12.6 Linear Correlation and Regression 827 If the absolute value of r, written r, is greater than the value given in the table under the specified α and appropriate sample size n, we assume that a correlation does exist between the variables. If r is less than the table value, we assume that no correlation exists. Returning to Example 1, if we want to determine whether there is a correlation at a 5% level of significance, we locate the critical value (or cutoff value) that corresponds to n 6 = (there are 6 pieces of bivariate data) and 0.05. α= The value to the right of n 6 = and under the 0.05 α= column is the critical value 0.811, circled in red in Table 12.9. From the formula, we had obtained r 0.922. = Since 0.922 0.811, > or 0.922 0.811, > we assume that a correlation between the variables exists. Note in Table 12.9 that the larger the sample size, the smaller is the value of r needed for a significant correlation. Also, Table 12.9 may be used only under certain conditions. If you take a statistics course, you will learn more about which critical values to use to determine whether a linear correlation exists. Example 2 Amount of Drug Remaining in the Bloodstream To test the length of time that an infection-fighting drug stays in a person’s bloodstream, a doctor gives 300 milligrams of the drug to 10 patients, labeled 1–10 in the table below. Once each hour, for 10 hours, one of the 10 patients is selected at random and that person’s blood is tested to determine the amount of the drug remaining in the bloodstream. The results are as follows. Patient 12345678910 Time (hr) 12345678910 Drug remaining (mg) 250 230 200 210 140 120 210 100 90 85 Determine at a level of significance of 5% whether a correlation exists between the time elapsed and the amount of drug remaining. Solution Let time be represented by x and the amount of drug remaining by y. We first draw a scatter diagram (Fig. 12.42). Time (hr) 250 225 200 175 150 125 100 75 12345678910 Drug Remaining (mg) Figure 12.42 The scatter diagram suggests that, if a correlation exists, it will be negative. We now construct a table of values and calculate r. Table 12.9 Critical Values for the Correlation Coefficient, r The derivation of this table is beyond the scope of this text. It shows the critical values of the Pearson correlation coefficient. n 005 . α = 001 . α = 4 0.950 0.990 5 0.878 0.959 6 0.811 0.917 7 0.754 0.875 8 0.707 0.834 9 0.666 0.798 10 0.632 0.765 11 0.602 0.735 12 0.576 0.708 13 0.553 0.684 14 0.532 0.661 15 0.514 0.641 16 0.497 0.623 17 0.482 0.606 18 0.468 0.590 19 0.456 0.575 20 0.444 0.561 22 0.423 0.537 27 0.381 0.487 32 0.349 0.449 37 0.325 0.418 42 0.304 0.393 47 0.288 0.372 52 0.273 0.354 62 0.250 0.325 72 0.232 0.302 82 0.217 0.283 92 0.205 0.267 102 0.195 0.254
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