12.6 Linear Correlation and Regression 825 The following formula is used to calculate r, the linear correlation coefficient. Linear Correlation Coefficient The formula to calculate the linear correlation coefficient, r, is as follows. = Σ − Σ Σ Σ − Σ Σ − Σ r n xy x y n x x n y y ( ) ( )( ) ( ) ( ) ( ) ( ) 2 2 2 2 In the formula to calculate the linear correlation coefficient, r, we use the Greek letter .Σ Recall from Section 12.3, that Σ is used to represent a sum. For example, the notation Σx refers to the sum of all x-values from the data. In this book, the linear correlation coefficient is sometimes just referred to as the correlation coefficient. To determine the correlation coefficient, r, and the equation of the line of best fit (to be discussed shortly), a statistical calculator may be used. In the Technology Tip box at the end of this section, we indicate the procedure to follow to use the computer software spreadsheet program Microsoft Excel, the TI-84 Plus calculator, and StatCrunch to determine the correlation coefficient. In Example 1, we show how to determine r for a set of bivariate data without the use of a statistical calculator. We will use the same set of bivariate data given on page 823 that was used to make the scatter diagram in Fig. 12.40. Example 1 Number of Absences Versus Number of Defective Parts Egan Electronics provided the following daily records about the number of assembly line workers absent and the number of defective parts produced for 6 days. Determine the correlation coefficient between the number of workers absent and the number of defective parts produced. Day 1 2 3 4 5 6 Number of workers absent 3 5 0 1 2 6 Number of defective parts 15 22 7 12 20 30 Solution We plotted this set of data on the scatter diagram in Fig. 12.40. We will call the number of workers absent x. We will call the number of defective parts produced y. In the table below, we list the values of x and y and calculate the necessary sums: Σ Σ Σ Σ Σ x y xy x y , , , , . 2 2 We determine the values in the column labeled x2 by squaring the x’s (multiplying the x’s by themselves). We determine the values in the column labeled y2 by squaring the y’s. We determine the values in the column labeled xy by multiplying each x-value by its corresponding y-value. Number of Workers Absent Number of Defective Parts x y x2 y2 xy 3 15 9 225 45 5 22 25 484 110 0 7 0 49 0 1 12 1 144 12 2 20 4 400 40 6 30 36 900 180 17 106 75 2202 387
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