1 4 Curve Fitting with Linear Models Objectives

1 -4 Curve Fitting with Linear Models Objectives Fit scatter plot data using linear models with and without technology. Use linear models to make predictions. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Vocabulary regression correlation line of best fit correlation coefficient Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Researchers, such as anthropologists, are often interested in how two measurements are related. The statistical study of the relationship between variables is called regression. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models A scatter plot is helpful in understanding the form, direction, and strength of the relationship between two variables. Correlation is the strength and direction of the linear relationship between the two variables. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models If there is a strong linear relationship between two variables, a line of best fit, or a line that best fits the data, can be used to make predictions. Helpful Hint Try to have about the same number of points above and below the line of best fit. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Example 1: Meteorology Application Albany and Sydney are about the same distance from the equator. Make a scatter plot with Albany’s temperature as the independent variable. Name the type of correlation. Then sketch a line of best fit and find its equation. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Example 1 Continued Step 1 Plot the data points. Step 2 Identify the correlation. o Notice that the data set is negatively correlated–as the temperature rises in Albany, it falls in Sydney. • • • • o Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Example 1 Continued Step 3 Sketch a line of best fit. o Draw a line that splits the data evenly above and below. • • • • o Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Example 1 Continued Step 4 Identify two points on the line. For this data, you might select (35, 64) and (85, 41). Step 5 Find the slope of the line that models the data. Use the point-slope form. Point-slope form. y – y 1= m(x – x 1) y – 64 = – 0. 46(x – 35) y = – 0. 46 x + 80. 1 Substitute. Simplify. An equation that models the data is y = – 0. 46 x + 80. 1. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Check It Out! Example 1 Make a scatter plot for this set of data. Identify the correlation, sketch a line of best fit, and find its equation. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Check It Out! Example 1 Continued Step 1 Plot the data points. Step 2 Identify the correlation. Notice that the data set is positively correlated–as time increases, more points are scored • • • Holt Mc. Dougal Algebra 2 • •

1 -4 Curve Fitting with Linear Models Check It Out! Example 1 Continued Step 3 Sketch a line of best fit. Draw a line that splits the data evenly above and below. • • • Holt Mc. Dougal Algebra 2 • •

1 -4 Curve Fitting with Linear Models Check It Out! Example 1 Continued Step 4 Identify two points on the line. For this data, you might select (20, 10) and (40, 25). Step 5 Find the slope of the line that models the data. Use the point-slope form. y – y 1= m(x – x 1) y – 10 = 0. 75(x – 20) y = 0. 75 x – 5 Point-slope form. Substitute. Simplify. A possible answer is p = 0. 75 x + 5. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models The correlation coefficient r is a measure of how well the data set is fit by a model. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models You can use a graphing calculator to perform a linear regression and find the correlation coefficient r. To display the correlation coefficient r, you may have to turn on the diagnostic mode. To do this, press and choose the Diagnostic. On mode. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Example 2: Anthropology Application Anthropologists can use the femur, or thighbone, to estimate the height of a human being. The table shows the results of a randomly selected sample. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Example 2 Continued a. Make a scatter plot of the data with femur length as the independent variable. The scatter plot is shown at right. Holt Mc. Dougal Algebra 2 • •

1 -4 Curve Fitting with Linear Models Example 2 Continued b. Find the correlation coefficient r and the line of best fit. Interpret the slope of the line of best fit in the context of the problem. Enter the data into lists L 1 and L 2 on a graphing calculator. Use the linear regression feature by pressing STAT, choosing CALC, and selecting 4: Lin. Reg. The equation of the line of best fit is h ≈ 2. 91 l + 54. 04. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Example 2 Continued The slope is about 2. 91, so for each 1 cm increase in femur length, the predicted increase in a human being’s height is 2. 91 cm. The correlation coefficient is r ≈ 0. 986 which indicates a strong positive correlation. Holt Mc. Dougal Algebra 2

1 -4 Curve Fitting with Linear Models Example 2 Continued c. A man’s femur is 41 cm long. Predict the man’s height. The equation of the line of best fit is h ≈ 2. 91 l + 54. 04. Use the equation to predict the man’s height. For a 41 -cm-long femur, h ≈ 2. 91(41) + 54. 04 Substitute 41 for l. h ≈ 173. 35 The height of a man with a 41 -cm-long femur would be about 173 cm. Holt Mc. Dougal Algebra 2