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Common Core Algebra Unit 7 Extension: Linear Regression 12/21/16Lesson 3: Correlation Coefficient and Residual PlotsObjective: SWBAT determine the meaning and interpret the correlation coefficient for a set of data and assess the trend of fit using a residual plot.Do Now: Do you think the number of hours you study for a test affects the grade you get? Explain.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Group Task:The following data represents 10 students in a college level sociology class at a community college. Their absences and final course grades have been recorded in the following table. 3200400139700a) Create a scatter plot with the data below. b) Is there a correlation? If so, is it negative or positive? c) What conclusion can be made in regards to the data presented?Correlation CoefficientsOne measure of strength and direction of a correlation is the correlation coefficient, denoted by r. The value of r ranges from -1 to 1.Model Example:Determine the line of best fit and the correlation coefficient for the previous example.One-time process (or after the calculator has been reset) ?2nd 0 (CATALOG) ?Scroll down to DIAGNOSTICS ON ?ENTER, ENTER I. Enter the bivariate data into List 1 (L1) & List 2 (L2) STAT #1 (EDIT) List distance into L1 and time into L2 II. Creating the Scatter Plot 2nd y = (STAT PLOT) #1 ENTER Turn On and Choose Scatter Plot ZOOM #9 (ZOOM STAT) III. Determining the Linear Regression Equation STAT arrow over to CALC #4 (LinReg (ax + b)) Discovering the Correlation Coefficient:Below are six graphs with their respective correlation coefficient measure. For each graph, state whether the correlation is strong/weak positive, strong/weak negative, or no correlation. r = 0.4966r = 0.9355r = -0.9439r = 0.7745r = 0.07r = 1Turn and Talk:Based on the correlation and the value of the correlation coefficient, what conclusion can you make about the number and the correlation?Check for Understanding:Based on this scatter plot, it would be reasonable to conclude: (1) Age and value have a coefficient of correlation less than zero. (2) Age and value have a coefficient of correlation that is equal to zero. (3) Age and value have a coefficient of correlation that is between zero and 0.5. (4) Age and value have a coefficient of correlation that is greater than 0.5.Group Task: Complete part (a) only.The table below displays the number of hours spent studying, x, and the grade on a math test, y by 9 students.(a) Create a scatter plot of the data. 251460068580(b) Determine the equation of the line of best fit using your calculator. (c) Using the correlation coefficient, describe the correlation of the data. (d) Calculate the predicted y-values for each x-value using the equation of the line of best fit and enter it into the third column of the chart below (round to the nearest tenth). Draw the line of best fit on the graph above using these values.22860048260(e) Find the difference between the actual y-values and the y-values on the line of best fit.(f) Create a residual plot by graphing the relationship between x and the residual values. (x, residual)25400-11430000Analyzing The Line of Best Fit using a Residual PlotResidual plots are used to determine whether a regression model is a good fit for the bivariate data. The residuals should add up to 0. A sum of zero tells you that the regression line was properly calculated.Residuals can be used to determine if a linear regression model is a good fit for a data set.Residuals are error distances. A residual is the difference between the observed y-value (from the scatterplot) and the predicted y-value (from the line of best fit). A residual plot is a graph of points whose x-coordinates are the values of the independent variable and whose y-coordinates are the corresponding residuals.How To Find Residuals on the Calculator:254004445Check for Understanding:Identify if the linear regression line is a good predictor of the data based on the residual plot. State why. Lesson Summary:A scatter plot represents a set of data between ____________variables. The relationship between the two variables is known as the __________________________________. The strength of the relationship is represented by a number known as the _______________ __________________. This number ranges from -1 to 1.Problem Set:1. Which graph represents data used in a linear regression that produces a correlation coefficient closest to -1. 2. Consider the correlation coefficients below that represent various sets of data. List them in order from strongest correlation to weakest correlation. -0.72 0.92 0.05 -0.5 3. Use the scatter plots below to answer the questions.a) If one of these scatter plots represents the height and weight for eight adults, which plot do you think it is and why? b) If one of these scatter plots represents the relationship between height and SAT math score for eight high school seniors, which scatter plot do you think it is and why? 25400-146304004. The table below displays the number of hours spent studying, x, and the grade on a math test, y by 9 students. (a) Create a scatter plot of the data. (b) Determine the equation of the line of best fit using your calculator.(c) Using the correlation coefficient, describe the correlation of the data. 5. The local ice cream shop keeps a daily record of how much ice cream they sell and the noon temperature on that day. Displayed below is the data for the last 9 days. 3200400154940a) Create a scatter plot for the set of data. (b) Determine the equation of the line of best fit using your calculator. (c) Predict the ice cream sales when the weather is 60 degrees. 6. 3657600444500A real estate agent plans to compare the price of a cottage in a town on the seashore to the number of blocks the cottage is from the beach. The table shows a random sample of sales and location data. a) Write a linear regression equation that relates the price of a cottage (y) to its distance from the beach (x). Round a and b to the nearest whole. b) What is the correlation coefficient to the nearest thousandth? What does this number tell you about the data? 7. (a) Complete the table using the given linear regression equation (round to one decimal place). Construct a residual plot. y = 0.5x (b) Calculate the sum of the residuals. What does this sum tell you?(c) Based on the residual plot, is the linear regression line a good predictor of the data? Explain.8. The table shows the percent of the United States population who did not receive needed dental care services due to increasing costs. Let 1999 be year zero.(a) Create a scatter plot of the data. Let x represent the number of years since 1999.(b) Using the calculator, write the equation of the line of best fit. Round a and b to the nearest hundredth.(c) Using the line of best fit equation or your calculator, calculate the residuals for the set of data (round to one decimal place). Construct a residual plot for the data.Name ____________________________Exit TicketCommon Core Algebra Unit 7 Extension: Linear Regression 12/21/16Lesson 3: Correlation Coefficient and Residual PlotsObjective: SWBAT determine the meaning and interpret the correlation coefficient for a set of data and assess the trend of fit using a residual plot.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. ................
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