1. 1.5 L INEAR M ODELS Objectives: 1. Algebraically fit a linear model. 2. Use a calculator to determine a linear model. 3. Find and interpret the correlation coefficient for a model. 4. Create and interpret a residual plot for a linear model.

2. E XAMPLE #1 L INEAR D ATA Estimated expenses for a company over the five-year period 1993-1997 are shown in the table below. Determine whether a line would be a good model for this data by drawing a scatter plot. Year 19931994199519961997 Expenses per share ($) 1.241.451.661.882.09 Letting x = 0 represent 1990 the scatter plot looks as follows: Because the points appear to follow a straight path, then a linear equation would be a good model for the data.

3. E XAMPLE #2 M ODELING D ATA The data below shows the monthly amount spent on marketing (in hundreds of dollars) x and the monthly sales revenue (in thousands of dollars) r of an online store over a seven-month period. Find two models for the data, each determined by a pair of data points. For one model, use points (1, 2) and (4, 4). For the other model, use (1, 2) and (5, 5). Then use residuals to see which model best fits the data. x0123456 r2233455

4. R ESIDUALS ARE A MEASURE OF THE ERROR BETWEEN THE ACTUAL VALUE OF THE DATA AND THE VALUE GIVEN BY THE MODEL. R ESIDUAL = O BSERVED Y – P REDICTED Y Model AModel B

5. Model A

6. S INCE THE SUM OF THE SQUARES OF THE RESIDUALS IS LESS ON M ODEL A THIS IMPLIES THAT M ODEL A IS A BETTER MODEL FOR THE DATA SET. Model B

7. C ORRELATION C OEFFICIENTS When finding the Least-Squares Regression Line, an r - value is given called the correlation coefficient. This is different than the r used when calculating residuals. The correlation coefficient is always given between -1 and 1 with positive 1 being a perfect correlation in the positive direction and - 1 being a perfect correlation in the negative direction. A value of 0 means now correlation at all.

8. E XAMPLE #3 M ODELING D ATA The lengths of the radii of the inscribed circles in regular polygons whose sides have length one unit are given as follows: A. Draw a scatter plot. B. Find the model that best fits the data using the regression feature on a calculator. C. What does the correlation coefficient indicate about the data?

9. L EAST -S QUARES R EGRESSION L INE – T HE LINE THAT MAKES THE SUM OF THE SQUARES OF THE VERTICAL DISTANCES OF THE DATA POINTS FROM THE LINE AS SMALL AS POSSIBLE. Scatter PlotEnter the data into the graphing calculator: STAT  EDIT  1:Edit Calculate the Linear Regression Line: STAT  CALC  4:LinReg(ax+b) The correlation coefficient r tells us that the line is a very good fit because it is very close to 1.

10. E XAMPLE #4 P REDICTION FROM A M ODEL The total number of farms (in millions) in selected years is shown in the table below. A. Use linear regression to find an equation that models the data. Use the equation to estimate the number of farms in 1975 and in 2005. B. According to the model, when will the number of farms be one million? C. Is a line the best model for the data? Year1900192019301940195019601970198519901995 Farms5.7406.5186.5466.3505.6483.9632.9492.2932.1462.196 This implies that in the year 2017 the number of farms will reach 1 million. Looking at the scatter plot and the r -value, a linear model doesn’t appear to be the best fit.