Unit 2.2 Linear Regression “If you’re not the lead dog the view’s the same.” -Coach Ivey What does this quote mean?

Unit 2.2 Linear Regression

Unit 2.2 Linear Regression Line of Best Fit A line that best approximates ALL the data in a scatterplot

Unit 2.2 Linear Regression Domain Range x values y values

Unit 2.2 Linear Regression Interpolation Extrapolation To predict y-values from given x-values To predict y-values outside the given set of bivariate data

Unit 2.2 Linear Regression Correlation Coefficient “r” A number between -1 & 1 used to indicate how close data is to a line of best fit -1 < r < 1

Unit 2.2 Linear Regression Strong Correlation r > 0.7 r < -0.7 Weak Correlation r < 0.3 r > -0.3

Unit 2.2 Linear Regression Steps to find Linear Regression of Data Points Step 1: Plot the data on a Coordinate Plane Step 2: Draw a “Line of Best Fit” through your data. Step 3: Use 2 of the best fitting data points to determine your slope Step 4: Use a 3rd different data point near your best fit line to determine y-intercept

Example 1 Find the equation of the linear regression line for Rachael’s scatterplot in Example 1 from Lesson 2-1. Round the slope and y-intercept to the nearest hundredth. The points are given below. (65, 102), (71, 133), (79, 144), (80, 161), (86, 191), (86, 207), (91, 235), (95, 237), (100, 243)

Unit 2.2 Linear Regression Step 1 Step 1: Plot the data on a Coordinate Plane *Make sure you use a proper x-y scale The more accurate your plot the more accurate your linear equation

Unit 2.2 Linear Regression Step 2 Step 2: Draw a “Line of Best Fit” through your data. You have to “eyeball” this and try to get the same # of data points on each side of your Line of Best Fit

Unit 2.2 Linear Regression Step 3 Step 3: Use 2 of the best fitting data points to determine your slope Slope = Rise = y = (y2 – y1) = m Run x (x2 – x1) Linear Equation is … y = mx + b where m = slope

Unit 2.2 Linear Regression Step 4 Now that you have slope … Step 4: Use a 3rd different data point near your best fit line to determine y-intercept y = mx + b if you know m all you need is one point to find b

CHECK YOUR UNDERSTANDING Find the equation of the linear regression line of the scatterplot defined by these points: (1, 56), (2, 45), (4, 20), (3, 30), and (5, 9). Round the slope and y-intercept to the nearest hundredth.

Example 2 Interpret the slope as a rate for Rachael’s linear regression line. Use the equation from Example 1.

CHECK YOUR UNDERSTANDING Approximately how many more water bottles will Rachael sell if the temperature increases 2 degrees?

EXAMPLE 3 Rachael is stocking her concession stand for a day in which the temperature is expected to reach 106 degrees Fahrenheit. How many water bottles should she pack?

CHECK YOUR UNDERSTANDING How many water bottles should Rachael pack if the temperature forecasted were 83 degrees? Is this an example of interpolation or extrapolation? Round to the nearest integer.

EXAMPLE 4 Find the correlation coefficient to the nearest hundredth for the linear regression for Rachael’s data. Interpret the correlation coefficient.

CHECK YOUR UNDERSTANDING Find the correlation coefficient to the thousandth for the linear regression for the data in Check Your Understanding for Example 1. Interpret the correlation coefficient.

EXTEND YOUR UNDERSTANDING Carlos entered data into his calculator and found a correlation coefficient of -0.28. Interpret this correlation coefficient.

Example 1 Find the equation of the linear regression line for Rachael’s scatterplot in Example 1 from Lesson 2-1. Round the slope and y-intercept to the nearest hundredth. The points are given below. (65, 102), (71, 133), (79, 144), (80, 161), (86, 191), (86, 207), (91, 235), (95, 237), (100, 243)

CHECK YOUR UNDERSTANDING Find the equation of the linear regression line of the scatterplot defined by these points: (1, 56), (2, 45), (4, 20), (3, 30), and (5, 9). Round the slope and y-intercept to the nearest hundredth.

Example 2 Interpret the slope as a rate for Rachael’s linear regression line. Use the equation from Example 1.

CHECK YOUR UNDERSTANDING Approximately how many more water bottles will Rachael sell if the temperature increases 2 degrees?

EXAMPLE 3 Rachael is stocking her concession stand for a day in which the temperature is expected to reach 106 degrees Fahrenheit. How many water bottles should she pack?

CHECK YOUR UNDERSTANDING How many water bottles should Rachael pack if the temperature forecasted were 83 degrees? Is this an example of interpolation or extrapolation? Round to the nearest integer.

EXAMPLE 4 Find the correlation coefficient to the nearest hundredth for the linear regression for Rachael’s data. Interpret the correlation coefficient.

CHECK YOUR UNDERSTANDING Find the correlation coefficient to the thousandth for the linear regression for the data in Check Your Understanding for Example 1. Interpret the correlation coefficient.

EXTEND YOUR UNDERSTANDING Carlos entered data into his calculator and found a correlation coefficient of -0.28. Interpret this correlation coefficient.