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BUAD 300Prof. RobinsonRank Correlation Following your text, the rank correlation coefficient is given by equation (1). This test is for paired data of n observations. r = 1 – 6di2n(n2-1) where di is the difference in the ranks.(1)The example in your text shows the “potential” for ten different salespeople as evaluated at the time of hire by some set of criteria. These “potentials” are ranked from 1 to 10, and the “actual sales” for a two-year period after hire is then also ranked for the same ten salespeople. The rankings of “potential” and the “actual sales” is given in the table below. Table 1: Rankings of “Sales Potential” and “Actual Sales”Sales PersonPotential RanksActual Sales Ranksdd2121112431137524416-525567-11634-11710100089811989-11105239di2 = 44r = 1 – 6di2n(n2-1) = 1 – 6(44)10(102-1) = .733In order to show the ability of the formula to fit the range of -1 ≤ r ≤ +1, two illustrations of the extremes are presented here. Suppose these same rankings were given below where Table 2 presents an obvious perfect positive correlation ( r = +1), and Table 3 gives a perfect inverse correlation (r = -1). Table 2: Ranks for Potential and SalesPotentialSalesdd211002200330044005500660077008800990010100 0 ____ Σdi2 = 0r = 1 – 6di2n(n2-1) = 1 – 6(0)10(100-1) = 1Table 3: Ranks for Potential and SalesPotentialSalesdd2110-98129-74938-52547-3956-1165+1174+3983+52592+749101+9 81 ____ Σdi2 = 330r = 1 – 6di2n(n2-1) = 1 – 6(330)10(100-1) = 1 – 1980990= 1 – 2 = -1These extreme example calculations of r indicate that the formula functions as we would want. The range for r is given by -1 ≤ r ≤ +1.Since the calculations are from sample data, the interesting question concerns “Is the population coefficient different from 0?” Using the sample estimate such of “.733” presented above, we calculate the z statistic as in equation (2).z = sample estimate of r - 0σr where σr = [1n-1]1/2 (2)As a result,σr = (1/n-1)1/2 = (1/9)1/2 = 1/3 and zcalc = sample estimate of r - 0σr = .733 - 01/3 ≈ 2.2H0: “Potential” and “actual sales” are unrelated.Since this is a two-tailed test, because r can be very different from 0 in either the positive or negative directions, then we would reject H0. The critical value for the two-tailed test iszcritical = 1.96. The decision rule is given below.Decision rule: If │zcalc │> zcritical then reject H0.Decision: Reject H0.Review problem due as assigned: Do problem 32, on page 903 of the 12th edition.Answer: r = .768, and zcalc = 2.43. ................
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