Objectives Fit scatter plot data using linear models with and without technology. Use linear models to make predictions.

Vocabulary regression correlation line of best fit correlation coefficient

Researchers, such as anthropologists, are often interested in how two measurements are related. The statistical study of the relationship between variables is called regression.

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.

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. Try to have about the same number of points above and below the line of best fit. Helpful Hint

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.

• • • • • • • • • • • Example 1 Continued Step 1 Plot the data points. Step 2 Identify the correlation. Notice that the data set is negatively correlated–as the temperature rises in Albany, it falls in Sydney. • • • • • • • • • • •

• • • • • • • • • • • Example 1 Continued Step 3 Sketch a line of best fit. o Draw a line that splits the data evenly above and below. • • • • • • • • • • •

Step 4 Identify two points on the line. 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. y – y1= m(x – x1) Point-slope form. y – 64 = –0.46(x – 35) Substitute. y = –0.46x + 80.1 Simplify. An equation that models the data is y = –0.46x + 80.1.

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.

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 • • • • • • • • • •

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. • • • • • • • • • •

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 – y1= m(x – x1) Point-slope form. y – 10 = 0.75(x – 20) Substitute. y = 0.75x – 5 Simplify. A possible answer is p = 0.75x + 5.

The correlation coefficient r is a measure of how well the data set is fit by a model.

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 DiagnosticOn mode.

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.

• • • • • • • • 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.

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 L1 and L2 on a graphing calculator. Use the linear regression feature by pressing STAT, choosing CALC, and selecting 4:LinReg. The equation of the line of best fit is h ≈ 2.91l + 54.04.

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.

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 ≈ 2.91l + 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.