Draw Scatter Plots 2.6 and Best-Fitting Lines

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1 Draw Scatter Plots 2.6 and Best-Fitting Lines Before You wrote equations of lines. Now You will fit lines to data in scatter plots. Wh? So ou can model sports trends, as in Ex. 27. Ke Vocabular scatter plot positive correlation negative correlation correlation coefficient best-fitting line A scatter plot is a graph of a set of data pairs (x, ). If tends to increase as x increases, then the data have a positive correlation. If tends to decrease as x increases, then the data have a negative correlation. If the points show no obvious pattern, then the data have approximatel no correlation. x x x Positive correlation Negative correlation Approximatel no correlation E XAMPLE 1 Describe correlation TELEPHONES Describe the correlation shown b each scatter plot. Cellular Phone Subscribers and Cellular Service Regions, Cellular service regions (thousands) x Subscribers (millions) Cellular Phone Subscribers and Corded Phone Sales, Corded phone sales (millions of dollars) x Subscribers (millions) The first scatter plot shows a positive correlation, because as the number of cellular phone subscribers increased, the number of cellular service regions tended to increase. The second scatter plot shows a negative correlation, because as the number of cellular phone subscribers increased, corded phone sales tended to decrease. 2.6 Draw Scatter Plots and Best-Fitting Lines 113

2 CORRELATION COEFFICIENTS A correlation coefficient, denoted b r, is a number from 21 to 1 that measures how well a line fits a set of data pairs (x, ). If r is near 1, the points lie close to a line with positive slope. If r is near 21, the points lie close to a line with negative slope. If r is near, the points do not lie close to an line. r 5 21 r 5 r 5 1 Points lie near line Points do not lie Points lie near line with a negative slope. near an line. with positive slope. E XAMPLE 2 Estimate correlation coefficients Tell whether the correlation coefficient for the data is closest to 21, 2.5,,.5, or 1. a. 15 b. 15 c x x x a. The scatter plot shows a clear but fairl weak negative correlation. So, r is between and 21, but not too close to either one. The best estimate given is r (The actual value is r ø 2.46.) b. The scatter plot shows approximatel no correlation. So, the best estimate given is r 5. (The actual value is r ø 2.2.) c. The scatter plot shows a strong positive correlation. So, the best estimate given is r 5 1. (The actual value is r ø.98.) GUIDED PRACTICE for Examples 1 and 2 For each scatter plot, (a) tell whether the data have a positive correlation, a negative correlation, or approximatel no correlation, and (b) tell whether the correlation coefficient is closest to 21, 2.5,,.5, or x x x BEST-FITTING LINES If the correlation coefficient for a set of data is near 61, the data can be reasonabl modeled b a line. The best-fitting line is the line that lies as close as possible to all the data points. You can approximate a best-fitting line b graphing. 114 Chapter 2 Linear Equations and Functions

3 KEY CONCEPT For Your Notebook Approximating a Best-Fitting Line STEP 1 STEP 2 STEP 3 STEP 4 Draw a scatter plot of the data. Sketch the line that appears to follow most closel the trend given b the data points. There should be about as man points above the line as below it. Choose two points on the line, and estimate the coordinates of each point. These points do not have to be original data points. Write an equation of the line that passes through the two points from Step 3. This equation is a model for the data. E XAMPLE 3 Approximate a best-fitting line ALTERNATIVE-FUELED VEHICLES The table shows the number (in thousands) of alternative-fueled vehicles in use in the United States x ears after Approximate the best-fitting line for the data. x STEP 1 Draw a scatter plot of the data. STEP 2 Sketch the line that appears to best fit the data. One possibilit is shown. STEP 3 Choose two points that appear to lie on the line. For the line shown, ou might choose (1, 3), which is not an original data point, and (7, 548), which is an original data point. STEP 4 Write an equation of the line. First find the slope using the points (1, 3) and (7, 548). Number of vehicles (thousands) (7, 548) 3 (1, 3) x Years since 1997 m } 5 } ø 41.3 Use point-slope form to write the equation. Choose (x 1, 1 ) 5 (1, 3) m(x 2 x 1 ) Point-slope form (x 2 1) Substitute for m, x 1, and 1. ø 41.3x Simplif. c An approximation of the best-fitting line is x at classzone.com 2.6 Draw Scatter Plots and Best-Fitting Lines 115

4 E XAMPLE 4 Use a line of fit to make a prediction Use the equation of the line of fit from Example 3 to predict the number of alternative-fueled vehicles in use in the United States in 21. Because 21 is 13 ears after 1997, substitute 13 for x in the equation from Example x (13) ø 796 c You can predict that there will be about 796, alternative-fueled vehicles in use in the United States in 21. LINEAR REGRESSION Man graphing calculators have a linear regression feature that can be used to find the best-fitting line for a set of data. E XAMPLE 5 Use a graphing calculator to find a best-fitting line Use the linear regression feature on a graphing calculator to find an equation of the best-fitting line for the data in Example 3. FIND CORRELATION If our calculator does not displa the correlation coefficient r when it displas the regression equation, ou ma need to select DiagnosticOn from the CATALOG menu. STEP 1 Enter the data into two lists. Press and then select Edit. Enter ears since 1997 in L 1 and number of alternative-fueled vehicles in L 2. L1 L L1(2)=1 L3 STEP 3 Make a scatter plot of the data pairs to see how well the regression equation models the data. Press [STAT PLOT] to set up our plot. Then select an appropriate window for the graph. STEP 2 Find an equation of the bestfitting (linear regression) line. Press, choose the CALC menu, and select LinReg(ax1b). The equation can be rounded to 5 4.9x LinReg =ax+b a= b= r= STEP 4 Graph the regression equation with the scatter plot b entering the equation 5 4.9x The graph (displaed in the window x 8 and 2 6) shows that the line fits the data well. Plot1 Plot2 Plot3 On Off Tpe XList:L1 YList:L2 Mark: + c An equation of the best-fitting line is 5 4.9x Chapter 2 Linear Equations and Functions

5 GUIDED PRACTICE for Examples 3, 4, and 5 4. OIL PRODUCTION The table shows the U.S. dail oil production (in thousands of barrels) x ears after x a. Approximate the best-fitting line for the data. b. Use our equation from part (a) to predict the dail oil production in 29. c. Use a graphing calculator to find and graph an equation of the best-fitting line. Repeat the prediction from part (b) using this equation. 2.6 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 9, 11, and 25 5 STANDARDIZED TEST PRACTICE Exs. 2, 16, 18, 21, and 28 5 MULTIPLE REPRESENTATIONS Ex VOCABULARY Cop and complete: A line that lies as close as possible to a set of data points (x, ) is called the? for the data points. 2. WRITING Describe how to tell whether a set of data points shows a positive correlation, a negative correlation, or approximatel no correlation. EXAMPLE 1 on p. 113 for Exs. 3 5 DESCRIBING CORRELATIONS Tell whether the data have a positive correlation, a negative correlation, or approximatel no correlation x x x 6. REASONING Explain how ou can determine the tpe of correlation for a set of data pairs b examining the data in a table without drawing a scatter plot. EXAMPLE 2 on p. 114 for Exs. 7 9 CORRELATION COEFFICIENTS Tell whether the correlation coefficient for the data is closest to 21, 2.5,,.5, or x x x 2.6 Draw Scatter Plots and Best-Fitting Lines 117

6 EXAMPLES 3 and 4 on pp for Exs BEST-FITTING LINES In Exercises 1 15, (a) draw a scatter plot of the data, (b) approximate the best-fitting line, and (c) estimate when x x x x x x x MULTIPLE CHOICE Which equation best models the data in the scatter plot? A 5 15 B } 2 x 1 26 C } 5 x 1 19 D } 5 x ERROR ANALYSIS The graph shows one student s approximation of the bestfitting line for the data in the scatter plot. Describe and correct the error in the student s work x x 18. MULTIPLE CHOICE A set of data has correlation coefficient r. For which value of r would the data points lie closest to a line? A r B r 5 C r 5.38 D r 5.5 EXAMPLE 5 on p. 116 for Exs GRAPHING CALCULATOR In Exercises 19 and 2, use a graphing calculator to find and graph an equation of the best-fitting line. 19. x x OPEN-ENDED MATH Give two real-life quantities that have (a) a positive correlation, (b) a negative correlation, and (c) approximatel no correlation. 22. REASONING A set of data pairs has correlation coefficient r 5.1. Is it logical to use the best-fitting line to make predictions from the data? Explain. 23. CHALLENGE If x and have a positive correlation and and z have a negative correlation, what can ou sa about the correlation between x and z? Explain. 5 WORKED-OUT SOLUTIONS 118 Chapter 2 Linear on p. Equations WS1 and Functions 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

7 PROBLEM SOLVING EXAMPLES 3, 4, and 5 on pp for Exs GRAPHING CALCULATOR You ma wish to use a graphing calculator to complete the following Problem Solving exercises. 24. POPULATION The data pairs (x, ) give the population (in millions) of Texas x ears after Approximate the best-fitting line for the data. (, 19.7), (1, 2.2), (2, 2.6), (3, 2.9), (4, 21.3), (5, 21.7), (6, 22.1), (7, 22.5) 25. TUITION The data pairs (x, ) give U.S. average annual public college tuition (in dollars) x ears after Approximate the best-fitting line for the data. (, 2271), (1, 236), (2, 243), (3, 256), (4, 2562), (5, 2727), (6, 2928) 26. PHYSICAL SCIENCE The diagram shows the boiling point of water at various elevations. Approximate the best-fitting line for the data pairs (x, ) where x represents the elevation (in feet) and represents the boiling point (in degrees Fahrenheit). Then use this line to estimate the boiling point at an elevation of 14, feet. 27. MULTIPLE REPRESENTATIONS The table shows the numbers of countries that participated in the Winter Olmpics from 198 to 22. Year Countries a. Making a List Use the table to make a list of data pairs (x, ) where x represents ears since 198 and represents the number of countries. b. Drawing a Graph Draw a scatter plot of the data pairs from part (a). c. Writing an Equation Write an equation that approximates the best-fitting line, and use it to predict the number of participating countries in EXTENDED RESPONSE The table shows manufacturers shipments (in millions) of cassettes and CDs in the United States from 1988 to 22. Year Cassettes CDs a. Draw a scatter plot of the data pairs (ear, shipments of cassettes). Describe the correlation shown b the scatter plot. b. Draw a scatter plot of the data pairs (ear, shipments of CDs). Describe the correlation shown b the scatter plot. c. Describe the correlation between cassette shipments and CD shipments. What real-world factors might account for this? 2.6 Draw Scatter Plots and Best-Fitting Lines 119

8 29. CHALLENGE Data from some countries in North America show a positive correlation between the average life expectanc in a countr and the number of personal computers per capita in that countr. a. Make a conjecture about the reason for the positive correlation between life expectanc and number of personal computers per capita. b. Is it reasonable to conclude from the data that giving residents of a countr more personal computers will lengthen their lives? Explain. MIXED REVIEW Solve the equation for. Then find the value of for the given value of x. (p. 26) 3. 2x 2 5 1; x x 5 25; x x ; x x ; x x 5 2; x x 2 4x 5 9; x 5 6 PREVIEW Prepare for Lesson 2.7 in Exs Evaluate the function for the given value of x. (p. 72) 36. f(x) 5 2x 1 7; f(9) 37. f(x) 5 24x 2 11; f(25) 38. f(x) x ; f(22) 39. f(x) 5 x 2 1 ; f(1) 4. f(x) x ; f(4) 41. f(x) 5 2x ; f(23) Graph the equation. (p. 89) x x x x x x QUIZ for Lessons Write an equation of the line that satisfies the given conditions. (p. 98) 1. m 5 25, b m 5 2, b m 5 4, passes through (23, 6) 4. m 5 27, passes through (1, 24) 5. passes through (, 7) and (23, 22) 6. passes through (29, 9) and (29, ) Write and graph a direct variation equation that has the given ordered pair as a solution. (p. 17) 7. (1, 2) 8. (22, 8) 9. (5, 216) 1. (12, 4) The variables x and var directl. Write an equation that relates x and. Then find when x 5 8. (p. 17) 11. x 5 4, x 5 23, x 5 4, x 5 12, CONCERT TICKETS The table shows the average price of a concert ticket to one of the top 5 musical touring acts for the ears Write an equation that approximates the best-fitting line for the data pairs (x, ). Use the equation to predict the average price of a ticket in 21. (p. 113) Years since 1999, x Ticket price (dollars), Chapter 2 EXTRA Linear Equations PRACTICE and for Functions Lesson 2.6, p. 111 ONLINE QUIZ at classzone.com