It describes building different regression models and evaluating their fit and significance.
This document discusses key concepts for collecting data and conducting research studies. It defines variables, data sets, and types of bias that can occur in data collection. Common sampling methods like simple random sampling, stratified sampling, and cluster sampling are described. The document also distinguishes between observational studies and experiments, noting that experiments allow researchers to control variables and determine causal effects. Key aspects of experimental design like treatments, placebos, and control groups are also explained.
This document provides an overview of random variables and probability distributions. It defines discrete and continuous random variables and gives examples of each. Discrete random variables have probabilities associated with each possible value, while continuous random variables are defined by probability density functions where the area under the curve equals the probability. The document discusses how to calculate the mean, variance and standard deviation of discrete random variables from their probability distributions. It also covers how the mean and variance are affected for linear transformations of random variables.
This document discusses methods for comparing two population or treatment means, including notation, hypothesis tests, and confidence intervals. Key points covered include:
1) Notation for comparing two means includes the sample size, mean, variance, and standard deviation for each population or treatment.
2) Hypothesis tests for comparing two means can use a z-test if the population standard deviations are known, or a two-sample t-test if the standard deviations are unknown.
3) Confidence intervals can be constructed for the difference between two population means using a t-distribution, assuming independent random samples of sufficient size or approximately normal populations.
1. A study examined survival times of patients with advanced cancers in different organs (stomach, bronchus, colon, ovary, or breast) treated with ascorbate.
2. An analysis of variance (ANOVA) was used to determine if survival times differed based on the affected organ. ANOVA compares the means of multiple groups and tests if they are equal.
3. The ANOVA test statistic, F, compares the variation between groups (mean square for treatments) to the variation within groups (mean square for error). If F exceeds a critical value, then at least one group mean is significantly different from the others.
1. The document discusses probability and chance experiments. It provides examples to illustrate key concepts such as sample space, events, and how to calculate probabilities.
2. One example examines student food preferences in a cafeteria, with the sample space consisting of all possible combinations of student gender and food line choice.
3. The document also covers conditional probability, explaining how to calculate the probability of an event given that another event has occurred. An example calculates the probability of nausea given being seated in the front of a bus.
This document discusses various numerical methods for describing data, including measures of central tendency (mean, median), variability (range, variance, standard deviation), and graphical representations (boxplots). It provides examples and formulas for calculating the mean, median, quartiles, interquartile range, variance, standard deviation, and constructing boxplots. Outliers are defined as observations more than 1.5 times the interquartile range from the quartiles.
1. The document discusses hypothesis testing using a single sample. It outlines the formal structure of hypothesis tests including the null and alternative hypotheses.
2. Common hypothesis tests are presented including tests of a population proportion, mean, and variability. Examples of hypotheses and solutions are provided.
3. The key steps in a hypothesis testing analysis are defined including stating the hypotheses, selecting the significance level, computing the test statistic and p-value, and making a conclusion. Large sample and small sample tests are described.
This document provides examples and explanations of various graphical methods for describing data, including frequency distributions, bar charts, pie charts, stem-and-leaf diagrams, histograms, and cumulative relative frequency plots. It demonstrates how to construct these graphs using sample data on student weights, grades, ages, and other examples. The goal is to help readers understand different ways to visually represent data distributions and patterns.
1. The document discusses categorical data analysis and goodness-of-fit tests. It introduces concepts such as univariate categorical data, expected counts, the chi-square test statistic, and assumptions of the chi-square test.
2. An example analyzes faculty status data from a university using a goodness-of-fit test to determine if the proportions are equal across categories. The test fails to reject the null hypothesis that the proportions are equal.
3. Tests for homogeneity and independence in two-way tables are described. Examples calculate expected counts and perform chi-square tests to compare populations' category proportions.
This document discusses the importance of statistics and introduces key concepts. It explains that statistics involves collecting, analyzing, and drawing conclusions from data. It also defines important statistical terms like population, sample, variable, and different types of data. Frequency distributions are introduced as a way to organize categorical data by displaying the categories and associated frequencies or relative frequencies. An example frequency distribution is provided using vision correction data from a classroom example.
This document provides an overview of simple linear regression and correlation. It defines key concepts such as the population regression line, the simple linear regression model equation, and assumptions of the model. Examples are provided to demonstrate calculating the least squares regression line, interpreting the slope and intercept, and evaluating goodness of fit using r-squared. Formulas are given for computing sums of squares, estimating the standard deviation of residuals, and constructing confidence intervals for the slope of the population regression line.
This document discusses sampling distributions and the central limit theorem. It defines key terms like population, statistic, and sampling distribution. It shows examples of how sampling distributions become more normal and less variable as the sample size increases. The central limit theorem states that for large sample sizes, the sampling distribution of the sample mean will be approximately normally distributed even if the population is not. It provides properties and rules for the sampling distributions of the sample mean and sample proportion.
This document discusses summarizing bivariate data using scatterplots and correlation. It provides an example of fare data from a bus company that is modeled using linear and nonlinear regression. Linear regression finds a strong positive correlation between distance and fare, but the relationship is better modeled nonlinearly using the logarithm of distance. The nonlinear model accounts for 96.9% of variation in fares compared to 84.9% for the linear model.
Correlation by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
The regression coefficients are 0.8 and 0.2.
The coefficient of correlation r is the geometric mean of the regression coefficients, which is:
√(0.8 × 0.2) = 0.4
Therefore, the value of the coefficient of correlation is 0.4.
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Chapter 10: Correlation and Regression
10.2: Regression
This document discusses correlation and regression analysis. It begins by outlining the chapter's objectives and providing an introduction to investigating relationships between variables using statistical analysis. The document then presents examples of collecting data to study potential relationships between variables like stone dimensions, human heights and weights, and sprint and long jump performances. It introduces various statistical measures for quantifying relationships in data, including covariance, Pearson's product moment correlation coefficient, and Spearman's rank correlation coefficient. Examples are provided to demonstrate calculating and interpreting these statistics. Limitations of correlation analysis are also noted.
Simple Regression presentation is a
partial fulfillment to the requirement in PA 297 Research for Public Administrators, presented by Atty. Gayam , Dr. Cabling and Mr. Cagampang
1) The document discusses simple linear regression using a scatter diagram and data from a study of employees' years of working experience and income.
2) It presents the scatter diagram and shows how to draw a trend line to roughly estimate dependent variable (income) values from the independent variable (years experience).
3) Equations for the least squares linear regression line are provided, including how to calculate the standard error of estimate, which is interpreted as the standard deviation around the regression line.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses linear regression and can analyze effects across multiple dependent variables.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r2, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both linear regression and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses linear regression and can analyze effects across multiple dependent variables.
Correlation & Regression for Statistics Social Sciencessuser71ac73
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r2, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both linear regression and multiple regression.
The document discusses regression analysis, including definitions, uses, calculating regression equations from data, graphing regression lines, the standard error of estimate, and limitations. Regression analysis is a statistical technique used to understand the relationship between variables and allow for predictions. The document provides examples of calculating regression equations from various data sets and determining the standard error of estimate.
Identify the independent and dependent variable;
Draw the best fit line on a scatter plot;
Calculate the slope and the y-intercept of the regression line;
Interpret the calculated slope and the y-intercept of the regression line;
Predict the value of the dependent variable given the value of the independent variable; and
Solve problems involving regression analysis.
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2. 2
A multivariate data set consists of measurements
or observations on each of two or more variables.
The classroom data set introduced in the slides for
Chapter 1 is a multivariate data set. The data set
includes observations on the variables: age, weight,
height, gender, vision (correction method), and
smoke (status). Age, weight and height are
numerical variables while gender, vision and smoke
are categorical variables.
Terms
3. 3
A bivariate data set consists of
measurements or observations on each
of two variables.
For the rest of this chapter we will
concentrate on dealing with bivariate data
sets where both variables are numeric.
Terms
4. 4
Scatterplots
A scatterplot is a plot of pairs of observed
values (both quantitative) of two different
variables.
When one of the variables is considered to be a
response variable (y) and the other an explanatory
variable (x). The explanatory variable is usually
plotted on the x axis.
5. 5
Example
A sample of one-way
Greyhound bus fares
from Rochester, NY to
cities less than 750 miles
was taken by going to
Greyhound’s website.
The following table
gives the destination
city, the distance and the
one-way fare. Distance
should be the x axis and
the Fare should be the y
axis.
Destination City Distance
Standard
One-Way
Fare
Albany, NY 240 39
Baltimore, MD 430 81
Buffalo, NY 69 17
Chicago, IL 607 96
Cleveland, OH 257 61
Montreal, QU 480 70.5
New York City, NY 340 65
Ottawa, ON 467 82
Philadelphia, PA 335 67
Potsdam, NY 239 47
Syracuse, NY 95 20
Toronto, ON 178 35
Washington, DC 496 87
7. 7
Comments
The axes need not intersect at (0,0).
For each of the axes, the scale should be
chosen so that the minimum and maximum
values on the scale are convenient and the
values to be plotted are between the two
values.
Notice that for this example,
1. The x axis (distance) runs from 50 to 650 miles
where the data points are between 69 and 607.
2. The y axis (fare) runs from $10 to $100 where the
data points are between $17 and $96.
8. 8
Further Comments
It is possible that two points might have the same
x value with different y values. Notice that
Potsdam (239) and Albany (240) come very
close to having the same x value but the y values
are $8 apart. Clearly, the value of y in not
determined solely by the x value (there are
factors other than distance that affect the fare.
In this example, the y value tends to increase a x
increases. We say that there is a positive
relationship between the variables distance and
fare.
It appears that the y value (fare) could be
predicted reasonably well from the x value
(distance) by finding a line that is close to the
points in the plot.
9. 9
Association
Positive Association - Two variables are
positively associated when above-average
values of one tend to accompany above-
average values of the other and below-
average values tend similarly to occur
together. (i.e., Generally speaking, the y
values tend to increase as the x values
increase.)
Negative Association - Two variables are
negatively associated when above-average
values of one accompany below-average
values of the other, and vice versa. (i.e.,
Generally speaking, the y values tend to
decrease as the x values increase.)
10. 10
The Pearson Correlation Coefficient
A measure of the strength of the linear
relationship between the two variables is
called the Pierson correlation coefficient.
The Pearson sample correlation coefficient is
defined by
( )( )yxx y
y y
s
x x
sz z
r
n 1 n 1
−−
= =
− −
∑∑
11. 11
Example Calculation
x y
240 39 -0.5214 -0.7856 0.4096
430 81 0.6357 0.8610 0.5473
69 17 -1.5627 -1.6481 2.5755
607 96 1.7135 1.4491 2.4831
257 61 -0.4178 0.0769 -0.0321
480 70.5 0.9402 0.4494 0.4225
340 65 0.0876 0.2337 0.0205
467 82 0.8610 0.9002 0.7751
335 67 0.0571 0.3121 0.0178
239 47 -0.5275 -0.4720 0.2489
95 20 -1.4044 -1.5305 2.1494
178 35 -0.8989 -0.9424 0.8472
496 87 1.0376 1.0962 1.1374
11.6021
x
x-x
s y
y-y
s x y
x-x y-y
s s
x
y
x 325.615
s 164.2125
y=59.0385
s 25.506
=
=
=
11.601
r 0.9668
13 1
= =
−
18. 18
Properties of r
The value of r does not depend on the unit of
measurement for each variable.
The value of r does not depend on which of the
two variables is labeled x.
The value of r is between –1 and +1.
The correlation coefficient is
a) –1 only when all the points lie on a
downward-sloping line, and
b) +1 only when all the points lie on an
upward-sloping line.
The value of r is a measure of the extent to
which x and y are linearly related.
19. 19
Consider the following bivariate data set:
An Interesting Example
x y
1.2 23.3
2.5 21.5
6.5 12.2
13.1 3.9
24.2 4.0
34.1 18.0
20.8 1.7
37.5 26.1
20. 20
An Interesting Example
Computing the Pearson correlation coefficient,
we find that r = 0.001
x y
1.2 23.3 -1.167 0.973 -1.136
2.5 21.5 -1.074 0.788 -0.847
6.5 12.2 -0.788 -0.168 0.133
13.1 3.9 -0.314 -1.022 0.322
24.2 4.0 0.481 -1.012 -0.487
34.1 18.0 1.191 0.428 0.510
20.8 1.7 0.237 -1.249 -0.296
37.5 26.1 1.434 1.261 1.810
0.007
r = 0.001
X
x x
s
−
y
y y
s
−
X y
x x y y
s s
− −
X y
1 x x y y 1
r (0.007) 0.001
n 1 s s 7
− −
= = = −
∑
x yx 17.488, s 13.951, y 13.838, s 9.721= = = =
21. 21
With a sample Pearson correlation
coefficient, r = 0.001, one would note that
there seems to be little or no linearity to the
relationship between x and y.
Be careful that you do not infer that there is
no relationship between x and y.
An Interesting Example
22. 22
Note (below) that there appears to be an
almost perfect quadratic relationship
between x and y when the scatterplot is
drawn.
An Interesting Example
Scatterplot
0.0
5.0
10.0
15.0
20.0
25.0
30.0
0 5 10 15 20 25 30 35 40
x
y
Scatterplot
0.0
5.0
10.0
15.0
20.0
25.0
30.0
0 5 10 15 20 25 30 35 40
x
y
23. 23
Linear Relations
The relationship y = a + bx is the equation
of a straight line.
The value b, called the slope of the line, is
the amount by which y increases when x
increase by 1 unit.
The value of a, called the intercept (or
sometimes the vertical intercept) of the
line, is the height of the line above the value
x = 0.
24. 24
Example
x
y
0 2 4 6 8
0
5
10
15
y = 7 + 3x
a = 7
x increases by 1
y increases by b = 3
25. 25
Example
y
y = 17 - 4x
x increases by 1
y changes by b = -4 (i.e., changes by –4)
0 2 4 6 8
0
5
10
15
a = 17
26. 26
Least Squares Line
The most widely used criterion for
measuring the goodness of fit of a line
y = a + bx to bivariate data (x1, y1),
(x2, y2),…, (xn, yn) is the sum of the of the
squared deviations about the line:
[ ]
[ ] [ ]
2
2 2
1 1 n n
y (a bx)
y (a bx ) y (a bx )
− +
= − + + + − +
∑
K
The line that gives the best fit to the data is the one
that minimizes this sum; it is called the least squares
line or sample regression line.
27. 27
Coefficients a and b
The slope of the least
squares line is
( ) ( )
( )
2
x x y y
b
x x
− − =
−
∑
∑
And the y intercept is a y bx= −
We write the equation of the least squares line as
where the ^ above y emphasizes that (read as y-hat)
is a prediction of y resulting from the substitution of a
particular value into the equation.
ˆy a bx= +
ˆy
30. 30
Calculations
From the previous slide, we have
The regression line is ˆy 10.138 0.150 x.18= +
Also n 13, x 4233 and y 768
4233 768
so x 325.615 and y 59.0385
13 13
This gives
a y - bx 59.0385- 0.15018(325.615) 10.138
= = =
= = = =
= = =
∑ ∑
( ) ( )
( )
( ) ( )
( )
2
2
x x y y 48596.19 and
x x 323589.08
So
x x y y 48596.19
b 0.15018
323589.08x x
− − =
− =
− − = = =
−
∑
∑
∑
∑
33. 33
Greyhound Example Revisited
Using the calculation formula we have:
Notice that we get the same result.
( )( )
( )
( ) ( )
( )
2 2
2
2
x y 4233 768
298506xy
13nb
4233x
1701919x
13n
485
n 13, x 4233, y 768
x 1701919, and xy 298506
so
Asbefore a y - bx 59.0385- 0.15018(325.
96.19
0.15018
323589.
615) 10.138
and the regression line i ˆys
1
−−
= =
−−
=
= = =
= =
= = =
=
∑ ∑
∑
∑
∑
∑ ∑
∑ ∑
10.138 0.15 x.018= +
34. 34
Three Important Questions
To examine how useful or effective the line
summarizing the relationship between x and y,
we consider the following three questions.
1. Is a line an appropriate way to summarize the relationship between the two variables?
2. Are there any unusual aspects of the data set that we need to consider before
proceeding to use the regression line to make predictions?
3. If we decide that it is reasonable to use the regression line as a basis for prediction,
how accurate can we expect predictions based on the regression line to be?
35. 35
Terminology
The predicted or fitted values result
from substituting each sample x value
into the equation for the least squares
line. This gives
=1st predicted value
=2nd predicted value
=nth predicted value
1 1
2 2
n n
ˆy a bx
ˆy a bx
...
ˆy a bx
= +
= +
= +
The predicted or fitted values result
from substituting each sample x value
into the equation for the least squares
line. This gives
=1st predicted value
=2nd predicted value
=nth predicted value
1 1
2 2
n n
ˆy a bx
ˆy a bx
...
ˆy a bx
= +
= +
= +
The residuals for the least squares line are the
values: 1 1 2 2 n n
y y ,y y , ...,y yˆ ˆ ˆ− − −
The residuals for the least squares line are the
values: 1 1 2 2 n n
y y ,y y , ...,y yˆ ˆ ˆ− − −
38. 38
0
x
Residual
Residual Plot - What to look for.
Isolated points or patterns indicate potential
problems.
Ideally the the points should be randomly
spread out above and below zero.
This residual plot indicates no systematic
bias using the least squares line to predict
the y value.
Generally this is the kind of pattern that you
would like to see.
Note:
1.Values below 0 indicate over prediction
2.Values above 0 indicate under prediction.
39. 39
6005004003002001000
10
0
-10
x
Residual
Residuals Versus x
(response is y)
The Greyhound example continued
For the Greyhound example, it appears that the line
systematically predicts fares that are too high for cities
close to Rochester and predicts fares that are too little for
most cities between 200 and 500 miles.
Predicted
fares are
too high.
Predicted
fares are
too low.
40. 40
1009080706050403020
10
0
-10
Fitted Value
Residual
Residuals Versus the Fitted Values
(response is y)
More Residual Plots
Another common type of residual plot is a scatter
plot of the data pairs ( , residual). The following plot
was produced by Minitab for the Greyhound data.
Notice, that this residual plot shows the same type of
systematic problems with the model.
yˆ
Another common type of residual plot is a scatter
plot of the data pairs ( , residual). The following plot
was produced by Minitab for the Greyhound data.
Notice, that this residual plot shows the same type of
systematic problems with the model.
yˆ
41. 41
Definition formulae
The total sum of squares, denoted by
SSTo, is defined as
2 2 2
1 2 n
2
SSTo (y y) (y y) (y y)
(y y)∑
= − + − + + −
= −
L
The residual sum of squares, denoted by
SSResid, is defined as
2 2 2
1 1 2 2 n n
2
SSResid (y y ) (y y ) (y y )ˆ ˆ ˆ
(y y)ˆ∑
= − + − + + −
= −
L
42. 42
Calculational formulae
SSTo and SSResid are generally found
as part of the standard output from most
statistical packages or can be obtained
using the following computational
formulas:
( )2
2 y
SSTo y
n
∑
∑= −
2
SSResid y a y b xy∑ ∑ ∑= − −
The coefficient of determination, r2, can be
computed as 2 SSResid
r 1
SSTo
= −
43. 43
Coefficient of Determination
The coefficient of determination,
denoted by r2
, gives the proportion of
variation in y that can be attributed to an
approximate linear relationship between x
and y.
Note that the coefficient of determination is
the square of the Pearson correlation
coefficient.
44. 44
Greyhound Example Revisited
2
n 13, y 768, y 53119, xy 298506
b 0.150179 and a 10.1380
∑ ∑ ∑= = = =
= =
( )2 2
2
2
y 768
SSTo y 53119 78072.2
n 13
SSResid y a y b xy
53119 10.1380(768) 0.150179(298506)
509.117
∑
∑
∑ ∑ ∑
= − = − =
= − −
= − −
=
45. 45
We can say that 93.5% of the variation in the
Fare (y) can be attributed to the least squares
linear relationship between distance (x) and
fare.
Greyhound Example Revisited
2 SSResid 509.117
r 1 1 0.9348
SSTo 7807.23
= − = − =
46. 46
More on variability
The standard deviation about the least
squares line is denoted se and given by
se is interpreted as the “typical” amount by
which an observation deviates from the least
squares line.
e
SSResid
s
n 2
=
−
47. 47
The “typical” deviation of actual fare from the
prediction is $6.80.
Greyhound Example Revisited
e
SSResid 509.117
s $6.80
n 2 11
= = =
−
48. 48
Minitab output for Regression
Regression Analysis: Standard Fare versus Distance
The regression equation is
Standard Fare = 10.1 + 0.150 Distance
Predictor Coef SE Coef T P
Constant 10.138 4.327 2.34 0.039
Distance 0.15018 0.01196 12.56 0.000
S = 6.803 R-Sq = 93.5% R-Sq(adj) = 92.9%
Analysis of Variance
Source DF SS MS F P
Regression 1 7298.1 7298.1 157.68 0.000
Residual Error 11 509.1 46.3
Total 12 7807.2
SSTo
SSResidse
r2
a b
Least squares
regression line
49. 49
The Greyhound problem
with additional data
The sample of fares and mileages from
Rochester was extended to cover a total of
20 cities throughout the country. The
resulting data and a scatterplot are given on
the next few slides.
50. 50
Extended Greyhound Fare Example
Distance
Standard
Fare
Buffalo, NY 69 17
New York City 340 65
Cleveland, OH 257 61
Baltimore, MD 430 81
Washington, DC 496 87
Atlanta, GE 998 115
Chicago, IL 607 96
San Francisco 2861 159
Seattle, WA 2848 159
Philadelphia, PA 335 67
Orlando, FL 1478 109
Phoenix, AZ 2569 149
Houston, TX 1671 129
New Orleans, LA 1381 119
Syracuse, NY 95 20
Albany, NY 240 39
Potsdam, NY 239 47
Toronto, ON 178 35
Ottawa, ON 467 82
Montreal, QU 480 70.5
52. 52
3000200010000
30
20
10
0
-10
-20
-30
Distance
Residual
Residuals Versus Distance
(response is Standard)
3000200010000
150
100
50
0
Distance
StandardFar
S = 17.4230 R-Sq = 84.9 % R-Sq(adj) = 84.1 %
Standard Far = 46.0582 + 0.0435354 Distance
Regression Plot
Minitab reports the correlation coefficient, r=0.921,
R2
=0.849, se=$17.42 and the regression line
Standard Fare = 46.058 + 0.043535 Distance
Notice that even though the correlation coefficient is
reasonably high and 84.9 % of the variation in the
Fare is explained, the linear model is not very
usable.
Extended Greyhound Fare
Example
53. 53
Nonlinear Regression Example
Distance Log10(distance)
Standard
Fare
Buffalo, NY 69 1.83885 17
New York City 340 2.53148 65
Cleveland, OH 257 2.40993 61
Baltimore, MD 430 2.63347 81
Washington, DC 496 2.69548 87
Atlanta, GE 998 2.99913 115
Chicago, IL 607 2.78319 96
San Francisco 2861 3.45652 159
Seattle, WA 2848 3.45454 159
Philadelphia, PA 335 2.52504 67
Orlando, FL 1478 3.16967 109
Phoenix, AZ 2569 3.40976 149
Houston, TX 1671 3.22298 129
New Orleans, LA 1381 3.14019 119
Syracuse, NY 95 1.97772 20
Albany, NY 240 2.38021 39
Potsdam, NY 239 2.37840 47
Toronto, ON 178 2.25042 35
Ottawa, ON 467 2.66932 82
Montreal, QU 480 2.68124 70.5
54. 54
From the previous slide we can see that the
plot does not look linear, it appears to have a
curved shape. We sometimes replace the one
of both of the variables with a transformation of
that variable and then perform a linear
regression on the transformed variables. This
can sometimes lead to developing a useful
prediction equation.
For this particular data, the shape of the curve
is almost logarithmic so we might try to replace
the distance with log10(distance) [the logarithm
to the base 10) of the distance].
Nonlinear Regression Example
55. 55
Minitab provides the following output.
Regression Analysis: Standard Fare versus Log10(Distance)
The regression equation is
Standard Fare = - 163 + 91.0 Log10(Distance)
Predictor Coef SE Coef T P
Constant -163.25 10.59 -15.41 0.000
Log10(Di 91.039 3.826 23.80 0.000
S = 7.869 R-Sq = 96.9% R-Sq(adj) = 96.7%
High r2
96.9% of the
variation attributed
to the model
Typical Error = $7.87
Reasonably good
Nonlinear Regression Example
56. 56
The rest of the Minitab output follows
Analysis of Variance
Source DF SS MS F P
Regression 1 35068 35068 566.30 0.000
Residual Error 18 1115 62
Total 19 36183
Unusual Observations
Obs Log10(Di Standard Fit SE Fit Residual St Resid
11 3.17 109.00 125.32 2.43 -16.32 -2.18R
R denotes an observation with a large standardized residual
The only outlier is Orlando and as you’ll see from the
next two slides, it is not too bad.
Nonlinear Regression Example
57. 57
Looking at the plot of the residuals against distance,
we see some problems. The model over estimates
fares for middle distances (1000 to 2000 miles) and
under estimates for longer distances (more than
2000 miles
3000200010000
10
0
-10
-20
Distance
Residual
Residuals Versus Distance
(response is Standard)
Nonlinear Regression Example
58. 58
When we look at how the prediction curve looks on
a graph that has the Standard Fare and
log10(Distance) axes, we see the result looks
reasonably linear.
3.53.02.52.0
150
100
50
0
Log10(Distance)
StandardFare
S = 7.86930 R-Sq = 96.9 % R-Sq(adj) = 96.7 %
Standard Fare = -163.246 + 91.0389 Log10(Distance)
Regression Plot
Nonlinear Regression Example
59. 59
When we look at how the prediction curve
looks on a graph that has the Standard Fare
and Distance axes, we see the result appears
to work fairly well.
By and large, this prediction model for the
fares appears to work reasonable well.
Nonlinear Regression Example
3000200010000
150
100
50
0
Distance
StandardFare
Prediction Model