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1 S.ID.6c: Use Residuals to Assess Fit of a Function S.ID.6c: Use Residuals to Assess Fit of a Function Summarize, represent, and interpret data on two categorical and quantitative variables 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. c. Informally assess the fit of a function by plotting and analyzing residuals. Overview of Lesson  activate prior knowledge  present vocabulary and/or big ideas associated with the lesson  connect assessment practices with curriculum  model an assessment problem and solution strategy  facilitate guided practice of student activity Optional: Provide or allow students to create additional problem sets  facilitate a summary and share out of student work Optional HW  Write the math assignment. Vocabulary A residual is the vertical distance between where a regression equation predicts a point will appear on a graph and the actual location of the point on the graph (scatterplot). If there is no difference between where a regression equation places a point and the actual position of the point, the residual is zero. A residual can also be understood as the difference in predicted and actual yvalues (dependent variable values) for a given value of x (the independent variable). A residual plot is a scatter plot that shows the residuals as points on a vertical axis (yaxis) above corresponding (paired) values of the independent variable on the horizontal axis (xaxis). Any pattern in a residual plot suggests that the regression equation is not appropriate for the data.residuals on the Graphing Calculator. A residual plot without a pattern and with a near equal distribution of points above and below the x axis suggests that the regression equation is a good fit for the data. Residuals are automatically stored in graphing calculators when regression equations are calculated. To view a residuals scatterplot in the graphing calculator, you must set the Y list variable to RESID, the use Zoom 9 to plot the residuals. REGENTS PROBLEMS
2 1. Which statistic would indicate that a linear function would not be a good fit to model a data set? a. c. b. d.
3 2. Use the data below to write the regression equation ( ) for the raw test score based on the hours tutored. Round all values to the nearest hundredth. Equation: Create a residual plot on the axes below, using the residual scores in the table above. Based on the residual plot, state whether the equation is a good fit for the data. Justify your answer.
4 3. The table below represents the residuals for a line of best fit. Plot these residuals on the set of axes below. Using the plot, assess the fit of the line for these residuals and justify your answer. 4. The residual plots from two different sets of bivariate data are graphed below. Explain, using evidence from graph A and graph B, which graph indicates that the model for the data is a good fit.
5 S.ID.6c: Use Residuals to Assess Fit of a Function Answer Section 1. ANS: C Strategy: Use knowledge of correlation coefficients and residual plots to determine which answer choice is not a good fit to model a data set. STEP 1. A correlation coefficient close to 1 or 1 indicates a good fit, so answer choices a and b can be eliminated. Both suggest a good fit. STEP 2. For a residual plot, there should be no observable pattern and a similar distribution of residuals above and below the xaxis. The residual plot in answer choice d shows a good fit, so answer choice d can be eliminated, leaving answer choice c as the correct answer. DIMS? Does it make sense? Yes. The clear pattern in answer choice c tells us that the linear function is not a good fit to model the data set. PTS: 2 REF: fall1303a1 NAT: S.ID.6c TOP: Correlation Coefficient and Residuals 2. ANS: Based on the residual plot, the equation is a good fit for the data because the residual values are scattered without a pattern and are fairly evenly distributed above and below the xaxis. Strategies: Use linear regression to find a regression equation that fits the first two columns of the table, then create a residuals plot using the first and third columns of the table to see if there is a pattern in the residuals. STEP 1. Input the data from the first two columns of the table into a graphing calculator.
6 STEP 2. Determine which regression strategy will best fit the data. The problem states that the regression equation should be in the form ( ), which means linear reression. The scatterplot produced by the graphing calculator also suggests linear regression. STEP 3. Execute the lionear regression strategy in the graphing calculator. Round all values to the nearest hundredth: STEP 4. Plot the residual values on the graph provided using data from the first and third columns of the table. The graph shows a near equal number of points above the line and below the line, and the graph shows no pattern. The regression equation appears to be a good fit. NOTE: The graphing calculator will also produce a residuals plot. DIMS: Ask the question, Does It Make Sense (DIMS)? Yes. The regression equation produces the same residuals as shown in the table. PTS: 4 REF: fall1314a1 NAT: S.ID.6c TOP: Correlation Coefficient and Residuals
7 3. ANS: The line is a poor fit because the residuals form a pattern. PTS: 2 REF: a1 NAT: S.ID.6c TOP: Correlation Coefficient and Residuals 4. ANS: Graph A is a good fit because it does not have a clear pattern, whereas Graph B does have a clear pattern.. PTS: 2 REF: AI NAT: S.ID.6b TOP: Correlation Coefficient and Residuals
8 START PART 1. PART 2. PART 3. PART 4. PART 5. PART 6. Standard Writing the Math Assignment Write your name, date, topic of lesson, and class on your paper. Copy the problem from the lesson and underline/highlight key words. State your understanding of what the problem is asking. Answer the problem. Explanation of strategy. Create a new problem that addresses the same mathematical understandings. State the answer to your new problem. Clearly label each of the six parts. Grading Rubric Each homework writing assignment is graded using a fourpoint rubric, as follows: Part 1. Statement of the problem. 1 point is awarded for correctly restating the 2 original problem. Part 2. Statement of what the problem is really asking. point is awarded for correctly identifying 1 2 what the problem is asking you to find or do. Part 3. Solution to the problem. 1 point is awarded for a correct solution to 2 the problem. Part 4. Written analysis of the mathematics Up to 1 point is awarded for an explanation of and solution strategy involved in the problem. the mathematics and solution strategy involved in the problem. Part 5. An alternative version of the problem. Up to 1 point is awarded for creating a new problem that involves similar mathematics and a similar solution strategy. Part 6. Solution to the alternative version of 1 the problem. point is awarded for correctly solving the 2 new problem that you have created. This assignment/activity is designed to incorporate elements of Polya s four step universal algorithm for problem solving with the idea that writing is thinking. Rationale for Assignment Each New York Regents Algebra I (Common Core) examination contains 13 open response problems. An analysis of the first three Algebra I examinations revealed that approximately 51% (20 out of 39) of these open response problems instructed students to: 1) describe; 2) state; 2) explain; 3) justify or otherwise write about their answers. It is theorized that students can benefit from explicit instruction and writing routines that are applicable to solving these problems.
9 EXEMPLAR OF A WRITING THE MATH ASSIGNMENT Student s Name Topic: Date: Class: Part 1. The Problem TOP Electronics is a small business with five employees. The mean (average) weekly salary for the five employees is $360. If the weekly salaries of four of the employees are $340, $340, $345, and $425, what is the salary of the fifth employee? Part 2. What is the problem asking? Find the salary of the fifth employee. Part 3. Answer The salary of the fifth employee is $350 per week. Part 4. Explanation of Strategy The arithmetic mean or average can be represented algebraically as: x1 x2... xn X n I put information from the problem into the formula. The problem says there are 5 employees, so n 5. The problem also gives the mean (average) salary and the salaries of 4 of the employees. These numbers can be substituted into the formula as follows: x x x x5 350 x Check: Part 5. A New Problem Joseph took five math exams this grading period and his average score on all of the exams is 88. He remembers that he received test scores of 78, 87, 94, and 96 on four of the examinations, but he has lost one examination and cannot remember what he scored on it. What was Joseph s score on the missing exam? Part 6. Answer to New Problem Joseph received a score of 85 on the missing examination. EVALUATION Name and Proper Heading Part 4. Explanation of strategy. 1 Point Part 1. Problem copied correctly? 1/2 Point Part 5. New problem. 1 Point Part 2. Understand the problem? 1/2 Point Part 6. Solution to new problem. 1/2 Point Part 3. Correct and complete answer? 1/2 Point TOTAL 4 Points
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