2-4 Scatterplots, Correlation, and Regression 77 P-Values for Determining Linear Correlation In Example 4, we used the computed value of the linear correlation coefficient r = 0.591 and compared it to the critical r values of {0.878 found from Table 2-11. (See Figure 2-17.) In the real world of statistics applications, the use of such tables is almost obsolete. Section 10-1 describes a more common approach that is based on “P-values” instead of tables. The Statdisk display accompanying Example 4 shows that the P-value is 0.29369, or 0.294 when rounded. P-values are first introduced in Chapter 8, but here is a preliminary definition suitable for the context of this section: TABLE 2-11 Critical Values of the Linear Correlation Coefficient r Number of Pairs of Data n Critical Value of r 4 0.950 5 0.878 6 0.811 7 0.754 8 0.707 9 0.666 10 0.632 11 0.602 12 0.576 0 −1 1 Correlation Correlation No correlation Sample Data: r = 0.591 r = 0.878 Critical Value r = −0.878 Critical Value FIGURE 2-17 Critical Values from Table 2-11 and the Computed Value of r DEFINITION If there really is no linear correlation between two variables, the P-value is the probability of getting paired sample data with a linear correlation coefficient r that is at least as extreme as the one obtained from the paired sample data. Based on Example 4 and the Statdisk displayed results showing a P-value of 0.294, we know that there is a 0.294 probability (or a 29.4% chance) of getting a linear correlation coefficient of r = 0.591 or more extreme, assuming that there is no linear correlation between shoe print length and height. (The values of r that are “at least as extreme” as 0.591 are the values greater than or equal to 0.591 and the values less than or equal to -0.591.) Interpreting a P-Value The P-value of 0.294 from Example 4 is high. It shows that there is a high chance of getting a linear correlation coefficient of r = 0.591 (or more extreme) by chance when there is no linear correlation between the two variables. Because the likelihood of getting r = 0.591 or a more extreme value is so high (29.4% chance), we conclude that there is not sufficient evidence to conclude that there is a linear correlation between shoe print lengths and heights of males. Only a small P-value, such as 0.05 or less (or a 5% chance or less), suggests that the sample results are not likely to occur by chance when there is no linear correlation, so a small P-value supports a conclusion that there is a linear correlation between the two variables.
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