Lesson 4.5 Topic/ Objective: To use residuals to determine how well lines of fit model data. To use linear regression to find lines of best fit. To distinguish between correlation and causation.

Y = -2x + 20 models the data in the table and graph. Step 1 Calculate the residuals. Organize your results in a table. Step 2 Use the points (x, residual) to make a scatter plot. The points are evenly dispersed about the horizontal axis. So, the equation y = −2x + 20 is a good fit.

The table shows the high temperatures on Monday for eight weeks The table shows the high temperatures on Monday for eight weeks. The equation y = 7x − 8 models the data. Is the model a good fit? Residual 48 - -1 = 49 43 – 6 = 37 42 – 13 = 29 40 – 20 = 20 45 – 27 = 18 56 – 34 = 22 67 – 41 = 26 76 – 48 = 28 No, The residuals are all positive. The residual points form a V-shaped pattern, which suggests the data are not linear. So, the equation y = 7x − 8 is not a good fit.

Linear regression Finding Lines of Best Fit Graphing calculators use a method called linear regression to find a precise line of fit called a line of best fit. This line best models a set of data. A calculator often gives a value r, called the correlation coefficient. This value tells whether the correlation is positive or negative and how closely the equation models the data. Values of r range from −1 to 1. When r is close to 1 or −1, there is a strong correlation between the variables. As r, gets closer to 0, the correlation becomes weaker.

Predict the cost per pound in 2008 Interpolation: Using a graph, table or equation to approximate a value between two known values is called interpolation Extrapolation: Using a graph, table or equation to predict a value outside the range of known values is called extrapolation. Year Meat price per pound 2001 2.50 2003 2.70 2005 3.00 2007 3.30 2009 3.50 Predict the cost per pound in 2008 About $3.40 per pound using interpolation Predict the cost per pound in 2012 About $3.87 per pound using Extrapolation

Correlation and Causation: When a change in one variable causes a change in another variable, it is called causation. Causation produces a strong correlation between the two variables. Tell whether a correlation is likely in the situation. If so tell whether there is a causal relationship. a. time spent exercising and the number of calories burned There is a positive correlation and a causal relationship because the more time you spend exercising, the more calories you burn b. the number of banks and the population of a city There may be a positive correlation but no causal relationship. Building more banks will not cause the population to increase