Math 142 Lecture Notes Section 1.3 – Linear Regression© Drost 8/9/2007 Page 1 of 5 Section 1-3 Math 142 Lecture Notes Section 1.3 – Linear Regression Mathematical Modeling: Definitions: Using mathematics to solve real-world problems. Three Steps: 1) Construct the mathematical model. 2) Solve the mathematical model. 3) Interpret the solution to the mathematical model. Slope as a Rate of Change When the variables x and y are related by the equation bmxy+=, then they have a linear relationship and the slope is xinchangeyinchangexxyym. =−−=1212 When looking at application problems this is called the rate of change of y with respect tox. Examples of rate of change 1) miles per hour = hoursofnumbermilesofnumber 2) revolutions per minute 3) price per pound 4) miles per gallon When the relationship is not linear, the ratio is called the average rate of change of y with respect to x. Example 1: Appropriate doses of medicine for both animals and humans are often based on body surface area, BSA. Since weight is much easier to determine than BSA, veterinarians use the weight of an animal to estimate BSA. The following linear equation expresses BSA for canines in terms of weight, , where is the BSA in square inches, and is the weight in pounds. Source: 6.37512.16 += waa wCalculus for Business, Economics, Life Sciences and Social Sciences by Barnett, Ziegler, Byleen, 11th edition, pg 30 Questions: 1. What is the slope of the linear equation? 2. Interpret. 3. What is the effect of a 1 pound increase in weight?© Drost 8/9/2007 Page 2 of 5 Section 1-3 Example 2: A 100-pound cargo of delicate electronic equipment is dropped from an altitude of 2,880 feet and lands 200 seconds later. Source: Calculus for Business, Economics, Life Sciences and Social Sciences by Barnett, Ziegler, Byleen, 11th edition, pg 30 Questions: 1. Find a linear model relating altitude a (measured in feet) and time in the air (measured in seconds). t 2. How fast is the cargo moving when it lands? Linear Regression Regression Analysis is the process of finding a function that fits the mathematical data. 1) also referred to as curve fitting 2) using a graphing calculator to find the best fit Example 3: Diamond Prices Prices for round-shaped diamonds taken from an online trader are given in Table 1 (shown below). Source: Calculus for Business, Economics, Life Sciences and Social Sciences by Barnett, Ziegler, Byleen, 11th edition, pg 31 Weight (carats) Price0.5 $2,7900.6 $3,1910.7 $3,6940.8 $4,1540.9 $5,0181 $5,898 A linear model for the data is given by 480140,6−= cp where p is the price of a diamond weighing carats. c Questions: 1. Create a scatter plot of the data and graph the model. 2. Interpret the slope of the model. 3. Use the graph to estimate the price of a diamond weighing 0.85 carats. 4. Use the graph to estimate the weight of a diamond whose price is $4,000.© Drost 8/9/2007 Page 3 of 5 Section 1-3 Linear Regression on your Calculator: Using the TI-84 or TI-83 to find a linear regression Source: Jenn Whitfield's TI-83 Tutorial for Business Calculus http://www.math.tamu.edu/~jwhitfld/TI_83/TI83_tutorial.html#top Regression (finding a best fitting model) 1. Activating the R2 o Press to access the catalog. o Arrow down to DiagnosticOn and press . This will take you to the home screen. o Press and the calculator will respond with done. 2. Press and clear all functions. 3. Follow the instructions for creating scatterplots to display the data. (See pg 4 of notes) 4. Press and to access the CALC menu. Select the number that corresponds with the type of regression needed. Once you have chosen the type of regression needed, your calculator will automatically go to the home screen. 5. Press and to Y-VARS. 6. Select 1:Function. Then select 1:Y1 and press . The regression information, along with the R2 value, will appear on the screen. 7. Press and the regression equation appears in Y1. 8. Press to see the data points and regression model displayed. You may need to adjust your window to see the graph. 9. To find other regression models for the same data, go back to step 5 and repeat the process.© Drost 8/9/2007 Page 4 of 5 Section 1-3 CREATING A SCATTERPLOT 1. Clear all data from lists by pressing and choosing 4:ClrList. This will take you back to the home screen. 2. Now press . Your screen should like the following. If it does then press and the calculator should respond with Done. 3. Press then select 1:Edit... 4. Input the x values into L1 and the y values into L2. 5. Press to access the scatterplot menu. Select 1:Plot 1... Make the screen look like the folllowing 6. Press and adjust your window so all data points can be seen on the screen. Note: The instructions above are for all types of regression formulas, although in this lesson we are primarily looking at linear regressions, or straight lines. Example 4: The table below shows the annual average price, median price and units sold for both single-family homes and condos/townhomes in Laguna Beach, California. Source: http://orangecoastrealestate.com/annual/lb.html© Drost 8/9/2007 Page 5 of 5 Section 1-3 Define t as number of years since 1990, and enter the data, as instructed above in your calculator in lines 1, 2, 3, and 4. First select stat, edit, then enter data. The first column will have to be redefined. Year Average Price Median Price Number Sold 1998 $ 764,547 $ 549,000 519 1999 $ 836,438 $ 618,250 483 2000 $1,043,451 $ 725,000 494 2001 $1,155,027 $ 802,000 380 2002 $1,129,204 $ 850,000 446 2003 $1,464,398 $1,050,000 451 2004 $1,698,760 $1,322,500 418 2005 $1,934,760 $1,597,000 336 2006 $2,521,450 $1,732,500 270 Questions: 1. Find a linear model that relates the average price and the year sold, defining t as the number of years since 1990 and the average price as P. a. What does the slope represent? b. When does this model predict the average