Question

The scores of 8 students on the midterm exam and final exam were as follows. Final Student Anderson Bailey Cruz DeSana Erickson Francis Gray Harris Midterm 100 98 96 96 87 80 78 73 85 94 93 71 99 76 88 95 Find the value of the (Spearman's) rank correlation coefficient test statistic that would be used to test the claim of no correlation between midterm score and final exam score. Round your answer to 3 places after the decimal point, if necessary. Test statistic: rs = Submit Question

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Answer 1

Solution:

The provided data are shown in the table below, along with the corresponding ranks for the variables X and Y X Rank(X) Y Rank(Y) 100 8 85 3 98 7 94 6 96 5.5 93 5 96 5.5 71 1 87 4 99 8 80 3 76 2 78 2 88 4 73 1 95 7 Also, the following calculations involving the ranks are needed to compute the Spearman correlation coefficient: Rank(X) *Rank(Y Rank(X) Rank(Y) Rank(X) Rank(y) 2 2 ) 8 3 24 64 9 7 6 42 49 36 5.5 5 27.5 30.25 25 5.5 1 5.5 30.25 1 4 8 32 16 64 3 2 6 9 4 2 4 8 4 16 1 7 7 1 49 Sum = 36 36 152 203.5 204

The Spearman correlation coefficient rs is computed using the same calculations as those used for the computation of Pearson's correlation, but in this case, the ranks are used instead of the actual values. The following formula is used: SSXY rs = VSSxxSSyy where 1 2 5880 = Rank X) Rank(m) - ( Rankex) ( Rankyo) 988 = § Rent(x:39 - ( Hanke x.) į Rank(Y) 1 2 n SSỹỹ Rank(Y;)? 1 п i=1 i=1 In this case, based on the data provided, we get that SSŤy = 152 (36 x 36) = 152 8 1 $$xx = 203.5 – (36)2 = 203.5 SSyy = 204 – (36) 1 (36)2 = 204 Therefore, based on this information, the sample Spearman correlation coefficient is computed as follows SSKỸ 152 rs = -0.24 VSSÝSSÝŤ 203.5 x 204