680 CHAPTER 13 Nonparametric Tests TABLE 13-1 Ranks and Costs of Smartphones Quality Rank 1 2 3 4 5 6 7 8 9 10 Cost (dollars) 1000 1100 900 1000 750 1000 900 700 750 600 TABLE 13-7 Ranks of All Data from Table 13-1 Quality Rank 12345678910 Cost Rank 8 105.5 8 3.5 8 5.5 2 3.5 1 Table 13-1 from the Chapter Problem lists ranks and costs (dollars) of smartphones (based on data from Consumer Reports). Lower ranks correspond to better smartphones. Find the value of the rank correlation coefficient and use it to determine whether there is sufficient evidence to support the claim of a correlation between quality and price. Use a 0.05 significance level. Based on the result, does it appear that you get a better quality smartphone by spending more? CP EXAMPLE 1 Do Better Smartphones Cost More? SOLUTION REQUIREMENT CHECK The sample data are a simple random sample from the smartphones that were tested. The data are ranks or can be converted to ranks. The quality ranks are consecutive integers and are not from a population that is normally distributed, so we use the rank correlation coefficient instead of the linear correlation coefficient to test for a relationship between quality and price. The null and alternative hypotheses are as follows: H0: rs = 0 1There is no correlation between quality and price.2 H1: rs ≠ 0 1There is a correlation between quality and price.2 Following the procedure of Figure 13-4, we begin by converting the costs in Table 13-1 into their corresponding ranks shown in Table 13-7. The lowest cost of $600 in Table 13-1 is assigned a rank of 1, the next lowest cost of $700 is assigned a rank of 2, and so on. When ties occur, each of the tied values is assigned the mean of the ranks involved in the tie. For example, there are two costs of $750 and they are tied for the ranks of 3 and 4, so they are both assigned a rank of 3.5. The ranks corresponding to the costs from Table 13-1 are shown in the second row of Table 13-7. Because there are ties among ranks, we must use Formula 10-1 to find that the rank correlation coefficient rs is equal to -0.796. FORMULA 10-1 rs = nΣxy - 1Σx21Σy2 2 n1Σx22 - 1Σx222n1Σy22 - 1Σy22 = 1012382 - 15521552 2 1013852 - 1552221013822 - 15522 = -0.796 Now we refer to Table A-9 to find the critical values of {0.648 (based on a = 0.05 and n = 10). Because the test statistic rs = -0.796 is outside of the range between the critical values of -0.648 and 0.648, we reject the null hypothesis. There is sufficient evidence to support a claim of a correlation between quality and cost. It appears that you do get better quality by paying more, but this conclusion incorrectly implies causation. YOUR TURN. Do Exercise 7 “Colombian Coffee.”
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