Regression and Median-Fit Lines (4-6) Objective: Write equations of best-fit lines using linear regression. Write equations of median-fit lines.

Best-Fit Lines You have learned how to find and write equations for lines of fit by hand. Many calculators use complex algorithms that find a more precise line of fit called the best-fit line. One algorithm is called linear regression. Your calculator may also compute a number called the correlation coefficient. This number will tell you if your correlation is positive or negative and how closely the equation is modeling the data. The closer the correlation coefficient is to 1 or -1, the more closely the equation models the data.

Example 1 The table shows Ariana’s hourly earnings for the years 2001-2007. Use a graphing calculator to write an equation for the best-fit line for the data. Name the correlation coefficient. Round to the nearest ten-thousandth. Let x be the number of years since 2000. Year Cost 2001 $10 2002 $10.50 2003 $11 2004 $13 2005 $15 2006 $15.75 2007 $16.50

Example 1 Before you begin, you need to make sure the Diagnostic setting on your calculator is on. You can find this setting under the CATALOG menu by pressing 2nd 0 on your calculator. Alpha is set when you are in this menu. Press D by using the x-1 button and then scroll down until the arrow on the left is pointing to DiagnosticOn. Press ENTER twice.

Example 1 Enter the data in the table into your calculator by pressing STAT. Choose 1: Edit. If there is data in the calculator already, you will need to clear it out. Scroll up until L1 is highlighted and press CLEAR and the ENTER. Repeat this with L2. Enter the independent variable (x) under L1 and the dependent variable (y) under L2. Year Cost 2001 $10 2002 $10.50 2003 $11 2004 $13 2005 $15 2006 $15.75 2007 $16.50

Example 1 You may then calculate the regression line by pressing STAT again. Right arrow over so that the heading CALC is highlighted. Scroll down to 4: LinReg(ax+b) and press ENTER twice. The “a” value represents the slope. The “b” value represents the y-intercept. The “r” value represents the correlation coefficient. Use “a” and “b” to write the equation for the best-fit line. y = 1.21x + 8.25 r = 0.9801

Check Your Progress Choose the best answer for the following. The table shows the average body temperature in degrees Celsius of nine insects at a given temperature. Use a graphing calculator to write the equation for the best fit line for that data. Name the correlation coefficient. y = 0.85x + 1.28; 0.8182 y = 0.95x + 1.53; 0.9783 y = 1.53x + 0.95; 0.9873 y= 1.95x + 1.95; 0.8783 Temperature (˚C) Air 25.7 30.4 28.7 31.2 31.5 26.2 30.1 18.2 Body 27.0 28.9 31.0 25.6 28.4 31.7 18.7

Interpolation and Extrapolation We can use points on the best-fit line to estimate values that are not in the data. When we estimate values that are between known values, this is called linear interpolation. When we estimate a number outside of the range of the data, it is called linear extrapolation.

Example 2 The table below shows the points earned by the top ten bowlers in a tournament. Estimate how many points the 15th-ranked bowler earned. y = -7.9x + 201.2 y = -7.9(15) + 201.2 y = -118.5 + 201.2 y = 83 Rank 1 2 3 4 5 6 7 8 9 10 Score 210 197 164 158 151 147 144 142 134 132

Median-Fit Lines A second type of fit line that can be found using a graphing calculator is a median-fit line. The equation of a median-fit line is calculated using the medians of the coordinates of the data points.

Example 3 Find the equation of a median-fit line for the data on the bowling tournament in Example 2. Then predict the score of the 20th ranked bowler. Enter the data into the calculator the same way using the STAT button. After the data has been entered, go back to STAT and right arrow over so that CALC is highlighted. Choose 3: Med-Med. Once again, the “a” value is the slope and the “b” value is the y-intercept. y = -9x + 209.5 y = -9(20) + 209.5 y = -180 + 209.5 y = 30

Air Taxi to Kelley’s Island Check Your Progress Choose the best answer for the following. An air taxi keeps track of how many passengers it carries to various islands. The table shows the number of passengers who have traveled to Kelley’s Island in previous years. Use a regression line to determine how many passengers should the airline expect to go to Kelley’s Island in 2015? 1186 passengers 1702 passengers 1890 passengers 2186 passengers Air Taxi to Kelley’s Island Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 Number of Passengers 1020 1115 1247 1657 1324 1568 1987 1562 1468 1693 y = 68.7x + 1154.9 y = 68.7(15) + 1154.9 y = 1030.5 + 1154.9

Check Your Progress Choose the best answer for the following. Use the data from the table and a median-fit line to estimate the number of passengers the airline will have in 2015. 1100 passengers 1700 passengers 1900 passengers 2100 passengers y = 63.9x + 1142.5 y = 63.9(15) + 1142.5 y = 958.5 + 1142.5