10-1 Correlation 511 Formulas for Calculating r FORMULA 10-1 r = n1Σxy2 - 1Σx21Σy2 2 n1Σx22 - 1Σx222n1Σy22 - 1Σy22 1Good format for calculations2 FORMULA 10-2 r = Σ1zx zy2 n - 1 1Good format for understanding2 where zx denotes the z score for an individual sample value x and zy is the z score for the corresponding sample value y. Rounding the Linear Correlation Coefficient r Round the linear correlation coefficient r to three decimal places so that its value can be directly compared to critical values in Table A-6. Interpreting the Linear Correlation Coefficient r • Using P-Value from Technology to Interpret r: Use the P-value and significance level a as follows: P@value … a: Supports the claim of a linear correlation. P@value 7 a: Does not support the claim of a linear correlation. • Using Table A-6 to Interpret r: Consider critical values from Table A-6 or technology as being both positive and negative, draw a graph similar to Figure 10-3 shown below and used in Example 4 on page 516, and then use the following decision criteria: Correlation If the computed linear correlation coefficient r lies in the left tail at or below the leftmost critical value or if it lies in the right tail at or above the rightmost critical value (that is, r Ú critical value), conclude that there is sufficient evidence to support the claim of a linear correlation. No Correlation If the computed linear correlation coefficient lies between the two critical values (that is, r 6 critical value), conclude that there is not sufficient evidence to support the claim of a linear correlation. Correlation 0 −1 1 No correlation Correlation Sample Data: r = 0.947 r = 0.666 Critical Value r = −0.666 Critical Value FIGURE 10-3 Critical r Values and the Computed r Value CAUTION Remember, the methods of this section apply to a linear correlation. If you conclude that there does not appear to be a linear correlation, it is possible that there might be some other association that is not linear, as in Figure 10-2(d). Always generate a scatterplot to see relationships that might not be linear.
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