10-1 Correlation 519 The linear correlation coefficient is r = 0.947 and n = 9 (because there are 9 pairs of sample data), so the test statistic is t = r B1 - r2 n - 2 = 0.947 B1 - 0.9472 9 - 2 = 7.800 With n - 2 = 7 degrees of freedom, Table A-3 shows that the test statistic of t = 7.800 yields a P-value that is less than 0.01. Technologies show that the P-value is 0.000 when rounded. Because the P-value of 0.000 is less than the significance level of 0.05, we reject H0. (“If the P is low, the null must go.” Yup, the P-value of 0.000 is low.) YOUR TURN. Do Exercise 17 “Taxis.” INTERPRETATION This formal hypothesis test results in the same conclusion as Example 4. It appears that there is a linear correlation between lottery jackpots amounts and numbers of tickets sold. It appears that higher jackpots correspond to more tickets sold. One-Tailed Tests The examples and exercises in this section generally involve twotailed tests, but one-tailed tests can occur with a claim of a positive linear correlation or a claim of a negative linear correlation. In such cases, the hypotheses will be as shown here. Claim of Negative Correlation (Left-Tailed Test) Claim of Positive Correlation (Right-Tailed Test) H0: r = 0 H0: r = 0 H1: r 6 0 H1: r 7 0 For these one-tailed tests, the P-value method can be used as in earlier chapters. Rationale for Methods of This Section We have presented Formulas 10-1 and 10-2 for calculating r and have illustrated their use. Those formulas are given below along with some other formulas that are “equivalent,” in the sense that they all produce the same values. FORMULA 10-1 r = nΣxy - 1Σx21Σy2 2 n1Σx22 - 1Σx222n1Σy22 - 1Σy22 FORMULA 10-2 r = Σ1zx zy2 n - 1 r = Σ1x - x21y - y2 1n - 12sx sy r = ac 1x - x2 sx 1y - y2 sy d n - 1 r = sxy 1sxx 1syy We will use Formula 10-2 to help us understand the reasoning that underlies the development of the linear correlation coefficient. Because Formula 10-2 uses z scores, the value of Σ1zxzy2 does not depend on the scale that is used for the x and y values. Figure 10-4 shows the scatterplot of the paired data from Table 10-1 after the original s Palm Reading Some people believe that the length of their palm’s lifeline can be used to predict longevity. In a letter published in the Journal of the American Medical Association, authors M. E. Wilson and L. E. Mather refuted that belief with a study of cadavers. Ages at death were recorded, along with the lengths of palm lifelines. The authors concluded that there is no correlation between age at death and length of lifeline. Palmistry lost, hands down.
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