. Scatter Plots and Lines of Fit Essential Question How can ou use a scatter plot and a line of fit to make conclusions about data? A scatter plot is a graph that shows the relationship between two data sets. The two data sets are graphed as ordered pairs in a coordinate plane. Finding a Line of Fit REASONING QUANTITATIVELY To be proficient in math, ou need to make sense of quantities and their relationships in problem situations. Work with a partner. A surve was taken of 179 married couples. Each person was asked his or her age. The scatter plot shows the results. a. Draw a line that approimates the data. Write an equation of the line. Eplain the method ou used. b. What conclusions can ou make from the equation ou wrote? Eplain our reasoning. Wife s age 85 8 75 7 65 6 55 5 5 35 3 Ages of Married Couples 3 35 5 5 55 6 65 7 75 8 Husband s age Work with a partner. The scatter plot shows the median ages of American women at their first marriage for selected ears from 196 through 1. a. Draw a line that approimates the data. Write an equation of the line. Let represent the number of ears since 196. Eplain the method ou used. b. What conclusions can ou make from the equation ou wrote? Finding a Line of Fit c. Use our equation to predict the median age of American women at their first marriage in the ear. Communicate Your Answer 3. How can ou use a scatter plot and a line of fit to make conclusions about data? Age Ages of American Women at First Marriage 8 6 18 196 197 198 199 1 Year. Use the Internet or some other reference to find a scatter plot of real-life data that is different from those given above. Then draw a line that approimates the data and write an equation of the line. Eplain the method ou used. Section. Scatter Plots and Lines of Fit 185
. Lesson What You Will Learn Core Vocabular scatter plot, p. 186 correlation, p. 187 line of fit, p. 188 Interpret scatter plots. Identif correlations between data sets. Use lines of fit to model data. Interpreting Scatter Plots Core Concept Scatter Plot A scatter plot is a graph that shows the relationship between two data sets. The two data sets are graphed as ordered pairs in a coordinate plane. Scatter plots can show trends in the data. Interpreting a Scatter Plot Calories 3 31 3 9 8 7 6 5 Smoothies 6 5 5 58 6 66 7 Sugar (grams) The scatter plot shows the amounts (in grams) of sugar and the numbers of calories in 1 smoothies. a. How man calories are in the smoothie that contains 56 grams of sugar? b. How man grams of sugar are in the smoothie that contains 3 calories? c. What tends to happen to the number of calories as the number of grams of sugar increases? SOLUTION a. Draw a horizontal line from the point that has an -value of 56. It crosses the -ais at 7. So, the smoothie has 7 calories. b. Draw a vertical line from the point that has a -value of 3. It crosses the -ais at 7. So, the smoothie has 7 grams of sugar. c. Looking at the graph, the plotted points go up from left to right. So, as the number of grams of sugar increases, the number of calories increases. Calories 3 31 3 9 8 7 6 5 Smoothies 6 5 5 58 6 66 7 Sugar (grams) Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. How man calories are in the smoothie that contains 51 grams of sugar?. How man grams of sugar are in the smoothie that contains 5 calories? 186 Chapter Writing Linear Functions
STUDY TIP You can think of a positive correlation as having a positive slope and a negative correlation as having a negative slope. Identifing Correlations between Data Sets A correlation is a relationship between data sets. You can use a scatter plot to describe the correlation between data. Positive Correlation Negative Correlation No Correlation As increases, increases. As increases, decreases. The points show no pattern. Identifing Correlations Tell whether the data show a positive, a negative, or no correlation. a. age and vehicles owned b. temperature and coat sales at a store Number of vehicles owned Age and Vehicles Owned 7 6 5 3 1 5 3 35 5 5 55 Person s age (ears) Coats sold per da Temperature and Coat Sales 7 6 5 3 1 35 5 5 55 6 65 7 Average dail temperature ( F) SOLUTION a. The points show no pattern. The number of vehicles owned does not depend on a person s age. So, the scatter plot shows no correlation. b. As the average temperature increases, the number of coats sold decreases. So, the scatter plot shows a negative correlation. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Make a scatter plot of the data. Tell whether the data show a positive, a negative, or no correlation. 3. Temperature ( F), 8 78 68 87 75 71 9 8 Attendees (thousands),.5. 1.7 5.5 3.8.9.7 5.3. Age of a car (ears), 1 3 5 6 7 8 Value (thousands), $ $1 $19 $18 $15 $1 $8 $7 Section. Scatter Plots and Lines of Fit 187
STUDY TIP A line of fit is also called a trend line. Using Lines of Fit to Model Data When data show a positive or negative correlation, ou can model the trend in the data using a line of fit. A line of fit is a line drawn on a scatter plot that is close to most of the data points. Core Concept Using a Line of Fit to Model Data Step 1 Make a scatter plot of the data. Step Decide whether the data can be modeled b a line. Step 3 Draw a line that appears to fit the data closel. There should be approimatel as man points above the line as below it. Step Write an equation using two points on the line. The points do not have to represent actual data pairs, but the must lie on the line of fit. Finding a Line of Fit The table shows the weekl sales of a DVD and the number of weeks since its release. Write an equation that models the DVD sales as a function of the number of weeks since its release. Interpret the slope and -intercept of the line of fit. Week, 1 3 5 6 7 8 Sales (millions), $19 $15 $13 $11 $1 $8 $7 $5 SOLUTION Sales (millions of dollars) 18 16 1 1 1 8 6 DVD Sales (5, 1) (6, 8) 1 3 5 6 7 8 9 Week Step 1 Make a scatter plot of the data. Step Decide whether the data can be modeled b a line. Because the scatter plot shows a negative correlation, ou can fit a line to the data. Step 3 Draw a line that appears to fit the data closel. Step Write an equation using two points on the line. Use (5, 1) and (6, 8). The slope of the line is m = 8 1 6 5 =. Use the slope m = and the point (6, 8) to write an equation of the line. 1 = m( 1 ) Write the point-slope form. 8 = ( 6) Substitute for m, 6 for 1, and 8 for 1. = + Solve for. An equation of the line of fit is = +. The slope of the line is. This means the sales are decreasing b about $ million each week. The -intercept is. The -intercept has no meaning in this contet because there are no sales in week. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 188 Chapter Writing Linear Functions 5. The following data pairs show the monthl income (in dollars) and the monthl car pament (in dollars) of si people: (1, 1), (165, 315), (195, 5), (15, 95), (5, ), and (18, 375). Write an equation that models the monthl car pament as a function of the monthl income. Interpret the slope and -intercept of the line of fit.
. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE When data show a positive correlation, the dependent variable tends to as the independent variable increases.. VOCABULARY What is a line of fit? Monitoring Progress and Modeling with Mathematics In Eercises 3 6, use the scatter plot to fill in the missing coordinate of the ordered pair. 3. (16, ). (3, ) 5. (, 1) 6. (, 17) 7. INTERPRETING A SCATTER PLOT The scatter plot shows the hard drive capacities (in gigabtes) and the prices (in dollars) of 1 laptops. (See Eample 1.) Price (dollars) 18 1 1 6 6 1 1 18 Laptops 16 1 1 1 8 6 6 8 1 1 Hard drive capacit (gigabtes) a. What is the price of the laptop with a hard drive capacit of 8 gigabtes? b. What is the hard drive capacit of the $1 laptop? c. What tends to happen to the price as the hard drive capacit increases? 8. INTERPRETING A SCATTER PLOT The scatter plot shows the earned run averages and the winning percentages of eight pitchers on a baseball team. Winning percentage.7.6.5..3..1 Pitchers 3 5 6 Earned run average a. What is the winning percentage of the pitcher with an earned run average of.? b. What is the earned run average of the pitcher with a winning percentage of.33? c. What tends to happen to the winning percentage as the earned run average increases? In Eercises 9 1, tell whether and show a positive, a negative, or no correlation. (See Eample.) 9. 11. 8 1. 1. 3 3 1 1 8 3 8 1 Section. Scatter Plots and Lines of Fit 189
In Eercises 13 and 1, make a scatter plot of the data. Tell whether and show a positive, a negative, or no correlation. 13. 1. 3.1..5 3.7 3.9 1.5.7. 1 1 3 3 5 6 7 8 9 1 67 67 5 33 5 1 19 15. MODELING WITH MATHEMATICS The table shows the world birth rates (number of births per 1 people) ears since 196. (See Eample 3.) 1 3 5 35. 33.6 8.3 7... a. Write an equation that models the birthrate as a function of the number of ears since 196. b. Interpret the slope and -intercept of the line of fit. 16. MODELING WITH MATHEMATICS The table shows the total earnings (in dollars) of a food server who works hours. 1 3 5 6 18 6 77 85 113 a. Write an equation that models the server s earnings as a function of the number of hours the server works. b. Interpret the slope and -intercept of the line of fit. 17. OPEN-ENDED Give an eample of a real-life data set that shows a negative correlation. 18. MAKING AN ARGUMENT Your friend sas that the data in the table show a negative correlation because the dependent variable is decreasing. Is our friend correct? Eplain. 1 1 1 8 6 1 1 5 19. USING TOOLS Use a ruler or a ardstick to find the heights and arm spans of five people. a. Make a scatter plot using the data ou collected. Then draw a line of fit for the data. b. Interpret the slope and -intercept of the line of fit.. THOUGHT PROVOKING A line of fit for a scatter plot is given b the equation = 5 +. Describe a real-life data set that could be represented b the scatter plot. 1. WRITING When is data best displaed in a scatter plot, rather than another tpe of displa, such as a bar graph or circle graph?. HOW DO YOU SEE IT? The scatter plot shows part of a data set and a line of fit for the data set. Four data points are missing. Choose possible coordinates for these data points. 1 8 8 1 16 3. REASONING A data set has no correlation. Is it possible to find a line of fit for the data? Eplain.. ANALYZING RELATIONSHIPS Make a scatter plot of the data in the tables. Describe the relationship between the variables. Is it possible to fit a line to the data? If so, write an equation of the line. If not, eplain wh. 1 9 7 3 1 15 76 5 15 1 1 5 6 7 9 15 5 37 5 9 6 Maintaining Mathematical Proficienc Evaluate the function when = 3,, and. (Section 3.3) 5. g() = 6 6. h() = 1 7. f() = 5 8 8. v() = 1 3 Reviewing what ou learned in previous grades and lessons 19 Chapter Writing Linear Functions