Objectives Fit scatter plot data using linear models. Use linear models to make predictions.

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. Negative correlation, negative slope

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.

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.

CLASSWORK Worksheet

Example 2 Make a scatter plot for this set of data. Identify the correlation, sketch a line of best fit, and find its equation.

• • • • • • • • • • Example 2 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 • • • • • • • • • •

• • • • • • • • • • Example 2 Continued Step 3 Sketch a line of best fit. Draw a line that splits the data evenly above and below. • • • • • • • • • •

Step 4 Identify two points on the line. Example 2 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.

What is the range of the correlation coefficient Quiz What is the range of the correlation coefficient

Example 3: 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 3 Continued a. Make a scatter plot of the data with femur length as the independent variable. • • • • • • • • The scatter plot is shown at right.

Example 3 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. The equation of the line of best fit is h ≈ 2.91(L) + 54.04.

Example 3 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 ≈ 1 which indicates a strong positive correlation.

Example 3 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.

Example 4 The gas mileage for randomly selected cars based upon engine horsepower is given in the table.

• • • • • • • • • • Example 4 Continued a. Make a scatter plot of the data with horsepower as the independent variable. • • • • • • • • The scatter plot is shown on the right. • •

Example 4 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. The equation of the line of best fit is y ≈ –0.15x + 47.5.

Example 4 Continued The slope is about –0.15, so for each 1 unit increase in horsepower, gas mileage drops ≈ 0.15 mi/gal. The correlation coefficient is r ≈ –1, which indicates a strong negative correlation.

Example 4 Continued c. Predict the gas mileage for a 210-horsepower engine. The equation of the line of best fit is y ≈ –0.15x + 47.5. Use the equation to predict the gas mileage. For a 210-horsepower engine, y ≈ –0.15(210) + 47.50. Substitute 210 for x. y ≈ 16 The mileage for a 210-horsepower engine would be about 16.0 mi/gal.

CLASSWORK Worksheet