# Linear regression problems worksheet

4. Quadratic regression produces a more accurate quadratic model than the procedure in Example 3 because it uses all the data points. 10) 5. A, B, C, D, E, F. Activity—Twizzlers Linear Regression Pre-Activity & Worksheet Answer Key 2 6. Like half the models in statistics, standard linear regression relies on an assumption of normality. Biology: Five additional weeks of sunshine the sugar concentration in vine grapes will rise by X %. 2. Let the model be Y = 0 + 1X 1 + 2X 2 + ", where E("jX 1;X 2) = 0, and assume that we have a sample AP Stats: Section 3. d. Linear regression estimates the regression coefficients β 0 and β 1 in the equation Y j =β 0 +β 1 X j +ε j where X is the independent variable, Y is the dependent 4. 3 Inferences on the Slope Rarameter ββββ1111 NIPRL 1 12. That is, the equation of the best linear t. Linear reg. 9. A regression with two or more predictor variables is called a multiple regression. 02 (Simple Linear Regression) is based on the identical data set to the paired t -test example above. 1. regression. 32 0. There are two common ways to deal with nonlinear relationships: 1. Multiple Linear Regression Model We consider the problem of regression when the study variable depends on more than one explanatory or independent variables, called a multiple linear regression model. Part 1. 15. 6. CHAPTER 5. The general model can be estimated by grid search or by non-linear maximization of the likelihood and a maximum likelihood estimate for a obtained. Quiz. . Use the regression model to predict the credit card volume in 2003 and in 2010. ac. Going back to our original data, we can try to fit a line through the points that we have; this is called a “trend line”, “linear regression” or “line of best fit” (as we said earlier, the line that’s the “closest fit” to the points – the best trend line). 8 – Trigonometry & Regression Linear Correlations 1. The estimated  Under Output Options, choose "New Worksheet Ply," then click OK. (2006) - Chpt 6 zQuinn & Keough (2002) - Chpt 5 zRowntree (1981) - Chpts 12 Question 1 - Simple linear regression Here is an example from Fowler, Cohen and Parvis (1998). To learn more about Nonlinear Regression with data linearization, see the Nonlinear Regression worksheet. In this section we will first discuss correlation analysis, which is used to quantify the association between two continuous variables (e. For the reason that we should supply everything you need in a true along with dependable origin, most people provide valuable details on many themes in addition to topics. Then we perform linear regression on the data. 2 HSS-ID. 355 Problems Predictor Coef SE Coef T P Constant 44. For each of the following tables, treat the left-hand column as the independent variable (input) and the right-hand column as the dependent variable (output), and answer each of the following questions, along with any additional questions related to the actual problem. ’ Here are some of the common Linear Regression Interview Questions that pop up in interviews all over the world. Linear and Quadratic . Regression analysis is the art and science of fitting straight lines to patterns of data. Make a table that shows data from If both the regression coefficients are negative, r would be negative and if both are positive, r would assume a positive value. So, in Excel, you do linear regression using the least squares method and seek coefficients a and b such that: y = bx + a Student Worksheets Created by Matthew M. 4 and 8 2) The difference of two numbers is 3. Linear regression simply refers to creating a best fit for a linear relationship between two variables from observed data. Last ride. 1751(𝑌𝑌𝑌𝑌𝑌𝑌/𝐴𝐴) 𝑊𝑊𝑊𝑊 𝑦𝑦 = −70. Word problems on ages. The equation to represent this data is . In the data, x is the number of seconds after the missile is launched and y is the number of feet above water for the missile. 5 Prediction Intervals for Future Response Values 12. Property 4 : The two lines of regression coincide i. The least squares method is generally used with a linear regression, but Regression is the process by which the relationship between two variables is determined. The problem of determining the best values of a and b involves the principle of 2 is the sum of squares due to the linear regression SSR, with mean square. The least squares method is generally used with a linear regression, but Aug 01, 2018 · The linear regression equation always has an error term because, in real life, predictors are never perfectly precise. image0. Regression is used to assess the contribution of one or more “explanatory” variables (called independent variables) to one “response” (or dependent ) variable. The data in the table below show different X (depth in feet) 50 Y (maximum dive time) 80 depths with the maximum dive times in minutes. 4¯6, s2 Example of a regression equation Y = $0 +$ 1 (Age - 40) + $2 Gender + , Salary = 50 + 1 (Age - 40) - 3 Gender + , Salary in$1,000s, Age in years and Gender = 0 if male and 1 if female What is the average salary for 50 year old males? Ave(Y) = 50 + 1 (50-40) - 3(0) = $60K following form: y=alpha+beta*x+epsilon (we hypothesize a linear relationship) • The regression analysis „estimates“ the parameters alpha and beta by using the given observations for x and y. 13 Feb 2014 Linear Regression and Correlation - Example. In your story, interpret the slope of the line, the y-intercept, and the x-intercept. In the final chapter, I'll show you how to make the most of your results by changing parameters by hand, performing sensitivity analysis and creating scenarios. Word problems on sets and venn diagrams. By comparing the values of, determine the function that best fits the data. Simple Linear Regression and Correlation 12. • The simplest form of estimating alpha and beta is called ordinary least squares (OLS) regression Problem 3: Let X and Y be two variables in a study. Find the residual amount for a person who is 42. 902. LAB ACTIVITIES FOR SIMPLE LINEAR REGRESSION: TWO VARIABLES 1. This one-page worksheet contains seven problems. A student who waits on tables at a restaurant recorded the cost of meals and the tip b. In case you need guidance on exponents or even radical expressions, Algebra1help. Worksheet 13. dta. Outcomes Students will use a graphing calculator to find a quadratic curve of best fit. Some of the worksheets for this concept are Linear equations work, Solving linear equations, Linear regression work 1, Depends y dependent variable x independent variable y m x, Writing linear equationslinear regression, Real world applications of linear equations, Slope intercept form word problems, Y mx b A statistics Worksheet: The student will calculate and construct the line of best fit between two variables. Next, students will use their calculator to fit a simple linear regression equation with poverty predicting obesity. Y. In this tutorial, […] In Chapter 2 you used a graphing calculator to perform linear regression on a data set in order to find a linear model for the data. The data is shown below. 7 p1 The regression equation is Sales = 116 - 97. 26721 × (8) = 2. Apr 16, 2018 · Note The Regression tool alerts you to this problem and does not continue. Regression interpretation 3. This part of the program will fit a linear function of the form: Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. This lesson focuses on two variables which have a linear association. 7. Round to two Statistical Analysis 6: Simple Linear Regression. The population (in thousands) for Alpha City, t years after January 1, 2004 is modeled by the quadratic function P (t) 0. Old Faithful. To better understand the future strategies, you can visually represent the predicted values in a line chart. Detailed instructions on how to use the calculator are provided. The distribution for each important later. Under some conditions for the observed data, this problem can be solved numerically. variable is (ya - 1)/a, so that with a = 1, the regression is linear, with a = 0, it is logarithmic, these cases being only two possibilities out of an infinite range as a varies. Use the Input Y Range text box to identify the worksheet range holding your Write the linear regression equation for these data where miles driven is the independent Explain what the y-intercept means in the context of the problem. (1998) -Chpts 14 & 15 zHolmes et al. Transform the data so that there is a linear relationship between the transformed variables. Please be sure to check your answers as part of your homework assignment and be prepared with questions for next class. Worksheets are Work 3, Chapter 9 correlation and regression solutions, Writing linear equationslinear regression, Work 1, Linear regression work 1, Linear reg correlation coeff work, Work on correlation and regression, Algebra ii exponential regression work value 10. Make a scatter plot and determine the value of r. not significantly different from zero we conclude that: a) X is a good predictor of Y b) there is no linear relationship between X and Y. 01165 + 0. Mintzberg’s classic book The Nature of Managerial Work (1973) identiﬁed the roles found in all managerial jobs. 1 Linear Predictors Before computers became fast, linear regression was almost the only way of at-tacking certain prediction problems. 1751(6. 827 4. 3. Loading. . e. 2. c) BIOSTATS 540 - Fall 2018 Simple Linear Regression and Correlation Page 1 of 54 Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ Synthesis Unit 12 Simple Linear Regression and Correlation “ Assume that a statistical model such as a linear model is a good first start only” Lab!10:!Exploring!Linear!Regression! Objective:!In! this! lab,! you! will examine! relationships! between! two! quantitative! variables! using! a! graphical!tool May 31, 2016 · Regression analysis makes use of mathematical models to describe relationships. The regression coefficient estimated with a linear regression equation y = a + b*x can then tell the researchers b the life expectancy (y) is when smoking x cigarettes a day. Worksheet 3 Universidad Carlos III de Madrid Worksheet 3 The Multiple Regression Model Note: In those problems that include estimations and have a reference to a data set the students should check the outputs obtained with Gretl. That is why it is also termed "Ordinary Least Squares" regression. Linear inequalities word problems. That is, there is lack of fit in the simple linear regression model. However, some programs, including Excel, do the error term calculation behind the scenes. They believe that the number of books that will ultimately be sold for any particular course is related to the number of students registered for the course when the books are ordered. a. Feb 29, 2016 - Explore theboss1000's board "LINEAR REGRESSION", followed by 431 people on Pinterest. In linear regression, you are looking for a hyperplane "near" most of the points; with SVMs, you will be looking for a thick hyperplane, as thin as possible, that contains all the observations. In linear equation in two variables distance problems you have to use two variables and you can solve using any method such as substitution or elimination. Lab Activity: Linear Regression and Correlation In this lab activity, you will collect sample data of two variables, determine if a linear correlation exists between the two variables, and perform linear regression. correlation coefficients, and obtain linear regression equations. For this linear regression worksheet, students solve linear regression problems using the TI-86 calculator. Last year, Walmart conducted a study as to the amount of waiting in time in checkout lanes its customers had to wait. Scaffolded questions that start relatively easy and end with some real challenges. 391 + 17. regression project worksheet: Did the student show the material learned in this course can be useful in a topic that is relevant to the individual students? (explain in one sentence how your data is either interesting to you personally OR related to you major of study) The following are tables of data to be used for linear regression exercises. Find the linear and quadratic regression equations and correlation coefficients. This data set has n=31 observations of boiling points (Y=boiling) and temperature (X=temp). Linear Regression Displaying all worksheets related to - Linear Regression. 1. There are 2 types of factors in regression analysis: Dependent variable (y) : It’s also called the ‘criterion variable’ , ‘response’ , or ‘outcome’ and is the factor being solved. g. A correlation analysis provides information on the strength and direction of the linear relationship between two variables, while a simple linear regression analysis estimates parameters in a linear equation that can be used to predict values of one variable based on Quadratic Models and Quadratic Regression Worksheet 1. P. Consider the following scatter plots: (a) Write the new regression model. Writing Linear Equations/Linear Regression Write the slope-intercept form of the equation of each line given the slope and y-intercept. Graph the linear equation by 2. Regression problems are supervised learning problems in which the response is continuous. A researcher has collected data on the price of gasoline from 1990 to 2010 and has found that the price in dollars after t years can be predicted using the equation: y xx− += +0. Problem 2. The user can STAT 2215 Worksheet 5 – Chapter 7: Simple Linear Regression Problem 23, page 198. If there is not a linear relationship between x and y, then $$\mu_{i} ≠ \beta_{0} + \beta_{1}X_{i}$$. ! 2 4 6 8 0 50 100 150 200 250 Calories vs Alcohol Content Alcohol Content (%) Worksheets that accompany this lesson can be located under related documents, worksheets, Data Analysis #1-#8. Figure #10. You may notice that Linear regression where the sum of vertical distances d1 + d2 + d3 + d4 between observed and predicted (line and its equation) values is minimized. introduce problems that are relevant to the ﬁtting of nonlinear regression func- Title: Linear Regression Grade: 8th Lesson Summary: This short lesson plan is to describe students learn how to find the best line to fit the data of two variables they collect and be able to predicate the data by using the regression equation. Simple linear regression is a statistical method that allows us to summarize and study relationships between two continuous (quantitative) variables. In these worksheets, problems are presented as word problems. Simple Linear Regression To describe the linear association between quantitative variables, a statistical procedure called regression often is used to construct a model. The Simple Linear Regression Model is summarized by the equation $y=\beta _1x+\beta _0+\varepsilon$ Identify the deterministic part and the random part. mtpfrom the CD-ROM. SCUBA divers have maximum dive times they cannot exceed when going to different depths. Find r2. Linear regression is a type of machine learning algorithm that is used to model the relation between scalar dependent and one or more independent variables. To use K-nearest neighbors regression, or KNN regression for short, we must start with a data set. Motivation and Objective: We’ve spent a lot of time discussing simple linear regression, but simple linear regression is, well, “simple” in the sense that there is usually more than one variable that helps “explain” the variation in the response variable. write a regression equation and interpret the meaning of the slope and y-intercepts in the context of the problem; make predictions based on the correct mathematical models; and; solve linear equations. Even those who problem. However, we only calculate a regression line if one of the vari-ables helps to explain or predict the other variable. Superimpose the regression curve on the scatter plot. see and learn about curve fitting for multiple linear regression using method of least errors is as small as possible. At home: Read Chapter 5 and work the problems at the end of each short section as you go through them. Estimate the Blood pressure for a person who is 50. Chapter 12. Aug 07, 2014 · Search this site. 7 p1 + 109 p2 Remember: -97. If Y denotes the Excel Linear Regression. (10 marks) C). In linear regression analysis, the dependent variable is thought to be related to the independent variable or variables in a linear way. We would expect the ratio MSLF/MSPE to be close to 1. Time and work word problems. Then, we graph the linear regression equation with the scatterplot data. They compute the percent of the variability. For our problem, we would need only one dummy variable (since K = 2), the dichotomous variable coding level of idealism. Go to Overview Index. In simple linear regression, when β is . Do the Linear Regression ws. a and b are the constants of the regression model. It is a staple of statistics and is often considered a good introductory machine learning method. The elements in X are non-stochastic, meaning that the Linear Regression and NORMAL Curve Advanced Placement AAP Review will be held in room 315 and 312 on Tuesdays and Thursdays. Feb 26, 2018 · Linear regression is used for finding linear relationship between target and one or more predictors. Rewrite the equation into finding the x- and y-intercepts. Multiple Linear Regression. If you're seeing this message, it means we're having trouble loading external resources on our website. Plot four points so that the regression line is horizontal. Customize the worksheets to include one-step, two-step, or multi-step equations, variable on both sides, parenthesis, and more. Students will interpret the r-value of the data and write a summary of its meaning. Big Ideas: Bivariate quantitative variables can be represented by a table, graph, and a prediction equation, and estimates can be made from each. GAISE Components One step equation word problems. ) Another very serious problem is the lack of any provision for forecasting from additional values of the independent variables . Throughout the module, you will find many real-world appli cations of these two important topics: least-squares regression line and the correlation coefficient. Assumptions in the Linear Regression Model 2. Some of the worksheets for this concept are Linear regression work 1, Writing linear equationslinear regression, Chapter 9 correlation and regression solutions, Work 1, Work 3, , Quadratic regression, Kuta software. Such an equation can be used for prediction: given a new x-value, this equation can predict the y-value that is consistent with the information known about the data. The sample must be representative of the population 2. Jun 01, 2020 · Statistics Q&A Library WORKSHEET 24 Linear Regression and Correlation Name: 15. 8. This lesson builds on students work in the 8th grade. problem to be solved is reduced to a quadratic programming problem in which the objective function is the residual sum of the squares in regression, and the constraints are linear ones imlx~ed on the regression coefficients. 3 times as important as Unconventional. b) Use the calculator and the equation of the linear regression line to complete the following table: Year 1960 1968 1988 1999 2005 2008 Time (s) c) Use the values in the table to draw the linear regression function on the scatter plot. To get a better feel for the regression line, try the following tasks. In Class: Practice Linear Regression HW: Obesity Problem (Hints included in key) r_9. EXTRAS. 45. Displaying all worksheets related to - Regression Analysis. Math 137 Quadratic Regression Classwork2 . Explain worksheet - regression inference 1. It allows the mean function E()y to depend on more than one explanatory variables Like correlation coefficients, linear regression analyzes the relationship between two variables, x and y. 1) Slope = −1, y-intercept = 0 y = −x 2) Slope = 1 4, y-intercept = 1 y = 1 4 x + 1 Write the slope-intercept form of the equation of the line through the given point with the given slope. Start by creating a visual model of the data. Bivariate Data - Correlation - Linear Regression - Correlation Coefficient r Jun 2, 2020 - Are you looking for Algebra 2 worksheets WITH answers? Making keys is super time consuming, so peruse this board for resources that have the hard part done for you!. 5 and 8 This is because the correlation value for the cubic regression is about 0. The multiple linear regression result implies that Reliable is around 1. The table below lists the total estimated numbers of United States AIDS cases, by year of diagnosis. 3582 1. The first is done using the Tools menu, and results in a tabular output that contains Shodor > Interactivate > Lessons > Linear Regression and Correlation omit the scatter plot worksheet; As a class, before splitting them into groups, have the If you have a continuous dependent variable, linear regression is probably the To address these problems, statisticians have developed several advanced variants: ordered data (assuming you record them in your worksheet in time order). A graphing calculator can also be used to perform quadratic regression. In this correlation and linear regression worksheet, students examine data to determine the statistic mean. 2 Fitting the Regression Line 12. Brandon Foltz 296,888 views Linear regression simply refers to creating a best fit for a linear relationship between two variables from observed data. Ratio and proportion word problems. Although a linear regression can be quite helpful in understanding data, it can sometimes be misleading, as Anscombe's Quartet shows . H. If the degree of correlation between variables is high enough, it can cause problems when you fit the model and interpret the results. (When we need to note the difference, a regression on a single predic-tor is called a simple regression. simple linear regression A college bookstore must order books two months before each semester starts. widely used; runs fast; easy to use (not a lot of tuning The linear regression model that I’ve been discussing relies on several assumptions. A. In other words, the SS is built up as each variable is added, in the order they are given in the command. 12A2 - HW Calendar Linear Regression Problems Q. LINEAR REGRESSION 6 1. The course website page REGRESSION AND CORRELATION has some examples of code to produce regression analyses in STATA. use functions fitted to data to solve problems in the context of the data. Model Summaries Worksheet Every workbook produced by RegressIt contains not only the data analysis and regression worksheets but also a model summaries worksheet that keeps an audit trail of all regression models fitted so far and allows side-by-side comparison of models fitted to the same dependent variable, suitable for framing. the meaning of slope in a linear regression equation the relationship of the slope to analyzed data whether or not the y-intercept of the linear regression equation is a relevant value EU 2 Econometrics. Enter the data to answer this question. com. To see why, consider a model such as this Y = β 0 +β 1eβ 2X + , (4. Curve Fitting Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. 48. It also plots the experimental points and the equation y = a x + b where a and b are given by the formulas above. 9) 𝑦𝑦 = 48. where a and b are given by. Round r to the nearest 3 decimal places. The case of having one independent variable is know as simple linear regression while the case of having multiple linear regression is known as multiple linear regression. Analysis of Variance, Goodness of Fit and the F test 5. 242 seconds for each additional foot of initial drop. Twenty five plants are selected, 5 each assigned to each of the fertilizer levels (12, 15, 18, 21, 24). Files for use with the TI-Nspire™ A non-linear method with comparable simplicity is known as K-nearest neighbors regression. Sketch and shade the squares of the residuals. In many applications, there is more than one factor that inﬂuences the response. Our predictors would then be idealism, misanthropy, and idealism x misanthropy (an interaction term). This is a quadratic model because the second differences are the differences that have the same value (4). THE MODEL BEHIND LINEAR REGRESSION 217 0 2 4 6 8 10 0 5 10 15 x Y Figure 9. Use the standard error of the slope, Sb, to calculate 95% confidence interval for the slope in ºC/century. Print off the worksheet if you How to graph the linear regression equation with the scatterplot data, how to generate a least squares linear regression model, How to create a line of best fit, 31 Jul 2016 12. \] Linear Regression Practice Worksheet 1. An agriculturalist was interested in the effects of a) Use your calculator to determine the linear regression function (y = A + Bx) that best models the data. There are two types of linear regression- Simple and Multiple. LINEAR REGRESSION WORKSHEET #1 Name_____ Date_____ Period_____ 1. The variable we base our predictions on is called the independent or predictor variable and is referred to as X. Use a graphing calculator to fit linear, quadratic, cubic, and power functions to At any time, you can complete this interactive online quiz to check how well you understand using linear regression. Click on pop-out icon or print icon to worksheet to print or download. b. The table lists the heights and weights of six wide receivers who played for the Atlanta Falcons during the 2010 football season. Instruction will be from 3:15 pm to 3:30 pm Compute the least squares regression line with the number of bidders present at the auction as the independent variable (x) and sales price as the dependent variable (y). Then find the Least Squares Line and use it to make a prediction. 2 problems on linear regression May 19, 2018 · 12 videos Play all Statistics PL14 - Simple Linear Regression Brandon Foltz Statistics 101: Logistic Regression Probability, Odds, and Odds Ratio - Duration: 13:03. The session will begin in room 315 with a brief review of the weekly topic. This worksheet contains the following data, with the list price in column C1 and the best price in the column C2. If we were to plot height (the independent or 'predictor' variable) as a function of body weight (the dependent or 'outcome' variable), we might see a very linear relationship, as illustrated Related posts of "Linear Regression Worksheet Answers" Scheme For Igneous Rock Identification Worksheet Answers In advance of preaching about Scheme For Igneous Rock Identification Worksheet Answers, you should realize that Instruction can be each of our factor to a greater down the road, and also understanding doesn't just stop right after the (RegressIt can fit linear regression models in Excel with over 200 independent variables on a PC or 125 variables on a Mac, and its R interface can be used to fit large models much faster. Estimator 3. 344 10. 35519 0. Height (inches) Weight (pounds) 75 192 76 220 71 200 74 210 69 185 72 189 a. A linear regression simply means that the equation will be the equation of a line . Ea How to compute the linear regression equation, y=ax+b, the linear correlation coefficient, r, and the coefficient of determination, r 2, using the TI-84 calculator, including turning the diagnostics on. It is a very simple regression algorithm, fast to train and can have great performance if the output variable for your data is a linear combination of your inputs. MR. c. This correlation is a problem because independent variables should be independent. Worksheet: Name the Song you can also use SVMs for regression. pdf: File Size: 1849 kb: File Type: pdf: Download File linear regression: An approach to modeling the linear relationship between a dependent variable, $y$ and an independent variable, $x$. jpg. Linear regression analysis, in general, is a statistical method that shows or predicts the relationship between two variables or factors. 1 Transformations in Linear Regression Create printable worksheets for solving linear equations (pre-algebra or algebra 1), as PDF or html files. Regression. Explain the slope in context of the problem. May 15, 2009 · Statistics and Regression Tools Review: Detailed Descriptions - Click OK to execute the regression. Predictthe!type!(positive,!negative,!no)!and!strength!of!correlation!(strong,!weak)!for!the!following! Summarize the four conditions that comprise the simple linear regression model. You don't have to believe everything it says. The week of March 30th we will be reviewing Linear Regression and NORMAL Curve. Mar 20, 2019 · Linear regression forecasting graph. through a process called linear regression. As with linear regression, the dataset must take of form of pairs of predictor variables x ⃗ i \vec{x}_i x i with resultant variables y i y_i y i . For instance, for an 8 year old we can use the equation to estimate that the average FEV = 0. Make a scatter plot for the data. If the answers to (l) and (m) are yes then using your simple linear regression equation predict the percentage of games won from a team that has a passing percentage of 6. Know what the unknown population variance $$\sigma^{2}$$ quantifies in the regression setting. com is simply the right site to go to! A simple linear regression model is a mathematical equation that allows us to predict a response for a given predictor value. ) We’d never try to find a regression by hand, and Linear regression is a process of drawing a line through data in a scatter plot. For each of the following, perform linear, quadratic, and exponential regressions. We are dealing with a more complicated example in this case though. Pythagorean theorem word problems. Further along in the paper, we will investigate some of the alternatives mentioned above, but this is also an opportunity Linear Regression Worksheet 1. b) According to the linear model, a coaster with a 200 foot initial drop is expected to last 139. 9 we’ll talk a lot more about how to check that these assumptions are being met, but first, let’s have a look at each of them. In a linear regression model, the variable of interest (the so-called “dependent” variable) is predicted from k other variables (the so-called “independent” variables) using a linear equation. In Section 15. A convenience store manager notices that sales of soft drinks are higher on hotter days, so he assembles the data in the following table. Answers provided. , between an independent and a dependent variable or between two independent variables). DOWNLOADS Note: This particular activity requires the use of TI-Nspire™ technology to be used successfully. In what year does Alpha ity’s population reach twice its initial (1/1/2004) population? 2. Do this in several different ways. It is the same Lagrange multiplier problem as above, with all the inequalities reversed. Contact Info. Worksheet 3 - Regression and linear models Linear regression references zFowler et al. sav. The results are shown in the table. Some of the worksheets for this concept are Chapter 9 correlation and regression solutions, I exploring regression, Scatter plots, Correlation coefficient, The united states of obesity, Concept 20 scatterplots correlation, Lecture 12 linear regression test Linear Equation Models. com delivers insightful information on linear regression free worksheet, basic concepts of mathematics and dividing fractions and other algebra topics. Regression (new) 4. 8: Regression - Distance from School (Worksheet) - Statistics LibreTexts Scatterplots & Regression on the TI-84 This video shows how to input two variable data and create a scatterplot with the TI-84 calculator. L Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Systems of Equations Word Problems Date_____ Period____ 1) Find the value of two numbers if their sum is 12 and their difference is 4. Now we’ll take a look at a data set for which the linear regression model is appropriate. A. This lesson SIMPLE LINEAR REGRESSION – DEMAND AS FUNCTION OF PRICE A new worksheet will appear revealing the results of your regression analysis. 4 Inferences on the Regression Line 12. Practice quiz 6. 05 to help answer this question. There are NINE problem types. Properties of the O. It works by estimating coefficients for a line or hyperplane that best fits the training data. This model generalizes the simple linear regression in two ways. A simple linear regression model is fit, relating plant growth over 1 year (y) to amount of fertilizer provided (x). If x is the independent variable and y the dependent variable, then we can use a regression line to predict y for a given value of x. 11719% In simple linear regression, we predict scores on one variable from the scores on a second variable. The line summarizes the data, which is useful when making predictions. 02 0. Normality. Using this analysis we can estimate the relationship between two or more variables. Complete details on this method (also known as the Potthoff method) are in Chapter 13 of K & K. Evaluating managerial success. Coming up for air. 6 The Analysis of Variance Table 12. If the truth is non-linearity, regression will make inappropriate predictions, but at least regression will have a chance to detect the non-linearity. Graph using the slope and the y-intercept. Neither regression nor correlation analyses can be Use the linear model to calculate the expected cholesterol for a person with pressure 160 mmHg. The results of the model fit are given below: Can we Problems 1. 7 Residual Analysis The first difference (the difference between any two successive output values) is the same value (3). Regression 10. Identify your Y and X values. Let's look at an example of linear regression by examining the data in the following table to discover the relationship between temperatures measured in Celsius (Centigrade) and Fahrenheit. uk March 17, 2011 1 Logarithmic transformations of variables Considering the simple bivariate linear model Yi = + Xi + i,1 there are four possible com- Algebra1help. Is the number $$\beta _1$$ in the equation $$y=\beta _1x+\beta _0$$ a statistic or a population parameter? The linear regression model explains the method used to take observed data and find a 'best fit' line to describe the relationship of two variables, and this quiz/worksheet pairing will test your Linear Regression Correlation Coeeficient. A Tomahawk Cruise ship in the South Pacific misfires a missile. Statistics – Linear Regression Worksheet - Solutions The busiest season for Walmart is the Christmas holiday and weekends see a tremendous number of customers. Multiple regression models thus describe how a single response variable Y depends linearly on a A. (SHOW WORK using algebra!) Linear mixture problems, non-linear mixture problems, transportation problems, personal scheduling problems and for something a little different, sports scheduling problems. 26721 × age. 4: The Regression Equation Carry out an appropriate test at a significance level of 0. You can use the Regression tool instead of the LINEST worksheet function. Word Problem Worksheet #2 (each correct answer = 1 extra ballot for the draw!) Linear Regression Worksheet - HAND IN :) April 9 - Data Unit Assignment. Recall that the least squares line minimizes the squares of the residuals. So, we have a sample of 84 students, who have studied in college. - Stats Worksheet #1 - Stats Worksheet #2. ¯3, y¯ = 46. 15 Apr 2015 Research question type: When using one variable to predict or explain another Simple Linear Regression – Additional Information worksheet. The least squares method is generally used with a linear regression, but 1 day ago · You will not require the options for "residuals" for this analysis. Using a catapult of their own design, students will model a parabolic relationship, collect data, and draw conclusions from data and the quadratic curve of best fit. Finally, the students will explore question 4 on the Activity Worksheet. simple linear regression, the sample correlation coefficient is the square root of the coefficient of determination, with the sign of the correlation coefficient being the same as the sign of b1, the coefficient of x1 in the estimated regression equation. The student will evaluate the relationship between two variables to determine if that … 12. 23 Apr 2011 In the table below, list the assumptions of multiple linear regression all 6 predictors into the one model, because of the collinearity problem. Consider the following hypothetical data set. analysis 5. The Multiple Regression Process Conceptually, multiple regression is a straight forward extension of the simple linear regression procedures. 000 Problems 0. A convenience store manager notices that sales of soft drinks are higher on hotter days, so he assembles the data in the table. Classification problems are supervised learning problems in which the response is categorical; Benefits of linear regression. Regression Analysis. 1: Mnemonic for the simple regression model. Create a scatter plot and approximate a trend Using your trend line, predict the 0-60 time for a car that costs$120 K? 2. These worksheets are especially meant for pre-algebra and algebra 1 courses (grades 7-9). For this question, we could swap the independent and dependent variables and still get reasonable results. The linear regression equation, also known as least squares equation has the following form: $$\hat Y = a + b X$$, where the regression coefficients $$a$$ and $$b$$ are computed by this regression That is, there is no lack of fit in the simple linear regression model. (a) Make a scatter plot of the data. 778 (or a value 0. 3t 2 6t 80 . Now, select Sheet 1 (by clicking on its tab at the bottom of the worksheet) and paste the information into cell F4. Worksheets are Linear regression work 1, Writing linear equationslinear regression, Work 1, Work 3, Linear reg correlation coeff work, Chapter 9 correlation and regression solutions,, Work regression. 1, Demand estimation using linear regression. In a simple linear regression model, we model the relationship between both variables by a straight line, formally $Y = b \cdot X + a. Fit linear, quadratic, cubic, exponential, quartic, and power functions to the data. 000 These are called the regression parameters in the simple linear regression equation (the equation is also known as the least squares regression equation or the trend equation or simply the regression). 2 Practice Worksheet 1. The regression line and the residuals are displayed in figure #10. Answers are included here to check your work. One of the favorite topics on which the interviewers ask questions is ‘Linear Regression. AP Stats: Section 3. The linear regression analysis can then 7 Aug 2017 The students can immediately understand that linear regression is on the worksheet eliminates many class time and design problems and 5 Mar 2020 Open a new workbook in Excel and make 3 worksheets: Data, Chart, and Saves. In Microsoft Office Excel 2007, you can find the Regression tool by clicking Data Analysis in the Analysis group on the Data tab. Find the equation of the regression line. In this example R2 = 0. For example, suppose that height was the only determinant of body weight. Linear Equation in Two Variables Distance Problems This worksheet is based on linear equation in two variables distance problems. Transit demand. Linear Regression Worksheet Answers with Valuable Issues. 5 Linear Regression Algebra II Name Guided Notes Date_____Block_____ Recap: Graphing Linear Functions 1. 7 is the affe ct on sales of a change in p1 with p2 held fixed !! 0 5 10 15 9 8 4 Sales I Simple Linear Regression Common Mistakes Statistics Tables Quiz: Cumulative Review A Quiz: Cumulative Review B Online Quizzes for CliffsNotes Statistics QuickReview A linear regression model corresponds to a linear regression model that minimizes the sum of squared errors for a set of pairs $$(X_i, Y_i)$$. Linear Algebra in Linear Regression Continue Suppose we have a column space in R 3 \mathbf{R}^3 R 3 , W W W , a vector b ⃗ \vec{b} b , and A x ⃗ A\vec{x} A x , the point closest to b ⃗ \vec{b} b on W . Worksheet – Regression The table below displays data on the temperature ( F) reached on a given day and the number of cans of soft drink sold from a particular vending machine in front of a grocery store. The least square regression line for the set of n data points is given by the equation of a line in slope intercept form: y = a x + b. Quiz (new) 7.$ For now, let us suppose that the function which relates test score and student-teacher ratio to each other is \[TestScore = 713 - 3 \times STR. The last page of this exam gives output for the following situation. Section 1: Input Data Below are the input parameters to begin the simulation. 2 in-depth answers. Linear Regression Correlation Coeeficient - Displaying top 8 worksheets found for this concept. 60 55 70 45 80 35 a. Save the workbook as Linear Regression - Brief Lesson, or something similar, into a logical file folder. Does the relationship appear to be linear? Why? Yes, the relationship appears to be linear because it seems to be decreasing in similar intervals after Unit 3 - Linear Functions & Linear Regression 3-1, Functions and Function Notation - Video , Notes , Worksheet 3-3, Linear Functions - Video , Notes , Worksheet Multiple Linear Regression So far, we have seen the concept of simple linear regression where a single predictor variable X was used to model the response variable Y. Percent of a number word problems. 4 Solving Real-Life Problems How can you use a linear equation in two variables to model and solve a real-life problem? Write a story that uses the graph at the right. and solve practical problems using models of linear, quadratic, and exponential functions. An observational study if 19 managers from a medium-sized manufacturing plant extended Mintzberg’s work by investigating which activities successful man- Linear regression is a method for modeling the relationship between one or more independent variables and a dependent variable. slope-intercept form. C. Linear regression is a technique that is useful for regression problems. A boat goes 30km upstream and 44km downstream in 10 hours. Quiz (new) 11. a) According to this model what was the price of gas in1990? Jan 17, 2013 · Introduction to Correlation and Regression Analysis. 9199 yields a coefficient of determination of 0. Simple Linear Regression Example 12. 0128 . 766, adjusted for Output 2: Regression output for the grade versus homework study Regression Analysis: CourseGrade versus Problems The regression equation is CourseGrade = 44. 2 MULTIVARIATE LINEAR REGRESSION Multiple linear regression with a single criterion variable is a straightforward generalization of linear regression. This is the only section that requires user input. We need to also include in CarType to our model. Displaying top 8 worksheets found for - Linear Equation Models. n. e. Linear Regression Interview Questions – Fundamental Questions. Materials: CD Player / Computer with CD drive. If you were a careful artist, you could take a ruler and draw a straight-line as close as possible to every point in Worksheet 2. The measure of how well this linear function ts the experimental points, is called regression analysis. • In a simple linear regression model, a single response measurement Y is related to a single predictor (covariate, regressor) X for each observation. Squaring −0. Worksheet for Correlation and Regression (February 1, 2013). In addition, after finding the equations, students are asked "extension questions" in which they must use the equation to answer questions a About This Quiz & Worksheet About This Quiz & Worksheet Simple linear regression builds on the concept of a regression line by allowing you to specifically make predictions based on the regression Practice linear regression with 10 Canadian data sets covering a range of topics. Simple linear regression is a bivariate situation, that is, it involves two dimensions, one for the dependent variable Y and one for the independent variable x. Students will determine the linear regression equation and correlation coefficient for their data using the graphing calculator. 21. Open or retrieve the worksheet Slr01. This leads to the Below is a plot of the data with a simple linear regression line superimposed. a) According to the linear model, the duration of a coaster ride is expected to increase by about 0. USING THIS MODULE vii Apr 27, 2017 · Non-Linear Relationships Not all relationships are linear. a) Find the least square regression line for the following set of data  Practice Worksheet: Linear Regression. 903, and because the graph of the cubic model is seen to be a closer match to the dots in the scatterplot than is the linear model. Chapter 8 Linear Regression 91 22. Excel completes the regression analysis Regression Analysis Worksheets- Includes math lessons, 2 practice sheets, homework sheet, and a quiz! Linear regression simply refers to creating a best fit for a linear relationship between two variables from observed data. This makes the line fit the points. In this problem we find the model by analyzing the data on femur length and height for the ten males given in the table. Consider the following diagram. 3: Setup for Linear Regression Test on TI-83/84  to fit a simple linear regression equation with poverty predicting obesity. Linear Regression Assumptions • Linear regression is a parametric method and requires that certain assumptions be met to be valid. the linear relationship through the correlation coefficient. 𝑦𝑦 = −70. Graphic calculators, such as the TI-83, have built in programs which allow us to nd the slope and the y intercept of the best tting line to a set of data points. On a piece of graph paper, create a scatter plot. Of easily catch up can understand this You  There are actually two ways to do a linear regression analysis using Excel. Practice-Regression 2 linear, quadratic, exponential A simple linear regression equation for this would be $$\hat{Price} = b_0 + b_1 * Mileage$$. Thus a linear model only explains 85% of the variation in women's world record 100 m dash times. Export problem to Excel (highlighted when problem is correctly entered). Identify the type of regression with the best fit, and answer the question using the type of regression that best fits the data. 433 seconds. It may be printed, downloaded or saved and used in your classroom, home school, or other educational Linear Regression. Linear Regression & Correlation Coefficient Worksheet Name _____ Hr _____ 0 2 4 6 8 10 0 2 4 6 1. Linear regression only supports regression type problems. Use the two plots to intuitively explain how the two models, Y!$0 %$ 1x %& and Why Linear Regression? •Suppose we want to model the dependent variable Y in terms of three predictors, X 1, X 2, X 3 Y = f(X 1, X 2, X 3) •Typically will not have enough data to try and directly estimate f •Therefore, we usually have to assume that it has some restricted form, such as linear Y = X 1 + X 2 + X 3 correlation, in linear regression. Femur Length (cm) (a) Make a scatter plot of the data. Our model will take the form of ŷ = b 0 + b 1 x where b 0 is the y-intercept, b 1 is the slope, x is the predictor variable, and ŷ an estimate of the mean value of the response variable for any value of the predictor Copy the regression coefficients onto the sheet with the actual population data. become identical when r = –1 or 1 or in other words, there is a perfect negative or positive correlation between the two variables under discussion. Problem-solving using linear regression has so many applications in business, digital customer experience, social, biological, and many many other areas. CAS'S WEBSITE - Home I don't think so. The regression line is the line that makes the square of the residuals as small as possible, so the regression line is also sometimes called the least squares line. This math worksheet was created on 2013-02-14 and has been viewed 62 times this week and 1,053 times this month. (1) Investigator #1 is interested in predicting Y from X, and fits and computes a regression line for this purpose. Using a similar approach, we may prove that S2 y = S 2 yˆ +S 2 e (5. There are also other regression modelling techniques for data not considered to be at continuous/interval/ratio level. The calculators also give 1. temperature 70 75 80 90 93 98 72 75 75 80 90 95 98 91 98 quantity 30 31 40 52 57 59 33 38 32 45 53 56 62 51 58 ¯x = 85. The regression model is linear in the unknown parameters. To make the residual plot, use “Graphs” and then type in the name of the explanatory variable. Each point of data is of the the form (x, y) and each point of the line of best fit using least-squares linear regression has the form   Excel displays the Regression dialog box. nate because the world is too complex a place for simple linear regression alone to model it. Find a quadratic model in standard form for the data. 2 Linear Regression If there is a \signi cant" linear correlation between two variables, the next step is to nd the equation of a line that \best" ts the data. 8462. Comments: Another SPSS output table – see Table 3 – gives a useful value 'R square', or the 'coefficient of determination'. c) the relationship between X and Y is quadratic d) there is no relationship between X and Y. The dependent variable must be of ratio/interval scale and normally distributed overall and normally distributed for each value of the independent variables 3. Apply the method of least squares (or maximum likelihood) with a non-linear function. A simple linear regression model to relate BP with age will be BP = regression estimate (b) * age + constant (a) + error term (å) The regression estimate (b) and the constant (a) will be derived from the data (using the method of least-squares(5)) and the error term is to factor in the situation that two persons with the same age need not have the same BP. L. State which model, linear or quadratic, best fits the data. Know how to obtain the estimate MSE of the unknown population variance $$\sigma^{2 }$$ from Minitab's fitted line plot and regression analysis output. A regression is a process that takes all the points and calculates the equation that best 'fits' those points. The Nonlinear Regression Model 1 Goals The nonlinear regression model block in the Weiterbildungslehrgang (WBL) in ange- wandter Statistik at the ETH Zurich should 1. are called the residuals. The estimated regression equation is that average FEV = 0. With this worksheet generator, you can make customizable worksheets for linear inequalities in one variable. Included are two versions of the optional placemat/worksheet to help students through the process of linear regression and analysis. a) Enter the data into two lists of your graphing. Show that in a simple linear regression model the point ( ) lies exactly on the least squares regression line. CD – Music compiled by teacher. Investigator #2 is interested in predicting X from Y, and computes his regression line for that purpose (note that in the real problem of “parallel-line bioassays, with X=log(dose) STATISTICS 110/201 PRACTICE FINAL EXAM KEY (REGRESSION ONLY) Questions 1 to 5: There is a downloadable Stata package that produces sequential sums of squares for regression. This result is smaller than suggested by any of the other analyses that I have conducted, and is most similar to the analysis with all of the variables except for each of Reliable and Unconventional. A study found that age and blood pressure are correlated. Practice Problems: Correlation and Linear Regression Researchers interested in determining if there is a relationship between death anxiety and religiosity conducted the following study. We talk about looking at the data to decide what type of regression is appropriate. is found. See more ideas about Linear regression, Regression, Algebra. (b) Find and graph a linear regression equation that models the data. Each of the data sets has 4 or 5 points and approximates a linear relationship. We can find the equation of the line of best fit through the data in the least squares sense, as follows. ﬁt linear regression models. Is there a linear relationship between how critics score the games and how users score the games? (Let x = critic score and y = user score). 2 Equation:_____ WORKSHEET GENERATORS. Categorical Data 12. A residual plot is displayed showing the deviation between the data and the calculated values of the dependent variable. expression the strength of a linear relationship between two variables are two of the desired outcomes of this module. 1 The Simple Linear Regression Model 12. Interpreting r and r 2. Word problems on constant speed. Oct 05, 2012 · The “Good” linear regression model. Remember to complete the five steps of hypothesis testing in the spaces provided on the worksheet. This means that this data can be modeled using a linear regression line. S. (b) What change in gasoline mileage is associated with a 1 cm3 change is engine displacement? 11-18. Correlation!Coefficient!&Linear!of!Best!Fit!HW! Name:!!_____! 8. The fit is not quite as tight for the women's times. The missile goes over the side of the ship and hits the water. SGDRegressor is well suited for regression problems with a large number of training samples (> 10;000), for other prob-lems we recommend Ridge, Lasso, or ElasticNet. Then they are able to apply the knowledge they learn in this project on the real life problems. Explain Jul 31, 2016 · State the three assumptions that are the basis for the Simple Linear Regression Model. Figure 2. 05898 6. If the data is curved, a line would not be the best equation to use. In most problems, more than one predictor variable will be available. than ANOVA. Research question type: When wanting to predict or explain one variable in WORKSHEETS\calcium. Find the numbers. Solve general word problems about real-world relationships that can be modeled by linear equations or functions. 999, which is closer to 1 than is the linear correlation value of 0. It is also a method that can be reformulated using matrix notation and solved using matrix operations. Use the linear regression equation determined in part "B" to calculate a set of "predicted y values" for each observed x value. Logarithmic Regression Problems. Regression 9. If the data on the scatter plot seems to represent a linear relationship, then linear regression can be used to find the line that best fits the data. Inference in the Linear Regression Model 4. The New Worksheet Ply default output option means that Excel locates the regression outputs on a new, separate worksheet that it creates and puts to the left of the worksheet that holds the original data. To do this, highlight the cell range A17:B18 and click on the COPY button. The plot to the right shows 5 data points and the least squares line. Interpret the meaning of the slope β ^ 1 of regression line in the context of problem. Use Stat > Regression > Regression to find the regression equation AND make a residual plot of the residuals versus the explanatory variable. Open the “Good” worksheet; this is a (made-up) data set showing the Height (independent variable) and Weight (dependent variable) values for a selection of people. Nov 26, 2014 · to linear regression . Old Faithful Geyser in Yellowstone National Park, Wyoming, derives its name and its considerable fame from the regularity (and beauty) of its eruptions. ) Using a graphing calculator and quadratic regression to find a model: A study compared the speed x, in miles per hour and the average fuel economy y (in miles per gallon) for cars. 3 (b) Find and graph a linear regression equation that models the data. In this exercise, you will gain some practice doing a simple linear regression using a data set called week02. Linear Regression and Correlation Introduction Linear Regression refers to a group of techniques for fitting and studying the straight-line relationship between two variables. Your output for this multiple regression problem should be similar to the results shown   Let's explore the problem with our linear regression example. To make the notation simpler, assume that the criterion variable Y and the p Lesson 21: Multiple Linear Regression Analysis . Derivation of linear regression equations The mathematical problem is straightforward: given a set of n points (Xi,Yi) on a scatterplot, find the best-fit line, Y‹ i =a +bXi such that the sum of squared errors in Y, ∑(−)2 i Yi Y ‹ is minimized The simple linear regression model Our goal is to obtain estimates ^ 0 and ^ 1 for 0 and 1 to de ne the regression line ^y = ^ 0 + ^ 1x that provides the best t for the data Example: Assume that the regression line of the previous example is: Cost = 15:65 + 1:29 Volume Regression Problems Math The data for these problems is from Math 142, Dr Lacey, Packet 118. Student Learning Outcomes By the end of this chapter, you should be able to do the following: Linear Regression Models with Logarithmic Transformations Kenneth Benoit Methodology Institute London School of Economics kbenoit@lse. The critical assumption of the model is that the conditional mean function is linear: E(Y|X) = α +βX. 7 This worksheet is designed to give students extra practice at using their graphing calculators to calculate Linear Regression Equations. Oct 21, 2019 · Some of the worksheets below are Correlation Coefficient Practice Worksheets, Interpreting the data and the Correlation Coefficient, matching correlation coefficients to scatter plots activity with solutions, classify the given scatter plot as having positive, negative, or no correlation, … Simple Linear Regression Model Only one independent variable, x Relationship between x and y is described by a linear function Changes in y are assumed to be caused by changes in x Fall 2006 – Fundamentals of Business Statistics 18 Types of Regression Models Positive Linear Relationship Negative Linear Relationship Relationship NOT Linear Free worksheets for solving or graphing linear inequalities. Here are data from four students on their Quiz 1 scores and their Quiz 5 scores and a graph where we connected the points by a line. Regression 8. Their sum is 13. Linear Regression is a statistical tool in excel that is used as a predictive analysis model to check the relationship between two sets of data of variables. Winking at Phoenix High School Sec 5. This three-page worksheet contains 16 problems. 8 + 0. Predict the number of aids cases for the year 2006. Linear Regression Equation - Displaying top 8 worksheets found for this concept. Compute S S E, the measure of the goodness of fit of the regression line to the sample Use Linear Regression Calculator and Grapher Given a set of experimental points, this calculator calculates the coefficients a and b and hence the equation of the line y = a x + b and the Pearson correlation coefficient r. HSF-LE. The above simple linear regression examples and problems aim to help you understand better the whole idea behind simple linear regression equation. Determine the slope of the linear regression line, b, in ºC/century and the correlation coefficient, r. Linear regression is nice, but it isn't a religion. 1) 45 Multicollinearity occurs when independent variables in a regression model are correlated. Put bite number on the x-axis and Twizzler length on the y-axis. Worksheet: Rate the Song. To answer this question the researcher would measure body weight and blood cholesterol level in various subjects. Multiple-choice. x, y ( ) points. Bascially, the least-squares regression line is the line that minimizes the squared "errors" between the actual points and the points on the line. 9. slcmath@pc. Students will use different methods to Sep 23, 2018 · This video explains you the basic idea of curve fitting of a straight line in multiple linear regression. Let us begin with a fundamental Linear Regression Interview Questions. In the 8th grade students solved problems in the context of bivariate measurement data. Finally, the students will explore the residuals for the regression equation and interpret the results from the activity in the context of the problem. The multiple regression of Sales on own price (p1) and competitor's price (p2) yi eld more intuitive signs: How does this happen ? The regression equation is Sales = 211 + 63. This Practice Problems: Correlation and Linear Regression Worksheet is suitable for 9th - 11th Grade. To draw a linear forecast graph like shown in the screenshot below, here's what you need to do: Copy the last historical data value to the Forecast In this example, we copy the value from B13 to C13 Select Viewport in the Trend Lines tab to perform linear regression for the sub-period. linear regression and modeling problems with answers. The variable we predict is called the dependent or outcome variable and is referred to as Y. Describe the correlation Use the line of best fit to make predictions for the following real-world problems. 10: Regression - Fuel Efficiency (Worksheet) · 13: F Distribution and One-Way ANOVA If a loan officer makes 95% of his or her goal, write the linear function that applies based 12. A random sample was taken as stated in the problem. Student will learn how to write a linear regression equation and use the equation to solve a  Make a scatterplot of the data, letting x represent the number of years since 1990. According to the linear model, what is the expected pressure for a person with cholesterol 270 mg/dl? Use the following sums: $\sum x_i=16960$ mg/dl, $\sum y_j=11160$ mmHg, $\sum x_i^2=3627200$ (mg/dl)$^2$, $\sum y_j^2=1576800$ mmHg$^2$ y \$\sum x_iy Welcome to The Systems of Linear Equations -- Two Variables (A) Math Worksheet from the Algebra Worksheets Page at Math-Drills. Inference on Prediction Assumptions 1. 126 Chapter 3 Writing Linear Equations and Linear Systems 3. Word problems on average speed Word problems Free worksheet(pdf) and answer key on the solving word problems based on linear equations and real world linear models. Equation: Created Date: 1/4/2016 8:11:44 PM A correlation or simple linear regression analysis can determine if two numeric variables are significantly linearly related. ## [1] 680 640 670 660 630 660 635. linear regression problems worksheet

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Does the convergence of some subsequences imply the convergence of a sequence?

I am considering a following problem: Does $$\\a_{2k}, a_{2k+1}, a_{3k} \rightarrow g$$ imply that $$a_{n} \rightarrow g ?$$ I know that if every subsequence goes to $g$ then also a sequence goes to $g$. My way of reasoning is following: we know that odd and even $k$ subsequences goes to $g$. But if we substitute for example $k=t^{2} -5$, then there is problem (at least for me). If I can reason that way, what must I do to prove more formally that the statement is false? And if I can not, what is going on with $a_{n}$? Thanks for any hints!

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If the even numbered terms and odd numbered terms both converge to the same limit $L$, then the limit of the sequence exists and is $L$.

To prove this, let $\epsilon > 0$ be arbitrary. We want to find $N \in \mathbb{N}$ such that $n > N$ implies $|a_n - L| < \epsilon$.

Since the even numbered terms are all within epsilon of $L$ after some $N_1 = 2j$ and the odd numbered terms are all within epsilon of $L$ after some $N_2 = 2k + 1$, just pick $N = \max\{N_1, N_2\}$.

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You may want to prove the next nice lemma:

Lemma: If $\,\{A_i\}_{i\in I},$ is some partition of $\,\Bbb N\,$ s.t. $\,|A_i|=\aleph_0=|\Bbb N|\,\,\,,\,\forall\,\,i\in I\,$ , and all the sets $\,A_i\,$ are well ordered, then for a real sequence $\,\{x_n\}\,$ it is true that

$$x_n\xrightarrow [n\to\infty]{} x\Longleftrightarrow (\,\forall\,\,i\in I\,\,,x_{n_i}\xrightarrow [n_i\to\infty\,,\,n_i\in A_i]{} x)$$

Thus, as B. wrote in his answer, it is enough and sufficient that the subsequence of odd indexes and the one of even indexes converge both to the same limit.

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Doesn't $I$ have to be a finite in order for this to work? –  Arthur Fischer Nov 1 '12 at 17:48
Well, not that you mention it...I'm not completely sure, I'll try to check this later. Thanks. –  DonAntonio Nov 1 '12 at 18:23
Take $I = \{ 1 \} \cup \{ p : p \text{ is prime} \}$. Define $A_1 = \{ 1 \} \cup \{ n : n \text{ is not a prime power} \}$, and $A_p = \{ p^k : k \geq 1 \}$ for prime $p$. Define the sequence $( x_n )_{n \geq 1}$ so that $x_n = 0$ for all $n \in A_1$ and $x_{p^k} = \frac{1}{k}$ for $p$ prime and $k \geq 1$. Then for each $i \in I$ the sequence $( a_n )_{n \in A_i}$ converges to $0$, but the sequence $( a_n )_{n \geq 1}$ does not converge as it takes the value $1$ infinitely often and has subsequences converging to $0$. –  Arthur Fischer Nov 1 '12 at 18:39
In the other answers you have seen that it $a_{2k},a_{2k+1}\to g$ then already $a_n\to g$. You don't even need $a_{3k}$. I would like to add that if you just require $a_{2k},a_{2k+1}$ and $a_{3k}$ to converge (a priori with different limits), then you have that all three limits coincide and that $a_n$ has the same limit.
To see this note that $a_{2n}$ and $a_{3n}$ as well as $a_{2n+1}$ and $a_{3n}$ have a commen subsequence. Then you only need that if a converging sequence has a subsequence with limit $g$ then the whole sequence has limit $g$.


# How to show $\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$?

I am able to evaluate the limit $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$ for a given $n$ using l'Hôspital's (Bernoulli's) rule.

The problem is I don't quite like the solution, as it depends on such a heavy weaponry. A limit this simple, should easily be evaluable using some clever idea. Here is a list of what I tried:

• Substitute $y = x - 1$. This leads nowhere, I think.
• Find the Taylor polynomial. Makes no sense, it is a polynomial.
• Divide by major term. Dividing by $x$ got me nowhere.
• Find the value $f(x)$ at $x = 1$ directly. I cannot as the function is not defined at $x = 1$.
• Simplify the expression. I do not see how I could.
• Using l'Hôspital's (Bernoulli's) rule. Works, but I do not quite like it.

If somebody sees a simple way, please do let me know.

Added later: The approach proposed by Sami Ben Romdhane is universal as asmeurer pointed out. Examples of another limits that can be easily solved this way:

• $\lim_{x \to 0} \frac{\sqrt[m]{1 + ax} - \sqrt[n]{1 + bx}}{x}$ where $m, n \in \mathbb{N}$ and $a, b \in \mathbb{R}$ are given, or
• $\lim_{x \to 0} \frac{\arctan(1 + x) - \arctan(1 - x)}{x}$.

It sems that all limits in the form $\lim_{x \to a} \frac{f(x)}{x - a}$ where $a \in \mathbb{R}$, $f(a) = 0$ and for which $\exists f'(a)$, can be evaluated this way, which is as fast as finding $f'$ and calculating $f'(a)$.

This adds a very useful tool into my calculus toolbox: Some limits can be evaluated easily using derivatives if one looks for $f(a) = 0$, without the l'Hôspital's rule. I have not seen this in widespread use; I propose we call this Sami's rule :).

• Please do not use titles consisting only of math expressions; these are discouraged for technical reasons -- see meta. Jan 15, 2015 at 12:36

Let $$f(x)=x+x^2+\cdots+x^n-n$$ then by the definition of the derivative we have

$$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1}= \lim_{x \to 1}\frac{f(x)-f(1)}{x - 1}=f'(1)\\[10pt] = \left[ \vphantom{\frac11} 1 + 2x + 3x^2 + \cdots + nx^{n-1} \right]_{x=1} = \frac{n(n + 1)}{2}$$

• It means that we should substitute $x$ by $1$.
– user63181
Jul 25, 2013 at 12:20
• Then this is truly an ingenious solution. Jul 25, 2013 at 12:30
• So the original is just the integral of a function summing the positive natural numbers? Pauling would be proud. Jul 25, 2013 at 15:29
• I almost cried when I saw this solution. Jul 25, 2013 at 17:44
• @RossMillikan shouldn't it not matter? if $f(x)$ contains the $-n$ then $f(1) = 0$, and if $f(x)$ does not, $f(1) = n$, but either way $f(x) - f(1)$ doesn't change and since $n$ is a constant, $f'(x)$ looks the same. Jul 25, 2013 at 23:22

Well, you can use that \begin{align} x-1&=\left(x-1\right)\cdot 1,\\ x^2-1&=\left(x-1\right)\cdot\left(x+1\right),\\ x^3-1&=\left(x-1\right)\cdot(x^2+x+1),\\ &{}\ \ \vdots\\ x^n-1&=\left(x-1\right)\cdot(x^{n-1}+\cdots+1),\\ \end{align} Now sum the left hand sides and the right hand sides, divide by $x-1$ and consider the limit $x\rightarrow 1$.

The given limit is $$\lim_{x\rightarrow 1}\frac{\sum_{k=1}^nx^k-n}{x-1}\\ =\lim_{x\rightarrow 1}\frac{\sum_{k=1}^n(x^k-1)}{x-1}\\ =\sum_{k=1}^n \lim_{x\rightarrow 1} \frac{(x^k-1)}{x-1}$$

Now, $$\lim_{x\rightarrow 1} \frac{(x^k-1)}{x-1}\\ =\lim_{x\rightarrow 1} (\sum_{j=0}^{k-1}x^j)=k$$

Hence the given limit becomes $$\sum_{k=1}^n k=\frac{n(n+1)}{2}$$

Hints:

$$x+x^2+\ldots+x^n-n=(x-1)+(x^2-1)+\cdots +(x^n-1)=(x-1)\left(1+(x+1)+\cdots\right)$$

Also

$$1+(x+1)+(x^2+x+1)+\cdots+(x^{n-1}+\cdots+x+1)\xrightarrow [x\to 1]{}1+2+\cdots+n=\frac{n(n+1)}2$$

You can use induction:

$$\frac{x + x^2 + \dots + x^n + x^{n+1} - (n+1)}{x - 1} =\\ \frac{x + x^2 + \dots + x^n - n}{x - 1} +\frac{x^{n+1}-1}{x-1}=\\ \frac{x + x^2 + \dots + x^n - n}{x - 1} +(1+x+x^2+\ldots+x^n)\xrightarrow[x\to 1]{} \frac{n(n + 1)}{2}+(n+1)=\frac{(n+1)(n + 2)}{2}.$$

• I am a little lost. How is that you are getting different result? Or is $\frac{(n + 1)(n + 2)}{2}$ only a part of the solution? Jul 25, 2013 at 12:28
• @DavidČepelík No, $\frac{(n+1)(n+2)}{2}$ is the limit for $\frac{x+x^2+\ldots+x^n\color{brown}{+x^{n+1}}-n\color{brown}{-1}}{x-1}$ and not for $\frac{x+x^2+\ldots+x^n-n}{x-1}$.
– P..
Jul 25, 2013 at 12:32
• Of course, that is the induction. Thank you! Jul 25, 2013 at 12:37
• So for $n = 1$ and $x \to 1$, we get $a_1 = \lim_{x \to 1} \frac{x - 1}{x - 1} = 1$ and $a_{n + 1} = a_n + (n + 1)$. The closed form for $a_n$ is then $\frac{n(n + 1)}{2}$. Correct? Jul 25, 2013 at 12:42
• @DavidČepelík: Yes that's right!
– P..
Jul 25, 2013 at 13:08

$$\frac{x + x^2 + \dots + x^n - n}{x - 1} =\frac{1+x + x^2 + \dots + x^n - n-1}{x - 1} =\frac{\frac{x^{n+1}-1}{x-1}-(n+1)}{x-1}$$

Putting $x-1=y,$

$$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \lim_{y\to0}\frac{\frac{(1+y)^{n+1}-1}y-(n+1)}y$$

$$=\lim_{y\to0}\frac{\frac{1+\binom {n+1}1y+\binom {n+1}2y^2+ O(y^3)-1}y-(n+1)}y\text{ (using Binomial Expansion)}$$

$$=\lim_{y\to0}\frac{(n+1)+\frac{n(n+1)}2y+ O(y^2)-(n+1)}y$$

$$=\frac{n(n+1)}2\text{ as }x\to1,y\to0\implies y\ne0$$

If you already know l'Hopital's rule, Sami Ben Romdhane's answer isn't telling you anything you don't already know. When you have $$\lim_{x\to a}\frac{f(x)}{x - a}$$ and $f(a) = 0$ (otherwise, the limit is infinite on either side of $a$), then l'Hopital's rule says that the limit is the same as $$\lim_{x \to a}f'(x)= f'(a).$$ The only hard part then, in this case, is evaluating $f'(a)$. Here, $f(x) = x + x^2 + \cdots + x^n - n$, so $f'(x) = 1 + 2x + \cdots + nx^{n - 1}$, so $f'(1) = 1 + 2 + \cdots + n = \frac{n(n + 1)}{2}$ by a famous identity.

In fact, you can think of the definition of the derivative as just a special case of l'Hopital's rule (it isn't really, because l'Hopital's rule depends on the definition of the derivative, not the other way around, but it's useful to think of it this way).

• I do not think this is how he approached the problem; my guess is he noticed that the function and the derivative of the numerator are bound together by the definition of derivative of a function. I did not realize that he used the same principle Bernoulli's rule is built upon, thanks for pointing it out. Jul 25, 2013 at 16:23

Your objection to using l'Hospital's rule is on the basis that it feels like it's "too powerful" a tool for the problem, right?

Lets go all the way to the other end, then, and prove the limit using just the definition.

$$\lim_{x\to a} f(x) = L \iff \forall \varepsilon \gt 0 \ \exists \delta \gt 0 \ni \left| x-a \right| \le \delta \Rightarrow \left| f(x)-L \right| \le \varepsilon$$

Now, all we need to do is figure out a way that we can always pick a $\delta$ small enough to keep the function within $\varepsilon$ of $L$.

The particular limit in question: $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$

From that, we can take: \begin{align*} a&=1\\ f(x)&=\frac{x + x^2 + \dots + x^n - n}{x - 1}\\ L&=\frac{n(n + 1)}{2} \end{align*}

So we need to solve $$\left| \frac{x + x^2 + \dots + x^n - n}{x - 1} - \frac{n(n + 1)}{2}\right| < \varepsilon$$ for $\left|x-1\right|$.

I really don't feel like doing any of that crunchwork solving that equation. If someone out there does feel like it, please do so, and edit it into my answer. For now, though, I am going to skip ahead bunch of steps, assume we solved it, and have our solution of $$\left|x-1\right| \ge g(\varepsilon)$$

Now, we know from the definition that $\left| x-1 \right| \le \delta$, so we can conclude that if we pick $\delta = g(\varepsilon)$, we ensure that the value of $f(x)$ is within $\varepsilon$ of $L$, satisfying our definition of the limit, and proving that $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$

• Thanks for your answer. You already know the limit exists and you know what it equals to ($\frac{n(n+1)}{2}$ was not my first guess). Finding what a limit equals to and proving the solution is correct by checking it satisfies the definition of the limit are two completely different exercises. (Similar to the difference between the P and NP classes in the theory of computational complexity.) I am aware I gave correct solution along with the limit, maybe that is why we did not understand each other. Sorry about it. Jul 26, 2013 at 20:01
• @DavidČepelík Personally, I think that if you prove the limit without using your super-powered tool, it doesn't matter if you used it to find the value of the limit, since the validity of your answer then does not depend on the validity of the method you applied. Jul 29, 2013 at 12:57


### 2 Do these iterative methods always work?

No. It is not difficult to invent examples where the iteration fails to approach the solution of $AX=B$ . The key point is related to matrix norms seen in the preceding Section.

The two iterative methods we encountered above are both special cases of the general form

$\phantom{\rule{2em}{0ex}}{X}^{\left(k+1\right)}=M{X}^{\left(k\right)}+N.$

1. For the Jacobi method we choose $M=-{D}^{-1}\left(L+U\right)$ and $N={D}^{-1}B$ .
2. For the Gauss-Seidel method we choose $M=-{\left(D+L\right)}^{-1}U$ and $N={\left(D+L\right)}^{-1}B$ .

The following Key Point gives the main result.

##### Key Point 13

For the iterative process ${X}^{\left(k+1\right)}=M{X}^{\left(k\right)}+N$ the iteration will converge to a solution if the norm of $M$ is less than 1 .

Care is required in understanding what Key Point 13 says. Remember that there are lots of different ways of defining the norm of a matrix (we saw three of them). If you can find a norm ( any norm ) such that the norm of $M$ is less than 1, then the iteration will converge. It doesn’t matter if there are other norms which give a value greater than 1, all that matters is that there is one norm that is less than 1.

Key Point 13 above makes no reference to the starting “guess" ${X}^{\left(0\right)}$ . The convergence of the iteration is independent of where you start! (Of course, if we start with a really bad initial guess then we can expect to need lots of iterations.)

Show that the Jacobi iteration used to approximate the solution of

$\phantom{\rule{2em}{0ex}}\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -5\hfill & \hfill -2\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]\left[\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \\ \hfill {x}_{3}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 3\hfill \end{array}\right]$

is certain to converge. (Hint: calculate the norm of $-{D}^{-1}\left(L+U\right)$ .)

The Jacobi iteration matrix is

$\begin{array}{rcll}-{D}^{-1}\left(L+U\right)& =& {\left[\begin{array}{ccc}\hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]}^{-1}\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0.25\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -0.2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.5\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]& \text{}\\ & =& \left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0.25\hfill & \hfill 0.25\hfill \\ \hfill -0.2\hfill & \hfill 0\hfill & \hfill 0.4\hfill \\ \hfill 0.5\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]& \text{}\end{array}$

and the infinity norm of this matrix is the maximum of $0.25+0.25$ , $0.2+0.4$ and $0.5$ , that is

$\phantom{\rule{2em}{0ex}}∥-{D}^{-1}\left(L+U\right){∥}_{\infty }=0.6$

which is less than 1 and therefore the iteration will converge.

#### 2.1 Guaranteed convergence

If the matrix has the property that it is strictly diagonally dominant , which means that the diagonal entry is larger in magnitude than the absolute sum of the other entries on that row, then both Jacobi and Gauss-Seidel are guaranteed to converge. The reason for this is that if $A$ is strictly diagonally dominant then the iteration matrix $M$ will have an infinity norm that is less than 1.

A small system is the subject of Example 20 below. A large system with slow convergence is the subject of Engineering Example 1 on page 62.

##### Example 20

Show that $A=\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -5\hfill & \hfill -2\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$ is strictly diagonally dominant.

##### Solution

Looking at the diagonal entry of each row in turn we see that

$\begin{array}{rcll}4& >& |-1|+|-1|=2\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}& \text{}\\ \left|-5\right|& >& 1+\left|-2\right|=3& \text{}\\ 2& >& \left|-1\right|+0=1& \text{}\end{array}$

and this means that the matrix is strictly diagonally dominant.

Given that $A$ above is strictly diagonally dominant it is certain that both Jacobi and Gauss-Seidel will converge.

#### 2.2 What’s so special about strict diagonal dominance?

In many applications we can be certain that the coefficient matrix $A$ will be strictly diagonally dominant. We will see examples of this in HELM booklet  32 and HELM booklet  33 when we consider approximating solutions of differential equations.

##### Exercises
1. Consider the system

$\phantom{\rule{2em}{0ex}}\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right]\left[\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 2\hfill \\ \hfill -5\hfill \end{array}\right]$

1. Use the starting guess ${X}^{\left(0\right)}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$ in an implementation of the Jacobi method to show that ${X}^{\left(1\right)}=\left[\begin{array}{c}\hfill 1.5\hfill \\ \hfill -3\hfill \end{array}\right]$ . Find ${X}^{\left(2\right)}$ and ${X}^{\left(3\right)}$ .
2. Use the starting guess ${X}^{\left(0\right)}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$ in an implementation of the Gauss-Seidel method to show that ${X}^{\left(1\right)}=\left[\begin{array}{c}\hfill 1.5\hfill \\ \hfill -3.25\hfill \end{array}\right]$ . Find ${X}^{\left(2\right)}$ and ${X}^{\left(3\right)}$ .

(Hint: it might help you to know that the exact solution is $\left[\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 3\hfill \\ \hfill -4\hfill \end{array}\right]$ .)

1. Show that the Jacobi iteration applied to the system

$\phantom{\rule{2em}{0ex}}\left[\begin{array}{cccc}\hfill 5\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 5\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 5\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 5\hfill \end{array}\right]\left[\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \\ \hfill {x}_{3}\hfill \\ \hfill {x}_{4}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 7\hfill \\ \hfill -10\hfill \\ \hfill -6\hfill \\ \hfill 16\hfill \end{array}\right]$

can be written

$\phantom{\rule{2em}{0ex}}{X}^{\left(k+1\right)}=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \end{array}\right]{X}^{\left(k\right)}+\left[\begin{array}{c}\hfill 1.4\hfill \\ \hfill -2\hfill \\ \hfill -1.2\hfill \\ \hfill 3.2\hfill \end{array}\right].$

2. Show that the method is certain to converge and calculate the first three iterations using

zero starting values.

(Hint: the exact solution to the stated problem is $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \\ \hfill 3\hfill \end{array}\right]$ .)

1. $\phantom{\rule{2em}{0ex}}2{x}_{1}^{\left(1\right)}=2-1{x}_{2}^{\left(0\right)}=2$

and therefore ${x}_{1}^{\left(1\right)}=1.5$

$\phantom{\rule{2em}{0ex}}2{x}_{2}^{\left(1\right)}=-5-1{x}_{1}^{\left(0\right)}=-6$

which implies that ${x}_{2}^{\left(1\right)}=-3$ . These two values give the required entries in ${X}^{\left(1\right)}$ . A second and third iteration follow in a similar way to give

$\phantom{\rule{2em}{0ex}}{X}^{\left(2\right)}=\left[\begin{array}{c}\hfill 2.5\\ \hfill -3.25\end{array}\right]\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{X}^{\left(3\right)}=\left[\begin{array}{c}\hfill 2.625\\ \hfill -3.75\end{array}\right]$

2. $\phantom{\rule{2em}{0ex}}2{x}_{1}^{\left(1\right)}=2-1{x}_{2}^{\left(0\right)}=3$

and therefore ${x}_{1}^{\left(1\right)}=1.5$ . This new approximation to ${x}_{1}$ is used straight away when finding a new approximation to ${x}_{2}^{\left(1\right)}$ .

$\phantom{\rule{2em}{0ex}}2{x}_{2}^{\left(1\right)}=-5-1{x}_{1}^{\left(1\right)}=-6.5$

which implies that ${x}_{2}^{\left(1\right)}=-3.25$ . These two values give the required entries in ${X}^{\left(1\right)}$ . A second and third iteration follow in a similar way to give

$\phantom{\rule{2em}{0ex}}{X}^{\left(2\right)}=\left[\begin{array}{c}\hfill 2.625\\ \hfill -3.8125\end{array}\right]\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{X}^{\left(3\right)}=\left[\begin{array}{c}\hfill 2.906250\\ \hfill -3.953125\end{array}\right]$

where ${X}^{\left(3\right)}$ is given to 6 decimal places

1. In this case $D=\left[\begin{array}{cccc}\hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill \end{array}\right]$ and therefore ${D}^{-1}=\left[\begin{array}{cccc}\hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill \end{array}\right]$ .

So the iteration matrix $M\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{D}^{-1}\left[\begin{array}{cccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}\begin{array}{cccc}\hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \end{array}\phantom{\rule{0.3em}{0ex}}\right]$

and that the Jacobi iteration takes the form

$\phantom{\rule{2em}{0ex}}{X}^{\left(k+1\right)}=M{X}^{\left(k\right)}+{M}^{-1}\left[\begin{array}{c}\hfill 7\hfill \\ \hfill -10\hfill \\ \hfill -6\hfill \\ \hfill 16\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \end{array}\right]{X}^{\left(k\right)}+\left[\begin{array}{c}\hfill 1.4\hfill \\ \hfill -2\hfill \\ \hfill -1.2\hfill \\ \hfill 3.2\hfill \end{array}\right]$

as required.

2. Using the starting values ${x}_{1}^{\left(0\right)}={x}_{2}^{\left(0\right)}={x}_{3}^{\left(0\right)}={x}_{4}^{\left(0\right)}=0$ , the first iteration of the Jacobi method gives $\begin{array}{rcll}{x}_{1}^{1}& =& 0.2{x}_{2}^{0}+1.4=1.4& \text{}\\ {x}_{2}^{1}& =& 0.2\left({x}_{1}^{0}+{x}_{3}^{0}\right)-2=-2& \text{}\\ {x}_{3}^{1}& =& 0.2\left({x}_{2}^{0}+{x}_{4}^{0}\right)-1.2=-1.2\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}& \text{}\\ {x}_{4}^{1}& =& 0.2{x}_{3}^{0}+3.2=3.2& \text{}\end{array}$

The second iteration is

$\begin{array}{rcll}{x}_{1}^{2}& =& 0.2{x}_{2}^{1}+1.4=1& \text{}\\ {x}_{2}^{2}& =& 0.2\left({x}_{1}^{1}+{x}_{3}^{1}\right)-2=-1.96& \text{}\\ {x}_{3}^{2}& =& 0.2\left({x}_{2}^{1}+{x}_{4}^{1}\right)-1.2=-0.96\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}& \text{}\\ {x}_{4}^{2}& =& 0.2{x}_{3}^{1}+3.2=2.96& \text{}\end{array}$

And the third iteration is

$\begin{array}{rcll}{x}_{1}^{3}& =& 0.2{x}_{2}^{2}+1.4=1.008& \text{}\\ {x}_{2}^{3}& =& 0.2\left({x}_{1}^{2}+{x}_{3}^{2}\right)-2=-1.992\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}& \text{}\\ {x}_{3}^{3}& =& 0.2\left({x}_{2}^{2}+{x}_{4}^{2}\right)-1.2=-1& \text{}\\ {x}_{4}^{3}& =& 0.2{x}_{3}^{2}+3.2=3.008& \text{}\end{array}$


# cumtrapz

Cumulative trapezoidal numerical integration

## Syntax

``Q = cumtrapz(Y)``
``Q = cumtrapz(X,Y)``
``Q = cumtrapz(___,dim)``

## Description

example

````Q = cumtrapz(Y)` computes the approximate cumulative integral of `Y` via the trapezoidal method with unit spacing. The size of `Y` determines the dimension to integrate along:If `Y` is a vector, then `cumtrapz(Y)` is the cumulative integral of `Y`.If `Y` is a matrix, then `cumtrapz(Y)` is the cumulative integral over each column.If `Y` is a multidimensional array, then `cumtrapz(Y)` integrates over the first dimension whose size does not equal 1.```

example

````Q = cumtrapz(X,Y)` integrates `Y` with respect to the coordinates or scalar spacing specified by `X`. If `X` is a vector of coordinates, then `length(X)` must be equal to the size of the first dimension of `Y` whose size does not equal 1.If `X` is a scalar spacing, then `cumtrapz(X,Y)` is equivalent to `X*cumtrapz(Y)`. ```

example

````Q = cumtrapz(___,dim)` integrates along the dimension `dim` using any of the previous syntaxes. You must specify `Y`, and optionally can specify `X`. If you specify `X`, then it can be a scalar or a vector with length equal to `size(Y,dim)`. For example, if `Y` is a matrix, then `cumtrapz(X,Y,2)` cumulatively integrates each row of `Y`.```

## Examples

collapse all

Calculate the cumulative integral of a vector where the spacing between data points is 1.

Create a numeric vector of data.

`Y = [1 4 9 16 25];`

`Y` contains function values for $\mathit{f}\left(\mathit{x}\right)={\mathit{x}}^{2}$ in the domain `[1 5]`.

Use `cumtrapz` to integrate the data with unit spacing.

`Q = cumtrapz(Y)`
```Q = 1×5 0 2.5000 9.0000 21.5000 42.0000 ```

This approximate integration yields a final value of 42. In this case, the exact answer is a little less, $41\frac{1}{3}$. The `cumtrapz` function overestimates the value of the integral because f(x) is concave up.

Calculate the cumulative integral of a vector where the spacing between data points is uniform, but not equal to 1.

Create a domain vector.

`X = 0:pi/5:pi;`

Calculate the sine of `X`.

`Y = sin(X');`

Cumulatively integrate `Y` using `cumtrapz`. When the spacing between points is constant, but not equal to 1, an alternative to creating a vector for `X` is to specify the scalar spacing value. In that case, `cumtrapz(pi/5,Y)` is the same as `pi/5*cumtrapz(Y)`.

`Q = cumtrapz(X,Y)`
```Q = 6×1 0 0.1847 0.6681 1.2657 1.7491 1.9338 ```

Cumulatively integrate the rows of a matrix where the data has a nonuniform spacing.

Create a vector of x-coordinates and a matrix of observations that take place at the irregular intervals. The rows of `Y` represent velocity data, taken at the times contained in `X`, for three different trials.

```X = [1 2.5 7 10]; Y = [5.2 7.7 9.6 13.2; 4.8 7.0 10.5 14.5; 4.9 6.5 10.2 13.8];```

Use `cumtrapz` to integrate each row independently and find the cumulative distance traveled in each trial. Since the data is not evaluated at constant intervals, specify `X` to indicate the spacing between the data points. Specify `dim = 2` since the data is in the rows of `Y`.

`Q1 = cumtrapz(X,Y,2)`
```Q1 = 3×4 0 9.6750 48.6000 82.8000 0 8.8500 48.2250 85.7250 0 8.5500 46.1250 82.1250 ```

The result is a matrix of the same size as `Y` with the cumulative integral of each row.

Perform nested integrations in the x and y directions. Plot the results to visualize the cumulative integral value in both directions.

Create a grid of values for the domain.

```x = -2:0.1:2; y = -2:0.2:2; [X,Y] = meshgrid(x,y);```

Calculate the function $\mathit{f}\left(\mathit{x},\mathit{y}\right)=10{\mathit{x}}^{2}+20{\mathit{y}}^{2}$ on the grid.

`F = 10*X.^2 + 20*Y.^2;`

`cumtrapz` integrates numeric data rather than functional expressions, so in general the underlying function does not need to be known to use `cumtrapz` on a matrix of data. In cases where the functional expression is known, you can instead use `integral`, `integral2`, or `integral3`.

Use `cumtrapz` to approximate the double integral

`$I\left(a,b\right)={\int }_{-2}^{b}{\int }_{-2}^{a}\left(10{x}^{2}+20{y}^{2}\right)\phantom{\rule{0.16666666666666666em}{0ex}}dx\phantom{\rule{0.16666666666666666em}{0ex}}dy.$`

To perform this double integration, use nested function calls to `cumtrapz`. The inner call first integrates the rows of data, then the outer call integrates the columns.

`I = cumtrapz(y,cumtrapz(x,F,2));`

Plot the surface representing the original function as well as the surface representing the cumulative integration. Each point on the surface of the cumulative integration gives an intermediate value of the double integral. The last value in `I` gives the overall approximation of the double integral, `I(end) = 642.4`. Mark this point in the plot with a red star.

```surf(X,Y,F,'EdgeColor','none') xlabel('X') ylabel('Y') hold on surf(X,Y,I,'FaceAlpha',0.5,'EdgeColor','none') plot3(X(end),Y(end),I(end),'r*') hold off```

## Input Arguments

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Numeric data, specified as a vector, matrix, or multidimensional array. By default, `cumtrapz` integrates along the first dimension of `Y` whose size does not equal 1.

Data Types: `single` | `double`
Complex Number Support: Yes

Point spacing, specified as `1` (default), a uniform scalar spacing, or a vector of coordinates.

• If `X` is a scalar, then it specifies a uniform spacing between the data points and `cumtrapz(X,Y)` is equivalent to `X*cumtrapz(Y)`.

• If `X` is a vector, then it specifies x-coordinates for the data points and `length(X)` must be the same as the size of the integration dimension in `Y`.

Data Types: `single` | `double`

Dimension to operate along, specified as a positive integer scalar. If you do not specify the dimension, then the default is the first array dimension of size greater than 1.

Consider a two-dimensional input array, `Y`:

• `cumtrapz(Y,1)` works on successive elements in the columns of `Y`.

• `cumtrapz(Y,2)` works on successive elements in the rows of `Y`.

If `dim` is greater than `ndims(Y)`, then `cumtrapz` returns an array of zeros of the same size as `Y`.

## Tips

• Use `trapz` and `cumtrapz` to perform numerical integrations on discrete data sets. Use `integral`, `integral2`, or `integral3` instead if a functional expression for the data is available.

• `trapz` reduces the size of the dimension it operates on to 1, and returns only the final integration value. `cumtrapz` also returns the intermediate integration values, preserving the size of the dimension it operates on.

## Version History

Introduced before R2006a