SECTION 3.2 Building Linear Models from Data 153 Desmos can also be used to produce a scatter plot and linear regression. Enter the data points in a table, and then type the general form of the line with arbitrary slope m and y -intercept b. Desmos provides the values of m and b along with the values of r2 and r, as shown in Figure 11. Now Work PROBLEMS 11(d) AND (f) Does the line of best fit appear to be a good fit? In other words, does it appear to accurately describe the relation between on-base percentage and runs scored? And just how “good” is this line of best fit? Look again at Figure 10(a).The last line of output is = r 0.939. This number, called the correlation coefficient , − ≤ ≤ r r , 1 1, is a measure of the strength of the linear relation that exists between two variables.The closer r is to 1, the more nearly perfect the linear relationship is. If r is close to 0, there is little or no linear relationship between the variables. A negative value of < r r , 0, indicates that as x increases, y decreases; a positive value of > r r , 0, indicates that as x increases, y does also. The data given in Table 6, which have a correlation coefficient of 0.939, indicate a linear relationship with positive slope. Figure 11 = − y x 57.35 1120.75 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 3.2 Assess Your Understanding 1. Plot the points ( ) ( ) ( ) ( ) 1, 5 , 2, 6 , 3, 9 , 1, 12 in the Cartesian plane. Is the relation ( ) ( ) ( ) ( ) { } 1,5 , 2,6 , 3,9 , 1,12 a function? Why? (p. 2 and p. 64) 2. Find an equation of the line containing the points ( ) 1, 4 and ( ) 3, 8 . (p. 39) Figure 10(b) = − y x 57.35 1120.75 29.5 34 575 850 (b) Figure 10(b) shows the graph of the line of best fit, along with the scatter plot. (c) The slope of the line of best fit is 57.35, which means that, for every 1 percent increase in the on-base percentage, runs scored increase by 57.35, on average. (d) Letting = x 32.2 in the equation of the line of best fit, we obtain = ⋅ − ≈ y 57.35 32.2 1120.75 726 runs 3. A is used to help us to see what type of relation, if any, may exist between two variables. 4. True or False The correlation coefficient is a measure of the strength of a linear relation between two variables and must lie between −1 and 1, inclusive. Concepts and Vocabulary In Problems 5–10, examine each scatter plot and determine whether the relation is linear or nonlinear. 5. 5 10 15 20 25 30 35 0 5 10152025303540 x y 6. 2 4 6 8 10 12 14 0 2 4 6 8 10121416 x y 7. –2 0 22 12 Skill Building 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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