**16. Gravetter/Wallnau/Forzano, Essentials - Chapter 14 - End-of-chapter question 16**Judge and Cable (2010) report the results of a study demonstrating a negative relationship between weight and income for a group of men. Following are data similar to those obtained in the study. To simplify the weight variable, the men are classified into five categories that measure actual weight relative to height, from 1 = thinnest to 5 = heaviest. Income figures are annual income (in thousands), rounded to the nearest $1,000.| Weight | Income ||--------|--------|| 4 | 151 || 5 | 88 || 3 | 52 || 2 | 73 || 1 | 49 || 3 | 92 || 1 | 56 || 5 | 143 |These data show a positive relationship between weight and income for a sample of men. However, weight was coded in five categories, which could be viewed as an ordinal scale rather than an interval or ratio scale. If so, a Spearman correlation is more appropriate than a Pearson correlation. Convert the weights and the incomes into ranks and complete the table that follows.| Ranks | X | Y | XY ||-------|---|---|----|| | 8 | | || | 7 | | || | 6 | | || | 5 | | || | 4 | | || | 3 | | || | 2 | | || | 1 | | | ### Educational Website Transcription**Understanding Spearman Correlation**These data show a positive relationship between weight and income for a sample of men. However, weight was coded in five categories, which could be viewed as an ordinal scale rather than an interval or ratio scale. If so, a Spearman correlation is more appropriate than a Pearson correlation. Convert the weights and the incomes into ranks and complete the table that follows.#### Table: Ranks```X Y XY------------8 87 76 65 54 43 32 21 1```**Instructions:**Compute the following based on the ranks:- ΣX- ΣY- ΣXY- SSx- SSy- SP- Spearman correlation (rₛ)**Question:**Is this correlation statistically significant? Use a two-tailed test with α = .05.#### Critical Values TableThis table provides the critical values for determining statistical significance at different significance levels for both one-tailed and two-tailed tests.| df = n - 2 | .10 | .05 | .02 | .01 ||------------|------|------|------|------|| 2 | .900 | .950 | .980 | .990 || 3 | .805 | .878 | .934 | .959 || 4 | .729 | .811 | .882 | .917 || 5 | .669 | .754 | .833 | .874 || 6 | .622 | .707 | .789 | .834 |This table helps to determine if the Spearman correlation is significant for a given degree of freedom by comparing the calculated correlation value against these critical values.**Analysis:**Evaluate the computed Spearman correlation against critical values to ascertain statistical significance.

Given information: Given data represents the values of the variables weight and income.Make a table of ranks to be used in a Spearman rank correlation: The table of ranks to be used in a Spearman rank correlation test are given below: Weight X Income Y XY (Xi – Xbar)2 (Yi – Ybar)2 (Xi – Xbar)* (Yi – Ybar) 4 6 151 8 48 2.25 12.25 5.25 5 7.5 143 7 52.5 9 6.25 7.5 3 4.5 92 6 27 0 2.25 0 5 7.5 88 5 37.5 9 0.25 1.5 2 3 73 4 12 2.25 0.25 0.75 1 1.5 56 3 4.5 9 2.25 4.5 3 4.5 52 2 9 0 6.25 0 1 1.5 49 1 1.5 9 12.25 10.5 Total 36 36 192 40.5 42 30 The mean of X is Xbar = 36/8 = 4.5. The mean of Y is Ybar = 36/8 = 4.5. The required values are as shown below: ∑X=36∑Y=36∑XY=192SSX=40.5SSY=42SP=30Spearman rank correlation coefficient: The spearman rank correlation coefficient is obtained as 0.7274 from the calculation given below: rs=SPSSX×SSY=3040.5×42=301701=0.7274 Thus, the spearman rank correlation coefficient is 0.7274. The level of significance is given as α = 0.05. The sample size is n = 8. The degrees of freedom = d.f = n - 2 = 8 - 2 =6 The critical value corresponding to d.f = 6 and α = 0.05 is 0.707. Here, the correlation coefficient is greater than the critical value. Therefore, the correlation is significant.