ACTIVITY: Representing Data by a Linear Equation
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- Emil Roland Hodges
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1 9.2 Lines of Fit How can ou use data to predict an event? ACTIVITY: Representing Data b a Linear Equation Work with a partner. You have been working on a science project for 8 months. Each month, ou measured the length of a bab alligator. The table shows our measurements. September April Month, Length (in.), Data Analsis In this lesson, ou will find lines of fit. use lines of fit to solve problems. Use the following steps to predict the bab alligator s length net September. a. Graph the data in the table. b. Draw a line that ou think best approimates the points. c. Write an equation for our line. d. MODELING Use the equation to predict the bab alligator s length net September. Length (inches) Month 378 Chapter 9 Data Analsis and Displas
2 2 ACTIVITY: Representing Data b a Linear Equation Work with a partner. You are a biologist and stud bat populations. You are asked to predict the number of bats that will be living in an abandoned mine after 3 ears. To start, ou find the number of bats that have been living in the mine during the past 8 ears. V I D E O The table shows the results of our research. Math Practice Use a Graph How can ou draw a line that fits the collection of points? How should the points be positioned around the line? Year, Bats (thousands), Use the following steps to predict the number of bats that will be living in the mine after 3 ears. a. Graph the data in the table. 7 ears ago this ear b. Draw a line that ou think best approimates the points. c. Write an equation for our line. d. MODELING Use the equation to predict the number of bats in 3 ears. Bats (thousands) Year 3. IN YOUR OWN WORDS How can ou use data to predict an event? 4. MODELING Use the Internet or some other reference to find data that appear to have a linear pattern. List the data in a table, and then graph the data. Use an equation that is based on the data to predict a future event. Use what ou learned about lines of fit to complete Eercise 4 on page 382. Section 9.2 Lines of Fit 379
3 9.2 Lesson Lesson Tutorials Ke Vocabular line of fit, p. 38 line of best fit, p. 38 A line of fit is a line drawn on a scatter plot close to most of the data points. It can be used to estimate data on a graph. Month, Stud Tip EXAMPLE Depth (feet), A line of fit does not need to pass through an of the data points. Finding a Line of Fit The table shows the depth of a river months after a monsoon season ends. (a) Make a scatter plot of the data and draw a line of fit. (b) Write an equation of the line of fit. (c) Interpret the slope and the -intercept of the line of fit. (d) Predict the depth in month 9. a. Plot the points in a coordinate River Depth plane. The scatter plot shows a negative linear relationship. 2 Draw a line that is close to the data 8 points. Tr to have as man points 6 above the line as below it. 4 b. The line passes through (5, ) and (6, 8). slope = rise run = 2 = 2 Because the line crosses the -ais at (, 2), the -intercept is 2. So, an equation of the line of fit is = c. The slope is 2, and the -intercept is 2. So, the depth of the river is 2 feet at the end of the monsoon season and decreases b about 2 feet per month. d. To predict the depth in month 9, substitute 9 for in the equation of the line of fit. = = 2(9) + 2 = 2 Depth (feet) The depth in month 9 should be about 2 feet. 2 (5, ) (6, 8) Month Eercises 5 and 6. The table shows the numbers of people who have attended a festival over an 8-ear period. (a) Make a scatter plot of the data and draw a line of fit. (b) Write an equation of the line of fit. (c) Interpret the slope and the -intercept of the line of fit. (d) Predict the number of people who will attend the festival in ear. Year, Attendance, Chapter 9 Data Analsis and Displas
4 Stud Tip You know how to use two points to find an equation of a line of fit. When finding an equation of the line of best fit, ever point in the data set is used. Graphing calculators use a method called linear regression to find a precise line of fit called a line of best fit. This line best models a set of data. A calculator often gives a value r called the coefficient. This value tells whether the is positive or negative, and how closel the equation models the data. Values of r range from to. When r is close to or, there is a strong between the variables. As r gets closer to, the becomes weaker. r r r Strong negative No Strong positive EXAMPLE 2 Finding a Line of Best Fit Using Technolog The table shows the worldwide movie ticket sales (in billions of dollars) from 2 to 2, where = represents the ear 2. Use a graphing calculator to find an equation of the line of best fit. Identif and interpret the coefficient. Year, Ticket Sales, Step : Enter the data from the table into our calculator. Step 2: Use the linear regression feature. slope -intercept Stud Tip The slope of.5 indicates that sales are increasing b about $.5 billion each ear. The -intercept of 6 represents the ticket sales of $6 billion for 2. Check An equation of the line of best fit is = The coefficient is about.982. This means that the relationship between ears and ticket sales is a strong positive and that the equation closel models the data. Use a graphing calculator to make a scatter plot and graph the line of best fit. 45 coefficient 2 Eercises 8 2. Use a graphing calculator to find an equation of the line of best fit for the data in Eample. Identif and interpret the coefficient. Section 9.2 Lines of Fit 38
5 9.2 Eercises Help with Homework. WRITING Eplain wh a line of fit is helpful when analzing data. 2. REASONING Tell whether the line drawn on the graph is a good fit for the data. Eplain our reasoning. 3. NUMBER SENSE Which coefficient indicates a stronger relationship:.98 or.9? Eplain. 9+(-6)=3 3+(-3)= 4+(-9)= 9+(-)= 4. BLUEBERRIES The table shows the weights of pints of blueberries. Number of Pints, Weight (pounds), a. Graph the data in the table. b. Draw a line that ou think best approimates the points. c. Write an equation for our line. d. Use the equation to predict the weight of pints of blueberries. e. Blueberries cost $2.25 per pound. How much do pints of blueberries cost? 5. HOT CHOCOLATE The table shows the dail high temperature ( F) and the number of hot chocolates sold at a coffee shop for eight randoml selected das. Temperature ( F), Hot Chocolates, a. Make a scatter plot of the data and draw a line of fit. b. Write an equation of the line of fit. c. Interpret the slope and the -intercept of the line of fit. d. Predict the number of hot chocolates sold when the high temperature is 2 F. 6. VACATION The table shows the distance ou are awa from home over a 6-hour period of our vacation. a. Make a scatter plot of the data and draw a line of fit. b. Write an equation of the line of fit. c. About how man miles per hour do ou travel? d. About how far were ou from home when ou started? e. Predict the distance from home in 7 hours. Hours, Distance (miles), Chapter 9 Data Analsis and Displas
6 7. REASONING A data set has no relationship. Is it possible to find a line of fit for the data? Eplain AMUSEMENT PARK The table shows the attendance (in thousands) at an amusement park from 24 to 23, where = 4 represents the ear 24. Use a graphing calculator to find an equation of the line of best fit. Identif and interpret the coefficient. Year, Attendance (thousands), SNOWSTORM The table shows the total snow depth (in inches) on the ground during a snowstorm hours after it began. Use a graphing calculator to find an equation of the line of best fit. Identif and interpret the coefficient. Use our equation to estimate how much snow was on the ground before the snowstorm began. Hours, Snow Depth (inches), TEXTING The table shows the numbers (in billions) of tet messages sent from 26 to 2, where = 6 represents the ear 26. a. Use a graphing calculator to find an equation of the line of best fit. Identif and interpret the coefficient. b. Interpret the slope of the line of best fit. Does the -intercept make sense for this problem? Eplain. c. Predict the number of tet messages sent in 25. Year, Tet Messages (billions), Modeling The table shows the height (in feet) of a baseball seconds after it was hit. a. Use a graphing calculator to find an equation of the line of best fit. Identif and interpret the coefficient. b. Predict the height after 5 seconds. c. The actual height after 5 seconds is about 3 feet. Wh do ou think this is different from our prediction? Seconds, Height (feet), Write the decimal as a fraction or a mied number. (Section 7.4) MULTIPLE CHOICE Which epression represents the volume of a sphere with radius r? (Section 8.3) A 3 πr 2 h B πr 2 h C 4πr 2 D 4 3 πr 3 Section 9.2 Lines of Fit 383
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