514 CHAPTER 10 Correlation and Regression TABLE 10-3 Calculating r with Formula 10-2 x (Jackpot) y (Tickets) zx zy zx ~ zy 334 54 1.6894 2.1172 3.57680 127 16 -1.2465 -0.8143 1.01502 300 41 1.2071 1.1143 1.34507 227 27 0.1718 0.0343 0.00589 202 23 -0.1828 -0.2743 0.05014 180 18 -0.4948 -0.6600 0.32657 164 18 -0.7218 -0.6600 0.47639 145 16 -0.9912 -0.8143 0.80713 255 26 0.5689 -0.0429 -0.02441 Σzx # zy = 7.57862 Use Formula 10-2 to find the value of the linear correlation coefficient r for the Powerball jackpot amounts and numbers of tickets listed in Table 10-1. CP YOUR TURN. Do Exercise 13 “Powerball Jackpots and Tickets Sold.” EXAMPLE 3 Finding r Using Formula 10-2 SOLUTION If manual calculations are absolutely necessary, Formula 10-1 is much easier than Formula 10-2, but Formula 10-2 has the advantage of making it easier to understand how r works. (See the rationale for r discussed later in this section.) As in Example 2, the variable x is used for the jackpot amount, and the variable y is used for the numbers of lottery tickets sold. In Formula 10-2, each sample value is replaced by its corresponding z score. For example, using unrounded numbers, the jackpots have a mean of x = 214.8889 and a standard deviation of sx = 70.5061, so the first x value of 334 is converted to a z score of 1.6894, as shown here: zx = x - x sx = 334 - 214.8889 70.5061 = 1.6894 Table 10-3 lists the z scores for all of the jackpot amounts (see the third column) and the z scores for all of the numbers of tickets (see the fourth column). The last column of Table 10-3 lists the products zx # zy. Using Σ1zx # zy2 = 7.57862 from Table 10-3, the value of r is calculated by using Formula 10-2, as shown below. r = Σ1zx # zy2 n - 1 = 7.57862 9 - 1 = 0.947 Is There a Linear Correlation? We know from the preceding three examples that the value of the linear correlation coefficient is r = 0.947 for the data in Table 10-1. We now proceed to interpret the meaning of r = 0.947 from the nine pairs of data listed in Table 10-1, and our goal in this section is to decide whether there appears to be a linear correlation between lottery jackpot amounts and numbers of tickets sold. Using the criteria given in the preceding Key Elements box, we can base our interpretation on a P-value or a critical value from Table A-6. (See the criteria for “Interpreting the Linear Correlation Coefficient r” given in the preceding Key Elements box on page 511.) C Go Figure 15 Billion: The number of years it would take the atomic clock in Boulder, Colorado, to be off by one second.
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