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Finding the Line of Best Fit 7 Graphing Calculator Lab (Use with Lesson 71) SSM_A1_NL_SBK_Lab07.inddM_A1_NL_SBK_Lab07.indd PagePage 464464 12/21/0712/21/07 6:12:256:12:25 PMPM epgepg //Volumes/ju102/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_LAB/SM_A1_SBK_Lab_07Volumes/ju102/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_LAB/SM_A1_SBK_Lab_07 LAB Finding the Line of Best Fit 7 Graphing Calculator Lab (Use with Lesson 71) A graphing calculator is often the easiest and most accurate method for making a scatter plot from a data set and then finding the equation of the line of best fit. The following table gives the total number y in billions of movie admissions for x years after 1993. Year (x) 12345678910 Admissions (y) 1.29 1.26 1.34 1.39 1.48 1.47 1.42 1.49 1.63 1.57 Find a line of best fit for the data. According to the model, about how many billion movie admissions were there in 2004 (x = 11)? Graphing Calculator Tip 1. Press and choose 1:Edit. Clear any old For help with entering data by pressing the key until the list data into a list, see the name is selected. Press and then . graphing calculator keystrokes in Graphing Calculator Lab 4 on 2. Enter the x-values as L1. Use the key to page 343. move to the next column and enter the y-values as L2. 3. Press and select 1:Plot1… to open the plot setup menu. 4. Press to turn Plot1 On. Then press the key once and then to select the first Type. 5. The default settings for Xlist: and Ylist: are L1 and L2, respectively. If these are not the current settings, then move the cursor next to Xlist:. Press . Select the list of independent variables, 1:L1. To change the Ylist: setting, press and select 2:L2, the list of dependent variables. Online Connection 6. Create a scatter plot by pressing and www.SaxonMathResources.com selecting 9:ZoomStat. 464 Saxon Algebra 1 SSM_A1_NL_SBK_Lab07.inddM_A1_NL_SBK_Lab07.indd PagePage 465465 12/21/0712/21/07 6:12:316:12:31 PMPM epgepg //Volumes/ju102/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_LAB/SM_A1_SBK_Lab_07Volumes/ju102/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_LAB/SM_A1_SBK_Lab_07 7. Find the linear regression for the scatter plot. Press and the key to highlight the CALC menu. Select 4:LinReg(ax+b). 8. Set the parameters for the linear regression. Press and choose 1:L1. Then press and choose 2:L2. 9. To view the line of best fit with the scatter plot, press and the key to highlight Y-VARS. Then press twice to select Y1 as the destination for the equation. 10. Press . To write the equation of the line of best fit, the values of a and b are 0.036 and 1.236, respectively. The equation for the line of best fit is y = 0.036x + 1.236. 11. Press to view the line of best fit with the scatter plot. 12. Predict the value of y when x = 11. Since x = 11 is outside the viewing window, press and the key and type 11. 13. Press and select 1:value. Type 11 and press . When x = 11, the value of y is 1.632. Therefore, according to the line of best fit, there would be about 1.632 billion movie admissions in the year 2004. Lab Practice The table gives solar energy cell capacity y in megawatts for x years after 1989. Use a graphing calculator to create a scatter plot and to graph the line of best fit for the data. What is an equation for the line of best fit? Use the model to predict the solar-energy cell capacity in 1996 (x = 7). Year (x) 123456 Capacity (y) 13.8 14.9 15.6 21.0 26.1 31.1 Lab 7 465 SSM_A1_NL_SBK_L071.inddM_A1_NL_SBK_L071.indd PPageage 446666 112/24/072/24/07 77:52:56:52:56 AAMM aadministratordministrator //Volumes/ju102/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L071Volumes/ju102/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L071 LESSON Making and Analyzing Scatter Plots 71 Warm Up 1. Vocabulary The constant m in the __________ form y = mx + b is the (49) slope of the line. 2. The equation of a line is y = -2x + 6. What is the slope of the line? (49) 3. Write the equation x - 2y = 4 in slope-intercept form. (49) _3 4. Write an equation in slope-intercept form for a line with slope = and (49) 8 y-intercept = -4. New Concepts One type of graph that relates two sets of data with plotted ordered pairs is called a scatter plot. Scatter plots are considered discrete graphs because their points are separate and disconnected. The points on scatter plots sometimes form a linear pattern. In these situations, a line can be drawn to model the pattern, which is called a trend line. A trend line is a line on a scatter plot, which shows the relationship between two sets of data. Example 1 Graphing a Scatter Plot and a Trend Line Caution Use the data in the table. A trend line does not x 123456 have to go through any of the points on the y 4 1012182329 scatter plot. It has to be drawn equally as close to one point as to another a. Make a scatter plot of the data. Then draw a trend line so that it models the on the scatter plot. approximate slope of the points. SOLUTION y Plot the points on a coordinate plane. 40 Then draw a straight line as near to as many of the points as possible. 30 20 Math Reasoning b. Find an equation for the trend line. 10 Predict Using the SOLUTION Use two points on or near the trend x table, the graph, or the line to write an equation for the line. equation from Example 1, 2 4 6 what do you think would _y2 - y1 be a reasonable y-value m = x - x for an x-value of 7? 2 1 29 - 4 = _ Find the slope of the line; use (1, 4) and (6, 29). 6 - 1 = 5 (y - y1) = m(x - x1) Write the equation for the line. (y - 4) = 5(x - 1) Substitute m = 5 and (1, 4) for (x , y ). Online Connection 1 1 www.SaxonMathResources.com y = 5x - 1 Solve for y. 466 Saxon Algebra 1 SSM_A1_NL_SBK_L071.inddM_A1_NL_SBK_L071.indd PPageage 446767 112/24/072/24/07 77:52:58:52:58 AAMM aadministratordministrator //Volumes/ju102/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L071Volumes/ju102/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L071 Trend lines may vary. However, a trend line that shows the linear relationship of a scatter plot the most accurately is called the line of best fit. The equation of the line of best fit can be calculated on a graphing calculator. The line of best fit is also referred to as the regression line. Example 2 Calculating a Line of Best Fit The table shows the averages for homework grades and test scores for nine students. Use a graphing calculator to find the equation of the line of best fit. Graphing Homework Grades and Test Scores Calculator Tip Average 75 85 94 88 91 95 76 84 90 Homework Grade For help with calculating a line of best fit, see Average Test 83 87 95 93 88 91 83 80 92 Graphing Calculator Scores Lab 7 on page 464. SOLUTION Step 1: Enter the data into your graphing calculator by pressing to EDIT the data. Enter the average homework grades into L1 and the average test scores into L2. Then return to the home screen. Hint Step 2: Calculate the equation for the line of best fit by pressing to access the CALC menu. Then press to choose LinReg (ax + b). The calculator uses the variable a instead of m Type in L1, L2 and press to calculate the values used for writing the to represent the slope in equation. the slope-intercept form of a line. Step 3: Round the values for a and b to the nearest thousandth and write the equation for the line of best fit in slope-intercept form: y = 0.558x + 39.792 Two sets of data may be related to each other. A correlation is a measure of the strength and direction of the association between data sets or variables. When the points tightly cluster in a linear pattern, then the correlation is strong. Data can be positively correlated or negatively correlated. There is a positive correlation when the data values for both variables increase. There is a negative correlation when the data values for one variable increase while the data values for the other variable decrease. Strong positive correlation Strong negative correlation No correlation between x and y between x and y between x and y y y y x x x The direction of the association can be determined for scatter plots that have correlation. If the slope of the trend line is positive, then there is a positive correlation between the data values. If the slope of the trend line is negative, then there is a negative correlation between the data values. Lesson 71 467 SSM_A1_NLB_SBK_L071.inddM_A1_NLB_SBK_L071.indd PagePage 468468 4/9/084/9/08 12:46:0712:46:07 AMAM elhielhi //Volumes/ju108/HCAC057/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L0Volumes/ju108/HCAC057/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L071 Example 3 Identifying Correlations State whether there is a positive correlation, a negative correlation, or no correlation between the data values. y a. x SOLUTION As the trend line rises from left to right, there is a positive correlation. Both sets of data values are increasing. b. x 1098765432 y 60 63 72 75 77 81 83 89 92 SOLUTION Negative correlation: As x-values decrease the y = values increase.
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