Multiple regression analysis allows modeling of a dependent variable (y) as a function of multiple independent variables (x1, x2,...xk). The model takes the form y = β0 + β1x1 + β2x2 +...+ βkxk + u. For classical hypothesis testing of the coefficients (β1, β2,...βk), the model assumes u is independent of the x's and normally distributed. The t-test can be used to test hypotheses about individual coefficients, such as H0: βj = 0, while the F-test allows jointly testing hypotheses about multiple coefficients, such as exclusion restrictions that several coefficients equal zero. P-values indicate the probability of observing test statistics at least
This document discusses concepts related to calculus including limits, continuity, and derivatives of functions. Specifically, it covers:
- Definitions and theorems related to limits, continuity, and derivatives of algebraic functions.
- Evaluating limits, determining continuity of functions, and taking derivatives of algebraic functions using basic theorems of differentiation.
- The objective is for students to be able to evaluate limits, determine continuity, and find derivatives of continuous algebraic functions in explicit or implicit form after discussing these calculus concepts.
This document provides instruction on applying derivatives to solve various types of application problems. It begins by outlining objectives of analyzing and solving application problems involving derivatives as instantaneous rates of change or tangent line slopes. Examples of application problems covered include writing equations of tangent and normal lines, curve tracing, optimization problems, and related rates problems involving time rates. The document then provides definitions and examples of using derivatives to find slopes of curves and tangent lines. It also covers concepts like concavity, points of inflection, maxima/minima, and solving optimization problems using derivatives. Finally, it gives examples of solving related rates problems involving time-dependent variables.
The document provides information on correlation and linear regression. It defines correlation as the association between two variables and discusses how the correlation coefficient r measures the strength of this linear association. It then discusses:
- Computing r from sample data
- Testing the hypothesis that r = 0 using a t-test
- Computing the linear regression equation and coefficient of determination
- Using the regression equation to make predictions when there is a significant linear correlation
Two examples are then provided to demonstrate computing r from data, testing for a significant correlation, finding the regression equation, and making a prediction.
The document outlines various formulae that students are expected to know, understand, or be able to use for the GCSE Mathematics exam. It presents formulae for the quadratic formula, circumference and area of a circle, Pythagoras' theorem, and trigonometry. It also lists formulae for perimeter, area, surface area, volume, compound interest, and probability that students should understand but won't be provided. Finally, it mentions kinematics formulae and calculators that may be provided or useful for questions involving various required formulae.
This document discusses the assumptions of multiple linear regression analysis and hypothesis testing using the t-test. It outlines the classical linear model assumptions including that errors are independent and normally distributed. It also describes how the t-statistic is used to test hypotheses about regression coefficients, such as H0: βj = 0, and how the t-statistic is compared to critical values from the t-distribution to determine whether to reject the null hypothesis.
This document provides information about regression analysis and linear regression. It defines regression analysis as using relationships between quantitative variables to predict a dependent variable from independent variables. Linear regression finds the best fitting straight line relationship between variables. The simple linear regression equation is given as Y = a + bX, where a and b are estimated parameters calculated from sample data. An example is worked through, showing how to calculate the regression equation from data, graph the relationship, and use the equation to estimate values.
This document provides an overview of simple linear regression analysis. It discusses estimating regression coefficients using the least squares method, interpreting the regression equation, assessing model fit using measures like the standard error of the estimate and coefficient of determination, testing hypotheses about regression coefficients, and using the regression model to make predictions.
Multiple regression analysis allows modeling of a dependent variable (y) as a function of multiple independent variables (x1, x2,...xk). The model takes the form y = β0 + β1x1 + β2x2 +...+ βkxk + u. For classical hypothesis testing of the coefficients (β1, β2,...βk), the model assumes u is independent of the x's and normally distributed. The t-test can be used to test hypotheses about individual coefficients, such as H0: βj = 0, while the F-test allows jointly testing hypotheses about multiple coefficients, such as exclusion restrictions that several coefficients equal zero. P-values indicate the probability of observing test statistics at least
This document discusses concepts related to calculus including limits, continuity, and derivatives of functions. Specifically, it covers:
- Definitions and theorems related to limits, continuity, and derivatives of algebraic functions.
- Evaluating limits, determining continuity of functions, and taking derivatives of algebraic functions using basic theorems of differentiation.
- The objective is for students to be able to evaluate limits, determine continuity, and find derivatives of continuous algebraic functions in explicit or implicit form after discussing these calculus concepts.
This document provides instruction on applying derivatives to solve various types of application problems. It begins by outlining objectives of analyzing and solving application problems involving derivatives as instantaneous rates of change or tangent line slopes. Examples of application problems covered include writing equations of tangent and normal lines, curve tracing, optimization problems, and related rates problems involving time rates. The document then provides definitions and examples of using derivatives to find slopes of curves and tangent lines. It also covers concepts like concavity, points of inflection, maxima/minima, and solving optimization problems using derivatives. Finally, it gives examples of solving related rates problems involving time-dependent variables.
The document provides information on correlation and linear regression. It defines correlation as the association between two variables and discusses how the correlation coefficient r measures the strength of this linear association. It then discusses:
- Computing r from sample data
- Testing the hypothesis that r = 0 using a t-test
- Computing the linear regression equation and coefficient of determination
- Using the regression equation to make predictions when there is a significant linear correlation
Two examples are then provided to demonstrate computing r from data, testing for a significant correlation, finding the regression equation, and making a prediction.
The document outlines various formulae that students are expected to know, understand, or be able to use for the GCSE Mathematics exam. It presents formulae for the quadratic formula, circumference and area of a circle, Pythagoras' theorem, and trigonometry. It also lists formulae for perimeter, area, surface area, volume, compound interest, and probability that students should understand but won't be provided. Finally, it mentions kinematics formulae and calculators that may be provided or useful for questions involving various required formulae.
This document discusses the assumptions of multiple linear regression analysis and hypothesis testing using the t-test. It outlines the classical linear model assumptions including that errors are independent and normally distributed. It also describes how the t-statistic is used to test hypotheses about regression coefficients, such as H0: βj = 0, and how the t-statistic is compared to critical values from the t-distribution to determine whether to reject the null hypothesis.
This document provides information about regression analysis and linear regression. It defines regression analysis as using relationships between quantitative variables to predict a dependent variable from independent variables. Linear regression finds the best fitting straight line relationship between variables. The simple linear regression equation is given as Y = a + bX, where a and b are estimated parameters calculated from sample data. An example is worked through, showing how to calculate the regression equation from data, graph the relationship, and use the equation to estimate values.
This document provides an overview of simple linear regression analysis. It discusses estimating regression coefficients using the least squares method, interpreting the regression equation, assessing model fit using measures like the standard error of the estimate and coefficient of determination, testing hypotheses about regression coefficients, and using the regression model to make predictions.
Ordinary least squares linear regressionElkana Rorio
Ordinary Least Squares Linear Regression is commonly used but often misunderstood and misapplied. It works by minimizing the sum of squared errors between predictions and actual values in the training data to determine coefficients for the linear regression equation. However, it is very sensitive to outliers in the data which can dramatically affect the determined coefficients and reduce prediction accuracy. Alternative regression techniques like least absolute deviations are more robust to outliers but less computationally efficient. Preprocessing data to remove or de-emphasize outliers can help address these issues with Ordinary Least Squares regression.
The document discusses simple linear regression and correlation methods. It defines deterministic and probabilistic models for describing the relationship between two variables. A simple linear regression model assumes a population regression line with intercept a and slope b, where observations may deviate from the line by some random error e. Key assumptions of the model are that e has a normal distribution with mean 0 and constant variance across values of x, and errors are independent. The slope b estimates the average change in y per unit change in x.
Applied Numerical Methods Curve Fitting: Least Squares Regression, InterpolationBrian Erandio
Correction with the misspelled langrange.
and credits to the owners of the pictures (Fantasmagoria01, eugene-kukulka, vooga, and etc.) . I do not own all of the pictures used as background sorry to those who aren't tagged.
The presentation contains topics from Applied Numerical Methods with MATHLAB for Engineers and Scientist 6th and International Edition.
The Kolmogorov-Smirnov test is used to test if an observed frequency distribution matches an expected theoretical distribution. It compares the cumulative distribution functions of the observed and expected distributions. The test statistic is the largest difference between these cumulative distributions. If this difference is larger than a critical value from tables, the null hypothesis of a good fit is rejected. An example calculates the test statistic for observed data compared to a normal distribution, and finds it is less than the critical value so the null hypothesis is accepted.
This document provides an overview of least-squares regression techniques including:
- Simple linear regression to fit a line to data
- Polynomial regression to fit higher order curves
- Multiple regression to fit surfaces using two or more variables
It discusses calculating regression coefficients, quantifying errors, and performing statistical analysis of the regression results including determining confidence intervals. Examples are provided to demonstrate applying these techniques.
I. The document discusses the method of ordinary least squares (OLS) regression analysis. OLS chooses estimates that minimize the sum of squared residuals between the actual and predicted y-values.
II. OLS provides point estimates for regression parameters and makes assumptions such as a linear relationship between variables, independent and homoscedastic errors, and no autocorrelation.
III. Monte Carlo experiments can test the statistical properties of OLS by repeatedly simulating the regression of randomly generated data on fixed x-values and checking if the average estimates equal the true parameter values.
The document discusses the least squares regression method for determining the line of best fit for a dataset. It explains that the least squares method finds the line that minimizes the sum of the squares of the distances between the observed responses in the dataset and the responses predicted by the linear approximation. The document provides steps to calculate the line of best fit, including calculating the slope and y-intercept. It also includes an example of applying the least squares method to find the line of best fit for a dataset relating t-shirt prices and number of t-shirts sold.
This document summarizes a lecture on randomized algorithms for approximating the median. It introduces a simple randomized algorithm called Rand-Approx-Median that takes an array as input and returns an element whose rank is approximately the median in O(log n log log n) time. The algorithm works by randomly sampling elements, sorting the samples, and returning the median of the sorted samples. The document analyzes the error probability of this algorithm using elementary probability theory and shows it has low error probability. It also emphasizes that designing and analyzing randomized algorithms requires insight into elementary probability concepts.
C2 st lecture 12 the chi squared-test handoutfatima d
Pearson's chi-squared test is used to determine if there is a relationship between two categorical variables. It has the following structure:
1) State the null and alternative hypotheses
2) Calculate the test statistic by finding residuals between observed and expected counts and summing their squares divided by expected values
3) Find the critical value based on degrees of freedom and significance level
4) Reject the null hypothesis if the test statistic exceeds the critical value, concluding the variables are dependent. Otherwise fail to reject, concluding independence.
Three examples are provided to demonstrate applying the chi-squared test to determine dependence between grades and attendance, height and nose size, and weather and season.
This document introduces sequences and series in mathematics. It defines a sequence as a set of numbers written in a particular order, with the n-th term written as un. A series is the sum of terms in a sequence. An arithmetic progression has terms where each new term is obtained by adding a constant difference to the preceding term. The n-th term of an arithmetic progression is a + (n - 1)d, where a is the first term and d is the common difference. A geometric progression multiplies each new term by a constant ratio r to obtain the next term, with the n-th term written as arn-1. Formulas are provided for finding the n-th term, sum of terms,
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.
This document introduces multiple linear regression models that have more than one explanatory variable. It discusses how to generalize the simple bivariate regression model to a multiple regression model with k explanatory variables, including a constant term. It also explains how to test multiple hypotheses about coefficients using an F-test, comparing the residual sum of squares from restricted and unrestricted regressions. The relationship between the t-test and F-test for hypothesis testing is also covered.
This document discusses non-linear regression. Non-linear regression uses regression equations that are non-linear in terms of the variables or parameters. Two main types are discussed: models that are nonlinear in variables but linear in parameters, and models that are nonlinear in both variables and parameters. Several non-linear regression methods are described, including direct computation, derivative, and self-starting methods. Examples of non-linear regression models and the differences between linear and non-linear regression are provided. Advantages of non-linear regression include applying differential weighting and identifying outliers.
I am Ronald G. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I have done Ph.D Statistics from New York University, USA. I have been helping students with their statistics assignments for the past 5 years. You can hire me for any of your statistics assignments.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with statistics.
This document provides an overview of probability concepts including:
- The three axioms of probability: probabilities are between 0 and 1, the probability of the sample space is 1, and the probability of the union of disjoint events equals the sum of the individual probabilities.
- Formulas for probability, conditional probability, independence, and complements.
- Discrete and continuous random variables and their properties including expected value and variance.
- Examples of probability mass functions for binomial and Poisson distributions.
If you are looking for business statistics homework help, Statisticshelpdesk is your rightest destination. Our experts are capable of solving all grades of business statistics homework with best 100% accuracy and originality. We charge reasonable.
The document provides information about regression analysis and calculating the coefficient of determination. It includes:
1) Instructions on how to perform a regression analysis using a calculator to find the least squares regression line, correlation coefficient, and residual plot from sample data.
2) An explanation of the coefficient of determination as a measure of how much variability in the variable y can be explained by its linear relationship with variable x.
3) A calculation example finding the coefficient of determination to be 0.83 for a dataset relating height and shoe size, meaning approximately 83% of the variation in shoe size can be explained by height.
1. The document discusses the simple linear regression model and how to derive the regression coefficients using the least squares method.
2. It uses a numerical example to show how to calculate the regression coefficients b1 and b2 by minimizing the sum of squared residuals.
3. The general method is then described for a model with n observations, where the regression coefficients b1 and b2 are the values that minimize the total sum of squared residuals.
This chapter discusses chi-square tests and nonparametric tests. It covers performing chi-square tests to compare two or more proportions, test independence between categorical variables using contingency tables, and introduce nonparametric tests including the Wilcoxon rank sum test and Kruskal-Wallis test. Examples are provided to demonstrate chi-square tests of equality of proportions, independence, and expected versus observed frequencies in contingency tables.
Country report on semi-structured interviews with temporary migrants - the Ne...EURA-NET project
This document summarizes semi-structured interviews conducted with 82 migrants in the Netherlands from various countries of origin. It finds that 17 migrants had lived in the Netherlands longer than their current residence permits allow. Many cited family/friends, education quality, and financial reasons for choosing the Netherlands. While technologies help them stay connected abroad, migration negatively impacts family relationships. The document also examines the migrants' employment status, language learning challenges, experiences with discrimination, and perspectives on Dutch culture. It concludes with recommendations to improve migration policies such as helping spouses of skilled workers find jobs and encouraging circular migration.
Ordinary least squares linear regressionElkana Rorio
Ordinary Least Squares Linear Regression is commonly used but often misunderstood and misapplied. It works by minimizing the sum of squared errors between predictions and actual values in the training data to determine coefficients for the linear regression equation. However, it is very sensitive to outliers in the data which can dramatically affect the determined coefficients and reduce prediction accuracy. Alternative regression techniques like least absolute deviations are more robust to outliers but less computationally efficient. Preprocessing data to remove or de-emphasize outliers can help address these issues with Ordinary Least Squares regression.
The document discusses simple linear regression and correlation methods. It defines deterministic and probabilistic models for describing the relationship between two variables. A simple linear regression model assumes a population regression line with intercept a and slope b, where observations may deviate from the line by some random error e. Key assumptions of the model are that e has a normal distribution with mean 0 and constant variance across values of x, and errors are independent. The slope b estimates the average change in y per unit change in x.
Applied Numerical Methods Curve Fitting: Least Squares Regression, InterpolationBrian Erandio
Correction with the misspelled langrange.
and credits to the owners of the pictures (Fantasmagoria01, eugene-kukulka, vooga, and etc.) . I do not own all of the pictures used as background sorry to those who aren't tagged.
The presentation contains topics from Applied Numerical Methods with MATHLAB for Engineers and Scientist 6th and International Edition.
The Kolmogorov-Smirnov test is used to test if an observed frequency distribution matches an expected theoretical distribution. It compares the cumulative distribution functions of the observed and expected distributions. The test statistic is the largest difference between these cumulative distributions. If this difference is larger than a critical value from tables, the null hypothesis of a good fit is rejected. An example calculates the test statistic for observed data compared to a normal distribution, and finds it is less than the critical value so the null hypothesis is accepted.
This document provides an overview of least-squares regression techniques including:
- Simple linear regression to fit a line to data
- Polynomial regression to fit higher order curves
- Multiple regression to fit surfaces using two or more variables
It discusses calculating regression coefficients, quantifying errors, and performing statistical analysis of the regression results including determining confidence intervals. Examples are provided to demonstrate applying these techniques.
I. The document discusses the method of ordinary least squares (OLS) regression analysis. OLS chooses estimates that minimize the sum of squared residuals between the actual and predicted y-values.
II. OLS provides point estimates for regression parameters and makes assumptions such as a linear relationship between variables, independent and homoscedastic errors, and no autocorrelation.
III. Monte Carlo experiments can test the statistical properties of OLS by repeatedly simulating the regression of randomly generated data on fixed x-values and checking if the average estimates equal the true parameter values.
The document discusses the least squares regression method for determining the line of best fit for a dataset. It explains that the least squares method finds the line that minimizes the sum of the squares of the distances between the observed responses in the dataset and the responses predicted by the linear approximation. The document provides steps to calculate the line of best fit, including calculating the slope and y-intercept. It also includes an example of applying the least squares method to find the line of best fit for a dataset relating t-shirt prices and number of t-shirts sold.
This document summarizes a lecture on randomized algorithms for approximating the median. It introduces a simple randomized algorithm called Rand-Approx-Median that takes an array as input and returns an element whose rank is approximately the median in O(log n log log n) time. The algorithm works by randomly sampling elements, sorting the samples, and returning the median of the sorted samples. The document analyzes the error probability of this algorithm using elementary probability theory and shows it has low error probability. It also emphasizes that designing and analyzing randomized algorithms requires insight into elementary probability concepts.
C2 st lecture 12 the chi squared-test handoutfatima d
Pearson's chi-squared test is used to determine if there is a relationship between two categorical variables. It has the following structure:
1) State the null and alternative hypotheses
2) Calculate the test statistic by finding residuals between observed and expected counts and summing their squares divided by expected values
3) Find the critical value based on degrees of freedom and significance level
4) Reject the null hypothesis if the test statistic exceeds the critical value, concluding the variables are dependent. Otherwise fail to reject, concluding independence.
Three examples are provided to demonstrate applying the chi-squared test to determine dependence between grades and attendance, height and nose size, and weather and season.
This document introduces sequences and series in mathematics. It defines a sequence as a set of numbers written in a particular order, with the n-th term written as un. A series is the sum of terms in a sequence. An arithmetic progression has terms where each new term is obtained by adding a constant difference to the preceding term. The n-th term of an arithmetic progression is a + (n - 1)d, where a is the first term and d is the common difference. A geometric progression multiplies each new term by a constant ratio r to obtain the next term, with the n-th term written as arn-1. Formulas are provided for finding the n-th term, sum of terms,
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.
This document introduces multiple linear regression models that have more than one explanatory variable. It discusses how to generalize the simple bivariate regression model to a multiple regression model with k explanatory variables, including a constant term. It also explains how to test multiple hypotheses about coefficients using an F-test, comparing the residual sum of squares from restricted and unrestricted regressions. The relationship between the t-test and F-test for hypothesis testing is also covered.
This document discusses non-linear regression. Non-linear regression uses regression equations that are non-linear in terms of the variables or parameters. Two main types are discussed: models that are nonlinear in variables but linear in parameters, and models that are nonlinear in both variables and parameters. Several non-linear regression methods are described, including direct computation, derivative, and self-starting methods. Examples of non-linear regression models and the differences between linear and non-linear regression are provided. Advantages of non-linear regression include applying differential weighting and identifying outliers.
I am Ronald G. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I have done Ph.D Statistics from New York University, USA. I have been helping students with their statistics assignments for the past 5 years. You can hire me for any of your statistics assignments.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with statistics.
This document provides an overview of probability concepts including:
- The three axioms of probability: probabilities are between 0 and 1, the probability of the sample space is 1, and the probability of the union of disjoint events equals the sum of the individual probabilities.
- Formulas for probability, conditional probability, independence, and complements.
- Discrete and continuous random variables and their properties including expected value and variance.
- Examples of probability mass functions for binomial and Poisson distributions.
If you are looking for business statistics homework help, Statisticshelpdesk is your rightest destination. Our experts are capable of solving all grades of business statistics homework with best 100% accuracy and originality. We charge reasonable.
The document provides information about regression analysis and calculating the coefficient of determination. It includes:
1) Instructions on how to perform a regression analysis using a calculator to find the least squares regression line, correlation coefficient, and residual plot from sample data.
2) An explanation of the coefficient of determination as a measure of how much variability in the variable y can be explained by its linear relationship with variable x.
3) A calculation example finding the coefficient of determination to be 0.83 for a dataset relating height and shoe size, meaning approximately 83% of the variation in shoe size can be explained by height.
1. The document discusses the simple linear regression model and how to derive the regression coefficients using the least squares method.
2. It uses a numerical example to show how to calculate the regression coefficients b1 and b2 by minimizing the sum of squared residuals.
3. The general method is then described for a model with n observations, where the regression coefficients b1 and b2 are the values that minimize the total sum of squared residuals.
This chapter discusses chi-square tests and nonparametric tests. It covers performing chi-square tests to compare two or more proportions, test independence between categorical variables using contingency tables, and introduce nonparametric tests including the Wilcoxon rank sum test and Kruskal-Wallis test. Examples are provided to demonstrate chi-square tests of equality of proportions, independence, and expected versus observed frequencies in contingency tables.
Country report on semi-structured interviews with temporary migrants - the Ne...EURA-NET project
This document summarizes semi-structured interviews conducted with 82 migrants in the Netherlands from various countries of origin. It finds that 17 migrants had lived in the Netherlands longer than their current residence permits allow. Many cited family/friends, education quality, and financial reasons for choosing the Netherlands. While technologies help them stay connected abroad, migration negatively impacts family relationships. The document also examines the migrants' employment status, language learning challenges, experiences with discrimination, and perspectives on Dutch culture. It concludes with recommendations to improve migration policies such as helping spouses of skilled workers find jobs and encouraging circular migration.
The document provides information about planning and conducting interviews for research purposes. It discusses different types of interviews including unstructured, semi-structured, and structured interviews. It explains the advantages and disadvantages of each type. The document also outlines steps for planning an interview such as preparing an interview schedule, piloting the questions, and selecting informants. Overall, the document serves as a guide for researchers on how to appropriately design, test, and conduct qualitative interviews.
A session on "Semi structured interviews for education research" faciltiated by Dr Ian Willis and Dr Debbie Prescott
as part of the CPD series on educational research
Academic Development, Centre for Lifelong Learning
University of Liverpool
5th November 2015
Qualitative interviews involve flexible, unstructured conversations to understand participants' perspectives. They are commonly used in combination with other methods like observation. Interviews can vary in structure from unstructured to semi-structured using an interview guide. Factors like number of interviews, use of visual aids, group settings, and recording methods are considered. Locating respondents, obtaining consent, and addressing issues like researcher effects are important planning considerations. Analyzing interviews involves transcription and identifying common themes.
The document discusses experimental research design. It covers key concepts like causality, conditions for causality, validity, and extraneous variables. It also describes different types of experimental designs including pre-experimental, true experimental, quasi-experimental, and statistical designs. Examples are provided to illustrate different designs like randomized block and Latin square designs. Limitations of experimentation are also briefly discussed.
Hadoop Interview Questions and Answers | Big Data Interview Questions | Hadoo...Edureka!
This Hadoop Tutorial on Hadoop Interview Questions and Answers ( Hadoop Interview Blog series: https://goo.gl/ndqlss ) will help you to prepare yourself for Big Data and Hadoop interviews. Learn about the most important Hadoop interview questions and answers and know what will set you apart in the interview process. Below are the topics covered in this Hadoop Interview Questions and Answers Tutorial:
Hadoop Interview Questions on:
1) Big Data & Hadoop
2) HDFS
3) MapReduce
4) Apache Hive
5) Apache Pig
6) Apache HBase and Sqoop
Check our complete Hadoop playlist here: https://goo.gl/4OyoTW
#HadoopInterviewQuestions #BigDataInterviewQuestions #HadoopInterview
This document provides an overview of multiple regression analysis and hypothesis testing using the classical linear model. It discusses the assumptions of the classical linear model and how they allow for hypothesis testing of regression coefficients. Specifically, it describes how to test hypotheses about individual coefficients using t-tests and hypotheses about linear combinations of coefficients or exclusion of multiple regressors using F-tests. Examples are provided to illustrate testing various null hypotheses about coefficients.
This document provides an overview of chi-square goodness-of-fit tests and tests of independence using contingency tables. It discusses using chi-square tests to determine if sample data fits an expected distribution, as well as testing if two attributes are independent. Examples are provided, including a test of whether technical support calls are uniformly distributed across days of the week, and a test of independence between hand preference and gender using a contingency table. Key steps of chi-square tests like calculating expected frequencies and the test statistic are outlined.
1. The document discusses statistical estimation and properties of estimators such as bias, variance, consistency, and asymptotic normality.
2. Key concepts covered include unbiasedness, mean squared error, relative efficiency, sufficiency, and properties of estimators like consistency, asymptotic unbiasedness, and best asymptotic normality.
3. Examples are provided to illustrate theoretical estimators for parameters like the variance of a distribution or coefficients in a linear regression model.
This document discusses univariate and multivariate regressions. It explains how to measure the accuracy of regression coefficient estimates using standard error, confidence intervals, and hypothesis testing. It then covers multiple linear regression, using methods like forward, backward, and mixed selection to identify important predictor variables. The document concludes by discussing model fitting metrics like RSE and R2, and factors to consider when making predictions with regression models.
Regression analysis is used to model relationships between variables. Simple linear regression involves modeling the relationship between a single independent variable and dependent variable. The regression equation estimates the dependent variable (y) as a linear function of the independent variable (x). The parameters β0 and β1 are estimated using the method of least squares. The coefficient of determination (r2) measures how well the regression line fits the data. Additional tests like the t-test, confidence intervals, and F-test are used to test if the independent variable significantly predicts the dependent variable. While these tests can indicate a statistically significant relationship, they do not prove causation.
This document provides an introduction and overview of logistic regression. It describes why logistic regression is used when the dependent variable is limited and not continuous. Maximum likelihood estimation is used to estimate the coefficients. The coefficients can be interpreted as odds ratios. The performance of the logistic regression model is evaluated using measures like the model chi-square, percent correct predictions, and pseudo R-squared. Potential problems like omitted variables, irrelevant variables, and structural breaks are also discussed.
This document discusses relationships between variables in experiments. It defines two types of relationships: functional and statistical. A functional relationship is a perfect mathematical relationship where each value of the independent variable corresponds to a single, unique value of the dependent variable. A statistical relationship is imperfect, with a range of possible dependent variable values for each independent variable value. The document also discusses simple linear regression analysis, how to estimate regression coefficients, and how to interpret them to understand the relationship between variables.
This document discusses statistical inference for multiple regression analysis. It begins by recapping the assumptions of the classical linear model (CLM), which are used to derive the sampling distributions of the OLS estimators. It then explains that the sampling distributions are needed to test hypotheses about the population parameters. Under the CLM assumptions, specifically the additional normality assumption, the sampling distributions are normal. This allows hypotheses about a single parameter to be tested using the t-distribution. The document provides an example of testing the null hypothesis that a parameter equals zero against a one-sided alternative hypothesis.
This document provides an overview and schedule for an advanced econometrics training using Stata. The training covers topics such as hypothesis testing, multiple regression, time series models, panel data models, and difference-in-differences. It discusses assumptions of classical linear regression models and how to perform statistical inference using estimates of variance, standard error, and hypothesis testing. The document explains how to construct t-statistics and compare them to critical values from the t-distribution to test hypotheses about population parameters.
This document provides an overview of hypothesis testing concepts including:
- A hypothesis is a claim about a population parameter that can be tested statistically. The null hypothesis states the claim to be tested, while the alternative hypothesis is what the researcher is trying to prove.
- The level of significance and critical values determine the rejection region where the null hypothesis would be rejected. Type I and Type II errors refer to incorrectly rejecting or failing to reject the null hypothesis.
- The key steps of hypothesis testing are stated as: 1) specify null and alternative hypotheses, 2) choose significance level and sample size, 3) determine test statistic, 4) find critical values, 5) collect data and compute test statistic, 6) make a decision
Two-Variable (Bivariate) RegressionIn the last unit, we covered LacieKlineeb
Two-Variable (Bivariate) Regression
In the last unit, we covered scatterplots and correlation. Social scientists use these as descriptive tools for getting an idea about how our variables of interest are related. But these tools only get us so far. Regression analysis is the next step. Regression is by far the most used tool in social science research.
Simple regression analysis can tell us several things:
1. Regression can estimate the relationship between x and y in their
original units of measurement. To see why this is so useful, consider the example of infant mortality and median family income. Let’s say that a policymaker is interested in knowing how much of a change in median family income is needed to significantly reduce the infant mortality rate. Correlation cannot answer this question, but regression can.
2. Regression can tell us how well the independent variable (x) explains the dependent variable (y). The measure is called the
R square.
Simple Two-Variable (Bivariate) Regression
Regression uses the equation of a line to estimate the relationship between x and y. You may remember back in algebra learning about the equation of a line. Some learned it as Y =s X + K or Y = mX + B. In statistics, we use a different form:
Equation 1: Y = B0 + B1X + u
Let’s define each term in the equation:
· Y is the dependent variable. It is placed on the Y (vertical) axis. In the example below, the dependent variable (Y) is the infant mortality rate.
· B0 is the Y intercept. B0 is also referred to as “the constant.” B0 is the point where the regression line crosses the Y axis. Importantly, B0 is equal to the
predicted value of Ywhen X=0. In most cases, B0 is does not get much attention for two reasons. First, the researcher is usually interested in the relationship between x and y. not the relationship between x and y at the single value of x=0. Second, often independent variables do not take on the value zero. Consider the AECF sample data. There are no states with low-birth-weight percentages equal to zero, so we would be extrapolating beyond what the data tell us.
· B1 is usually the main point of interest for researchers. It is the slope of the line relating x to y. Researchers usually refer to B1 as a slope coefficient, regression coefficient or simply a coefficient.
B1 measures the change in Y for a one-unit change in x. We represent change by the symbol ∆.
B1 =
· u is the error term. The error term is the distance between the regression line and the dots on the scatterplot. Think about it, regression estimates a single line through the cloud of data. Naturally, the line does not hit all the data points. The degree to which the line “misses” the data point is the error. u can also be thought of as
all the other factors that affect the infant mortality rate besides X. Importantly, we
assume that u is totally random given X.
The ...
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.
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C) The p-value represents the probability of observing the test statistic or a more extreme value if H0 is true. Small p-values provide evidence against H0. Conventionally, p ≤ 0.05 is considered significant
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The document discusses one-sided t-tests for hypotheses about regression coefficients. It explains that one-sided tests provide more power than two-sided tests when the alternative hypothesis can be restricted to values on one side of the null value. Special cases for testing if a coefficient is different from zero are presented, where a one-sided test requires the estimated coefficient to be further from zero than a two-sided test to reject the null hypothesis. An example of testing if price inflation is lower than wage inflation using a one-sided test is provided.
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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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The document discusses genome editing in agriculture, focusing on challenges and opportunities in the seed industry sector. It covers topics such as genome editing technologies, regulation, edited crops and traits, and challenges. Some key challenges discussed are issues around access to technology and intellectual property, divergent regulatory approaches between regions, difficulties detecting genome edits, and varying public views. The document also provides classifications for different types of genome edits and examines regulatory approaches to genome edited crops in countries like India.
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3. Economics 20 - Prof. Anderson 3
Assumptions of the Classical
Linear Model (CLM)
Given the Gauss-Markov assumptions,
OLS is BLUE,
In order to do classical hypothesis testing,
we need to add another assumption (beyond
the Gauss-Markov assumptions)
Assume that u is independent of x1, x2,…, xk
and u is normally distributed with zero
mean and variance s2: u ~ Normal(0,s2)
4. Economics 20 - Prof. Anderson 4
CLM Assumptions (cont)
Under CLM, OLS is not only BLUE, but is
the minimum variance unbiased estimator
We can summarize the population
assumptions of CLM as follows
y|x ~ Normal(b0 + b1x1 +…+ bkxk, s2)
While for now we just assume normality,
clear that sometimes not the case
Large samples will let us drop normality
5. Economics 20 - Prof. Anderson 5
.
.
x1 x2
The homoskedastic normal distribution with
a single explanatory variable
E(y|x) = b0 + b1x
y
f(y|x)
Normal
distributions
6. Economics 20 - Prof. Anderson 6
Normal Sampling Distributions
errorstheofncombinatiolinearais
itbecausenormallyddistributeisˆ
0,1Normal~ˆ
ˆ
thatso,ˆ,Normal~ˆ
st variableindependentheofvaluessamplethe
onlconditionas,assumptionCLMUnder the
jb
b
bb
bbb
j
jj
jjj
sd
Var
7. Economics 20 - Prof. Anderson 7
The t Test
1:freedomofdegreestheNote
ˆbyestimatetohavewebecause
normal)(vsondistributiaisthisNote
~ˆ
ˆ
sassumptionCLMUnder the
22
1
j
kn
t
t
se kn
j
j
ss
b
bb
8. Economics 20 - Prof. Anderson 8
The t Test (cont)
Knowing the sampling distribution for the
standardized estimator allows us to carry
out hypothesis tests
Start with a null hypothesis
For example, H0: bj=0
If do not reject null, then xj has no effect on
y, controlling for other x’s
9. Economics 20 - Prof. Anderson 9
The t Test (cont)
0
ˆj
H,hypothesisnullaccept the
owhether tdeterminetorulerejectiona
withalongstatisticourusethenwillWe
ˆ
ˆ
:ˆforstatisticthe""
formtoneedfirsteour test wperformTo
t
se
tt
j
j
j
b
b
b b
10. Economics 20 - Prof. Anderson 10
t Test: One-Sided Alternatives
Besides our null, H0, we need an alternative
hypothesis, H1, and a significance level
H1 may be one-sided, or two-sided
H1: bj > 0 and H1: bj < 0 are one-sided
H1: bj 0 is a two-sided alternative
If we want to have only a 5% probability of
rejecting H0 if it is really true, then we say
our significance level is 5%
11. Economics 20 - Prof. Anderson 11
One-Sided Alternatives (cont)
Having picked a significance level, a, we
look up the (1 – a)th percentile in a t
distribution with n – k – 1 df and call this c,
the critical value
We can reject the null hypothesis if the t
statistic is greater than the critical value
If the t statistic is less than the critical value
then we fail to reject the null
12. Economics 20 - Prof. Anderson 12
yi = b0 + b1xi1 + … + bkxik + ui
H0: bj = 0 H1: bj > 0
c0
a1 a
One-Sided Alternatives (cont)
Fail to reject
reject
13. Economics 20 - Prof. Anderson 13
One-sided vs Two-sided
Because the t distribution is symmetric,
testing H1: bj < 0 is straightforward. The
critical value is just the negative of before
We can reject the null if the t statistic < –c,
and if the t statistic > than –c then we fail to
reject the null
For a two-sided test, we set the critical
value based on a/2 and reject H1: bj 0 if
the absolute value of the t statistic > c
14. Economics 20 - Prof. Anderson 14
yi = b0 + b1Xi1 + … + bkXik + ui
H0: bj = 0 H1: bj > 0
c0
a/21 a
-c
a/2
Two-Sided Alternatives
reject reject
fail to reject
15. Economics 20 - Prof. Anderson 15
Summary for H0: bj = 0
Unless otherwise stated, the alternative is
assumed to be two-sided
If we reject the null, we typically say “xj is
statistically significant at the a % level”
If we fail to reject the null, we typically say
“xj is statistically insignificant at the a %
level”
16. Economics 20 - Prof. Anderson 16
Testing other hypotheses
A more general form of the t statistic
recognizes that we may want to test
something like H0: bj = aj
In this case, the appropriate t statistic is
teststandardfor the0
where,ˆ
ˆ
j
j
jj
a
se
a
t
b
b
17. Economics 20 - Prof. Anderson 17
Confidence Intervals
Another way to use classical statistical testing is
to construct a confidence interval using the same
critical value as was used for a two-sided test
A (1 - a) % confidence interval is defined as
ondistributiain
percentile
2
-1theiscwhere,ˆˆ
1
kn
jj
t
sec
a
bb
18. Economics 20 - Prof. Anderson 18
Computing p-values for t tests
An alternative to the classical approach is
to ask, “what is the smallest significance
level at which the null would be rejected?”
So, compute the t statistic, and then look up
what percentile it is in the appropriate t
distribution – this is the p-value
p-value is the probability we would observe
the t statistic we did, if the null were true
19. Economics 20 - Prof. Anderson 19
Stata and p-values, t tests, etc.
Most computer packages will compute the
p-value for you, assuming a two-sided test
If you really want a one-sided alternative,
just divide the two-sided p-value by 2
Stata provides the t statistic, p-value, and
95% confidence interval for H0: bj = 0 for
you, in columns labeled “t”, “P > |t|” and
“[95% Conf. Interval]”, respectively
20. Economics 20 - Prof. Anderson 20
Testing a Linear Combination
Suppose instead of testing whether b1 is equal to a
constant, you want to test if it is equal to another
parameter, that is H0 : b1 = b2
Use same basic procedure for forming a t statistic
21
21
ˆˆ
ˆˆ
bb
bb
se
t
22. Economics 20 - Prof. Anderson 22
Testing a Linear Combo (cont)
So, to use formula, need s12, which
standard output does not have
Many packages will have an option to get
it, or will just perform the test for you
In Stata, after reg y x1 x2 … xk you would
type test x1 = x2 to get a p-value for the test
More generally, you can always restate the
problem to get the test you want
23. Economics 20 - Prof. Anderson 23
Example (cont):
Now you get a standard error for b1 – b2 =
q1 directly from the basic regression
Any linear combination of parameters
could be tested in a similar manner
Other examples of hypotheses about a
single linear combination of parameters:
b1 = 1 + b2 ; b1 = 5b2 ; b1 = -1/2b2 ; etc
24. Economics 20 - Prof. Anderson 24
Multiple Linear Restrictions
Everything we’ve done so far has involved
testing a single linear restriction, (e.g. b1 = 0
or b1 = b2 )
However, we may want to jointly test
multiple hypotheses about our parameters
A typical example is testing “exclusion
restrictions” – we want to know if a group
of parameters are all equal to zero
25. Economics 20 - Prof. Anderson 25
Testing Exclusion Restrictions
Now the null hypothesis might be
something like H0: bk-q+1 = 0, ... , bk = 0
The alternative is just H1: H0 is not true
Can’t just check each t statistic separately,
because we want to know if the q
parameters are jointly significant at a given
level – it is possible for none to be
individually significant at that level
26. Economics 20 - Prof. Anderson 26
Exclusion Restrictions (cont)
To do the test we need to estimate the “restricted
model” without xk-q+1,, …, xk included, as well as
the “unrestricted model” with all x’s included
Intuitively, we want to know if the change in SSR
is big enough to warrant inclusion of xk-q+1,, …, xk
edunrestrictisurandrestrictedisr
where,
1
knSSR
qSSRSSR
F
ur
urr
27. Economics 20 - Prof. Anderson 27
The F statistic
The F statistic is always positive, since the
SSR from the restricted model can’t be less
than the SSR from the unrestricted
Essentially the F statistic is measuring the
relative increase in SSR when moving from
the unrestricted to restricted model
q = number of restrictions, or dfr – dfur
n – k – 1 = dfur
28. Economics 20 - Prof. Anderson 28
The F statistic (cont)
To decide if the increase in SSR when we
move to a restricted model is “big enough”
to reject the exclusions, we need to know
about the sampling distribution of our F stat
Not surprisingly, F ~ Fq,n-k-1, where q is
referred to as the numerator degrees of
freedom and n – k – 1 as the denominator
degrees of freedom
29. Economics 20 - Prof. Anderson 29
0 c
a1 a
f(F)
F
The F statistic (cont)
reject
fail to reject
Reject H0 at a
significance level
if F > c
30. Economics 20 - Prof. Anderson 30
The R2 form of the F statistic
Because the SSR’s may be large and unwieldy, an
alternative form of the formula is useful
We use the fact that SSR = SST(1 – R2) for any
regression, so can substitute in for SSRu and SSRur
edunrestrictisurandrestrictedisr
againwhere,
11 2
22
knR
qRR
F
ur
rur
31. Economics 20 - Prof. Anderson 31
Overall Significance
A special case of exclusion restrictions is to test
H0: b1 = b2 =…= bk = 0
Since the R2 from a model with only an intercept
will be zero, the F statistic is simply
11 2
2
knR
kR
F
32. Economics 20 - Prof. Anderson 32
General Linear Restrictions
The basic form of the F statistic will work
for any set of linear restrictions
First estimate the unrestricted model and
then estimate the restricted model
In each case, make note of the SSR
Imposing the restrictions can be tricky –
will likely have to redefine variables again
33. Economics 20 - Prof. Anderson 33
F Statistic Summary
Just as with t statistics, p-values can be
calculated by looking up the percentile in
the appropriate F distribution
Stata will do this by entering: display
fprob(q, n – k – 1, F), where the appropriate
values of F, q,and n – k – 1 are used
If only one exclusion is being tested, then F
= t2, and the p-values will be the same