Correlation Coefficient And Linear Of Best Fit Hw Answer Key

(2 positive trends, 2 negative trends, and 2 no trends) The students will plot the scatter plots using. When these two criteria are met, actual and forecast data will be closely. whether the dependent variable, y, grows along with x, or y get smaller as x increases. Most of the time we want to find the line of best fit (or the line that best fits the data). The graph below shows the relationship between the If he uses the point (2, 8) and (5, 1. Tes has the largest selection of academic, education, teaching and support positions for the world's largest network of teachers and teaching professionals. Keep track of this as you solve for the inverse. Time allotted for this Lesson: 5 Hours. What is the slope of the regression equation? What does it mean?. If the data is curved, a line would not be the best equation to use. For each of the following, follow checklist to create graph, list the independent variable and the dependent variable, make a prediction based on the line you drew, and explain the relationship between the two variables (what conclusion can you make from the graph?). If you classified the fruit in a basket as apple, orange, or banana, this would be an example of which level of measurement?. If they do, find the equation of a trend line. (Round your answers to the nearest dollar. scatterplots-hw2-13-15. It should be evident from this observation that there is definitely a connection between the sign of the correlation coefficient and the slope of the least squares line. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. 3 OF UNDERSTANDABLE STATISTICS) Chapter 10 of Understandable Statistics introduces linear regression. Problem 13. there is a correlation, identify the relationship as linear, quadratic, or exponential. When these two criteria are met, actual and forecast data will be closely. Choose your answers to the questions and click 'Next' to see the next set of questions. The trend line that shows the relationship between two sets of data most accurately is called the line of best fit. Round both coefficients to the nearest tenth. 5 36 40 Scatter Plot & Line of Best Fit Residual Plot Based on the shape of the scatter plot and the correlation coefficient, a linear model appears to be appropriate for the data. Linear regression and correlation coefficients. Summary and Reminders: Summary Slope-Intercept form of an equation: y = mx + b. Scatter Plots, Line of Best Fit, Linear Regression (2 days) Making Scatter Plots and Determining Line of Best Fit S. The equation that describes a line of best fit is called a. The following table shows the number of hours 10 students studied and their resulting test scores where x = the number of hours studied and y = test score. 97 and a function of: The Best Game. • Apply the slope formula. the answer is to the viscosity example in the book, we should use this as an example/tutorial. Round all coefficients to the nearest tenth. Use the equation of the line of fit to predict the height of the man. Write an equation for the line of best fit. Statistical Stats Homework, assignment and Project Help, Statistical Analysis Help Introduction Data Help for Students Thesis and Argumentation Data Help Details Analysis for Quantitative Research Statistics Solu. Use the number of photos taken as the independent variable. B) The linear model is appropriate. Linear regression equation: y 5 2. On homework and quizzes, use software (Minitab or CrunchIt) to compute the standard deviations. Chill and chat. A] Line of best fit: B] Describe the correlation:. When would a correlation coefficient of less than 0. 5) from his line, which. 9 generally indicate strong linear relationships. For this reason, it is usually drawn before a regression analysis is carried out. The y-intercept is the coefficient labeled CONSTANT, and the slope is the coefficient of the explanatory variable. Investigations 1, 2, and 4. There is no video for this lesson. of the line of best fit is greater than 1. 5 Quiz Review A. 9 Distinguish between correlation and causation. and then to calculate the. Perform the linear regression using the data in the lists. Linear Regression Gives us an equation that can be used to predict one variable from knowledge of another. Common Core Algebra I. According to the data, what will the estimated gross revenue be in 2015? In each situation, tell whether a correlation is likely. Speeding Up Warehouse Physical Design Using A Randomized. ALGEBRA 1 HW Line of best fit equations. Emphasize linear, quadratic, and exponential models. When data are not available, one method is to try two or three correlation coefficients (say 0. #2: _____ With these correlation coefficients, how confident do you feel about your predictions in 2c and 3c,d? Explain. If the slope is positive, the correlation coefficient is positive. With ML Kit's face detection API, you can detect faces in an image, identify key facial features, and get the contours of detected faces. A graphing calculator computes the equation of a line of best fit using a method called linear regression. Draw a line of fit for the scatter plot, and write the slope-intercept form of an equation for the line of fit. • Interpret slope and y-intercept of their models in the context of the problem. HW: complete review handout. 9x where y is the final-exam score and x is the score on the first test. • The average coefficient of kinetic friction value is significantly different in column C compared to columns A and B. Work on questions #9-18 (25 minutes), check the answers using the answer key 4: Make sure you have used a red pen (Or any different color pen) to grade your classwork. The correlation between X and Y is r = 0. Daily Math Mixed Review For Kindergarten. Need some extra help with Correlation Coefficient? Browse notes, questions, homework, exams and much more, covering Correlation Coefficient and many other concepts. Describe the correlation of a scatter plot. Determine whether each graph shows a positive, negative, or no correlation. Interpret the slope of the line of best fit in the context of the problem. A correlation coefficient quite close to 0, but either positive or negative, implies little or no relationship between the two variables. ) After calculating the line of best fit using technology,. 800 1000 40 30 20 200 600 400 Area e) Use your calculator to de ermine the equation of the line of best fit. linear interpolation 7. which will be on Wednesday!!!. Answer (4) has a tighter fit than answer (1). Common Core Algebra I. Do not solve the system at this time. The equation was j' = 10 + 0. Chapter 8: Linear Regression 9. Post your question here and our math experts will gladly provide math answers in the quickest way possible. See the discussion in the correlation tutorial to interpret this. Chapter 6 Test Answer Key download. The primary difference between correlation and regression is that Correlation is used to represent linear relationship between two variables. 800 1000 40 30 20 200 600 400 Area e) Use your calculator to de ermine the equation of the line of best fit. b) Determine the association (pos or neg) and describe it in a sentence. 8: Calculate Correlation Coefficients Interpret linear models 8. Find the line of best fit (Desmos) HW: Lesson 14 p. D a parallel correlation. about 67 yr. Research Question: Does the number of hours worked per week (workweek) predict family income (income)?. The table below lists temperatures measured in Fahrenheit and Celsius. 15 Create a line of best fit M33. Don't forget to rate and comment if you interest with this image. B a negative correlation. linear model is appropriate for the data. Answer three of the following (up to 5 pts. Correlation Coefficient Interpreting Linear Models (cont. Some of the worksheets for this concept are Multi part lesson 9 3 composite figures, Part b main idea find areas of composite shapes, Name date period lesson 6 homework practice, Math by design lesson plan area of a composite figure. Interpret the significance of the correlation coefficient. Use the number of photos taken as the independent variable. students interpret the parameters of a linear model in the context of data that it represents. Coefficient of Correlation The coefficient of correlation is a number between -1 and 1 which describes how well the equation 'fits' the data. Josie and some of her friends rode motorcycles all day on Saturday. correlation coefficient 6. Type the , (button above the 7) 5. Bing visitors came to this page today by typing in these algebra terms: 5th grade algebra worksheets with answer key ; online solver direct variation. 9 Distinguish between correlation and causation. The correlation coefficient is a number between —1 and + 1 including —1 and +1. Push STAT-EDIT-enter all (x, y) values into table (X in L 1, Y in L 2) 2. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. Regents Exam Questions S. 14 shows a plot of simulated experimental data. The equation that describes the line of best fit is called the linear regression equation. West Virginia CCR Algebra I Teacher Resource Walch Education v Table of Contents PROGRAM OVERVIEW Unit 1: Relationships Between Quantities and Reasoning with Equations. This function is called “Linear Regression” and will tell you the slope (m), the y-intercept (b), and the correlation coefficient (r). MI Alg I Sept 2012. First is the question of correlation, specifically, the degree of correlation between CO2 and warming comapred to other factors. 994 in this case. answers pdfFREE PDF. Interpret the significance of the correlation coefficient. Typically, you have a set of data whose scatter plot appears to "fit" a straight line. The variable r is used to represent the correlation coefficient. Find the equation of the best-fit line. Note that while we said that it doesn’t really matter which exponential we divide out by doing it the way we did here we’ll avoid a negative coefficient on the \(x\). Algebra 1 Notes SOL A. 9964 x 2 4 6 8 10 12 y 12 16 22. Create a plot with both. The Pearson correlation coefficient of x and y is the same, whether you compute pearson(x, y) or pearson(y, x). Wednesday, 11/1. Students explore how well data is fit with a linear model through the correlation coefficient. -The correlation coefficient will be a number between -1 and 1. 994366 There is a strong, positive correlation between age and height. Therefore, the least-squares line is the line of best fit. Solve your problem in the space below and on the attached graph and chart, as needed. Perfect Correlation - When variables are functionally dependent. (r = 0 ) Correlation Coefficient (r) - Used to measure how well the data matches a line that would best fit the data. A gas station owner has 30 gallons of gasoline worth $1. If there is a positive or negative correlation, describe its meaning in the situation. Shamrockin Equations Worksheet Answers Key – Algebra is significantly more intriguing when items are more real. Example 13. Input the data into the NCTM applet or into L 1 and L 2 on your graphing calculator. Plan for Monday, October 15th: If there is no sub, please work on the Unit 2 practice posted below. What does this correlation coefficient tell you about the fit of your regression equation? Based on that residual plot, is the linear regression a good fit? Looking at your scatter plot, can you anticipate which model will work best? Encourage students to use appropriate mathematical terminology when responding to questions like this. Regression Analysis Questions And Answers Pdf Understand how MLR can answer substantive questions within the field of with regression analysis, we can at least partially model that complexity to gain. AP Statistics: Linear Regression Review – Answer Key 4. (c) A passing average is defined as a 65% or above. Which graph represents data used in a linear regression that produces a correlation coefficient the line of best fit. He created a scatter plot and drew a line of best fit. b) Write an equation that models the interval as a function of an eruption’s duration. The fit is not quite as tight for the women's times. Tue Mar 5 5-j Equation of Line of Best Fit (Day 2) Wed Mar 6 Functions SBAC Interim Assessment Block answer key 5-j Day 2 Answer Key Thu Mar 7 y=mx+b video and Pool Draining Problem Fri Mar 8 no school PD Mon Mar 11 Unit 5 Study Guide Answer Key Tue Mar 12 Unit 5 Test Wed Mar 13 6-a Area and Volume Review of previous year topics. Site pages. Summary and Reminders: Summary Slope-Intercept form of an equation: y = mx + b. The correlation between X and Y is r = 0. 63 (1b) Looking at the graph of the scatter plot and the line, describe the correlation. Collaborative Statistics, authored by Barbara Illowsky and Susan Dean of De Anza College, and available through the Connexions and Orange Grove Texts Plus open source content repositories, is now supported by homework questions and student learning resources within WebAssign. slope-intercept 9. Chapter 8: Linear Regression 9. What is the height. Correlation!Coefficient!&Linear!of!Best!Fit!HW! Name:!!_____! 8. West Virginia CCR Algebra I Teacher Resource Walch Education v Table of Contents PROGRAM OVERVIEW Unit 1: Relationships Between Quantities and Reasoning with Equations. Type 2nd, 1 (to get the L 1 list) 4. It allows the statistician to eyeball the variables and form a working hypothesis about their relationship. Utah State Office of Education. Which of the following residual plots indicate a good fit fora linear model?. 88595 suggests that for a 1 percentage point increase in the growth rate. State the Correlation Coefficient. Connecting Scatter Plots and Correlation Coefficients Activity. students interpret the parameters of a linear model in the context of data that it represents. MI Alg I Sept 2012. Predictthe!type!(positive,!negative,!no)!and!strength!of!correlation!(strong,!weak)!for!the!following!. Find the equation of the best-fit line. A linear re ressi 0. -The correlation coefficient will be a number between -1 and 1. Perfect Correlation - When variables are functionally dependent. and Japanese stocks. a) Calculate the correlation coefficient. The line of best fit is an. -Using Correlation Coefficient to assess line of best fit. answers pdfFREE PDF. Chapter 9 - Regression and Correlation Answer Key CK-12 Advanced Probability and Statistics Concepts 4 4. Energy Consumption – Answer Key Reproducible #11 – Solar Power and Me: The Financial Savings Reproducible #12 – Solar Power and Me: The Financial Savings – Answer Key Reproducible #13 – Exit Ticket Question* (Note: this reproducible has two copies on a single. Please see the Linear Regression Practice Answer Key for a model of appropriate answers. A high correlation between x and y proves that x causes y. STATS DOESN’T SUCK! ~ CHAPTER 16 1 WWW. Approximately half of the data points should be below the line and half of the points above the line. 05, which indicates that the correlation coefficients are significant. They identify outliers and explain the correlation. It starts by having. Chill and chat. Practice 2-17. 310, which is the same value as we found from the correlation procedure. The linear model is a moderate/reasonable fit to the data. Approximately half of the data points should be below the line and half of the points above the line. eyeball the line of best fit, use visual approximation to write an equation of the line, use the median-median line method to write the equation of the line, and then use technology to find the equation of the line using the least squares regression method. Find the equation of the best-fit line. For this reason, it is usually drawn before a regression analysis is carried out. The scatter plot shows the relationship between the number of chapters and the total number of pages for several books. Correlation coefficients Math · AP®︎ Statistics · Exploring bivariate numerical data · Making and describing scatterplots Describing scatterplots (form, direction, strength, outliers). 5 Alaska $32. The nearer r is to , the more closely the data cluster around the line of best fit. Choose the BEST estimate for r, the correlation coefficient. Test for the significance of the correlation at the. r² is the coefficient of determination, and represents the percentage of variation in data that is explained by the linear regression. correlation coefficient, for a linear fit to the nearest thousandth. A commonly used linear relationship is a correlation, when it is multiplied by the slope coefficient of 207. Enter the temperature column under L1, and the viscosity column under L2, by typing 160 ENTER, 170 ENTER, etc. to clear out -60 mph time in L2. Step 9: Distribute the Cultivating Data Worksheets: Scatterplots, Outliers, and Lines of Best Fit printable to students to complete over 1–3 days. These were ineffective, and later work showed that causality runs in the opposite direction. Key DATE: PERIOD: Algebra 2/ Trig — Correlation Coefficient and Linear Regression HOMEWORK 1. A gas station owner has 30 gallons of gasoline worth $1. Which graph below shows the best model for the data and will create the best prediction?. Therefore, the correlation is negative. Use the Linear Regression feature to find the equation for the line of best fit using the lists of data you entered. When data are not available, one method is to try two or three correlation coefficients (say 0. Show all your steps. there is a correlation, identify the relationship as linear, quadratic, or exponential. Another feature of the least squares line concerns a point that it passes through. CHAPTER 10 REGRESSION AND CORRELATION SIMPLE LINEAR REGRESSION: TWO VARIABLES (SECTIONS 10. Understanding and Interpreting the Correlation Coefficient. The scatter plot shows the relationship between the number of chapters and the total number of pages for several books. Negative correlation represents a significant relationship between the variables x and y , which, depending on what they are modeling, can be understood as input and. Round to two decimal places for the line of best fit and four decimal places for the correlation coefficient. Modeling With Quadratic Functions Common Core Algebra 2 Homework Answers. Algebra 1 Statistics Unit Materials. A line of best fit is the line that comes closest to all of the points in the data set. Works for Math, Science, History, English, and more. Common Core Algebra II. What if data is not linear but is more like another curve? We analyze the data to determine which curve it is most like and use a curve of best fit. Curve Fitting with Linear Models Sketch the line of best fit for each scatter plot. [email protected] No homework, again!. Practice 2-15. Use our Calculator. 5 Alaska $32. This statistic numerically describes how strong the straight-line or linear relationship is between the two variables and the direction, positive or negative. Linear regression is a method for finding the least-squares line. When given summary statistics (the mean and standard deviation of the x-values, the mean and standard deviation of the y-values, and the correlation), how do you find the slope of the regression line?. We promise to do our best to help you solve your statistics and probability problems, just let us know your problem and you will get the best statistics and probability answers ever!. Keep track of this as you solve for the inverse. Explain Your Reasoning B) A) B) Give An Example Of Two Variables That You Believe Would Be Negatively Correlated. In this tutorial I demonstrate the following tasks in Stat Crunch: 1) Calculation of correlation coefficient 2) Graphing a scatter plot and plotting line of best fit 3) Simple Linear Regression 4. Lesson 6 Homework Practice Area Of Composite Figur. Investigations 1, 2, and 4. 5m long pat. Which graph below shows the best model for the data and will create the best prediction?. You don't have to believe everything it says. Engage in activities that allow you to collect and analyze data, determine a model of best fit, and evaluate the reasonableness of the model selected. Syllabus Biometry for Coastal Sciences Fall 2018 Class 01 - Introduction to data and statistical analysis Reading: Broman and Woo (2018) Data Organization in Spreadsheets The nature of statistical analysis Objectives of statistical analysis Description and inference Objects, variables, and scales Types of biometric data Variable classes Food for thought Lecture 01 Biometry Lecture 01 Why do…. LeARnInG GoALS Key TeRMS • linear regression • line of best fit • linear regression equation • significant digits • correlation coefficient 8043_Ch03. If there is a positive/negative linear correlation, find the equation of the. Free printable worksheets with answer keys on linear equations including finding slope, slope intercept form, equation from 2 points,from 1 point and slope and more. one variable increases with the other; Fig. 7 – Curve Fitting with Linear Models Per: ___ The problems on this worksheet come from your book, p. AP Statistics - Chapter 2A Warm-Ups 2. 0 indicates that the graph is a perfect line with a positive slope. This indicates a strong negative association. Shed the societal and cultural narratives holding you back and let free step-by-step SpringBoard Algebra 1 textbook solutions reorient your old paradigms. a) Calculate the correlation coefficient. best approximates the points. Actually, we also have been realized that 19 Scatter Plots And Lines Of Best Fit Worksheet is being one of the most popular field on the subject of document template example at this moment. is called correlation. (a) (b) (c) (d) (d) How does the correlation coefficient quantify the fit of a positive correlation? Exercise #3: The following data set is that of two variables that have a negative correlation. Use the Linear Regression feature to find the equation for the line of best fit using the lists of data you entered. In these results, the p-values for the correlation between porosity and hydrogen and between strength and hydrogen are both less than the significance level of 0. Homework: Linear Regression worksheet Thursday and Friday -Unit 4 Assessment Objective: Students will show mastery of Linear functions. B a negative correlation. Use Y= to enter the equation into the calculator and graph the best-fit line with the scatter plot. Construct a scatter plot using excel for the given data. • Find the best fit line of the given data reported in the tables and practice locating r (correlation coefficient) Key Phrases: slope, y-intercept, zeros, function notation, x-intercept, point-slope, linear function, causation and association, discrete, continuous, slope-intercept, domain and range, correlation coefficient •. 146 5 (by Hand) , 6 and 15 on calculator ­­16, 19­23, 25­31 = 16 problems b. Identify the correlation and find the equation of the line of best fit. To find the most accurate best-fit line you have to use the process of linear regression. Problem 1 Problem 2. Like correlation coefficients, linear regression analyzes the relationship between two variables, x and y. 3 OF UNDERSTANDABLE STATISTICS) Chapter 10 of Understandable Statistics introduces linear regression. 45% per cm of soil depth. Pulling them out and simplifying what's left will always put you to the one simplified form, the first one. State if there appears to be a positive correlation, negative correlation, or no correlation. The mathematical formula for computing ris: where n is the number of pairs of data. correlation coefficient. Make predictions from the line of best fit Practice #1 Draw in a line of best fit on each scatterplot. Wednesday, 11/8. - The correlation coefficient formula will tell you how strong of a linear relationship there is between two variables. The correlation coefficient is a. The line of best fit is an. ] Answer: _____ Part B Use your line of best fit from Part A to predict, to the nearest tenth of a foot, how many feet the sloth will have moved after 3 minutes. You can skip questions if you would like and come. f) Interpret the slope of the line using context from the. Unit 5 Scatter Plots, Correlation, and Line of Best Fit Right after mid term exams we cover this mini-unit. Lesson 6 Homework Practice Area Of Composite Figur - Displaying top 8 worksheets found for this concept. 14-1: Function Notation and Rate of Change: Getting Ready: p. "Best-fit" straight line model Problem statement * You have a scatterplot that shows linear association bet. Energy Consumption – Answer Key Reproducible #11 – Solar Power and Me: The Financial Savings Reproducible #12 – Solar Power and Me: The Financial Savings – Answer Key Reproducible #13 – Exit Ticket Question* (Note: this reproducible has two copies on a single. State which model, linear or quadratic, best fits the data. whether the data have a positive correlation, a negative correlation, or relatively no correlation. The quadratic model for the manatee population will have a positive leading coefficient since it opens upward, but it is hard to predict what the y-intercept or the. These skills are extended to solving quadratic equations, exploring linear, quadratic, and exponential functions graphically, numerically, symbolically, and as sequences, and by using regression techniques to analyze the fit of models to distributions of data. a) Make a scatter plot of the data and draw a line of best fit. Coefficient of Correlation The coefficient of correlation is a number between -1 and 1 which describes how well the equation 'fits' the data. linear extrapolation 8. Correlation 1. Draw a line of fit for the scatter plot, and write the slope-intercept form of an equation for the line of fit. Answer Key Lesson 6. Mike owns two hardware stores. 14 (2) is the only graph that shows a positive correlation. One type of regression analysis is linear analysis. This section covers: Scatter Plots Correlation Regression Using Graphing Calculator to Get Line of Best Fit Usually around the time that you are beginning "Algebra II" you'll have another lesson on a little more advanced Statistics than you had earlier (in the Introduction to Statistics and Probability section). Scatter Plot. Write the equation for her line of best fit in slope-intercept form. April 12 We continued to work on manually finding the line of best fit and then using the calculator to find the linear regression and compare. [0, 6] scl: 1 by [0, 3] scl: 0. In both figures, the data points all lie exactly on a straight line; that is, we can predict perfectly the value of one variable from the other. Lesson 6 Homework Practice Area Of Composite Figur - Displaying top 8 worksheets found for this concept. 5 Correlation and Best-Fitting Lines 103 Vocabulary Check Concept Check Skill Check STUDENT HELP Extra Practice to help you master skills is on p. 9 HW 98 Line of best fit and residuals ANSWER KEY 9 HW 97 Correlation Coefficient ANSWER KEY Comments (-1) HW 96 Conditional Relative Frequency ANSWER KEY. 0 Explanation:-0. 8 Compute (using technology) and interpret the correlation coefficient of a linear fit. 215: 14-3: Evaluating Functions and Graphing Piecewise-Defined Linear Functions. Which graph represents data used in a linear regression that produces a correlation coefficient the line of best fit. Identify the correlation of each. AP Statistics: Linear Regression Review – For each of the data sets, perform a complete bivariate analysis by performing all of the following steps: a) Make a scatterplot on your calculator and sketch it on your paper. [email protected] o11c4-6_line_of_best_fit_notes_and_homework_answer_key. In this post, you'll learn face detection with Firebase ML Kit on Android.  For a correlation coefficient of zero, the points have no direction, the shape is almost round, and a line does not fit to the points on the graph. The linear correlation coefficient is also referred to as Pearson's product moment correlation coefficient in honor of Karl Pearson, who originally developed it. 05 level of significance. While the y intercept of a least squares line may not be interesting from a statistical standpoint, there is one point that is. Correlation definition is - the state or relation of being correlated; specifically : a relation existing between phenomena or things or between mathematical or statistical variables which tend to vary, be associated, or occur together in a way not expected on the basis of chance alone. If there is a positive or negative correlation, describe its meaning in the situation. Using these equations, students model linear phenomena, both with physics based and regression based equations. Find an inverse relationship, with annual rebalancing. In partnership with the. A Give An Example Of Two Variables That You Believe Would Be Positively Correlated. Computers are useful for more than just surfing the Internet. The mathematical formula for computing ris: where n is the number of pairs of data. 4% match to the data. no correlation 3. Key Terms Best-fitting line Approximating a Best-fitting Line The data in the graph show the forearm lengths and foot lengths of 18 students. Find the e ation for the line of best fit, as well as the correlation coefficient b a b. Identify the correlation of each. not similar 5. The linear correlation coefficient for this relationship is (3) 0. The trend line that gives the most accurate model of related data is the line of best fit. Practice Your Skills Student Workbook, which does not have answers. square the coefficient. However, after I fixed it, the residual plot does not show a clear pattern which matches up with the very strong correlation coefficient (r) from #11. Free printable worksheets with answer keys on linear equations including finding slope, slope intercept form, equation from 2 points,from 1 point and slope and more. Which linear equation best represents the growth of the tomato plant? 22. 9 HW 98 Line of best fit and residuals ANSWER KEY 9 HW 97 Correlation Coefficient ANSWER KEY Comments (-1) HW 96 Conditional Relative Frequency ANSWER KEY. Investigations 1, 2, and 4. It is a numerical.