Statistics for People Who (Think They) Hate Statistics
Statistics for People Who (Think They) Hate Statistics
6th Edition
ISBN: 9781506333830
Author: Neil J. Salkind
Publisher: SAGE Publications, Inc
Question
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Chapter 5, Problem 8TP
To determine

The correlation between the variables and formulate a matrix.

Expert Solution & Answer
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Answer to Problem 8TP

The correlation between XY, YZ and ZX are rXY=0.055719 , rXZ=0.15432 and rYZ=0.389126 respectively. And correlation matrix is,

The matrix showing relationship between variables is,

X Y Z
1
Y 0.055719 1
Z –0.15432 0.389126 1

Explanation of Solution

Given info:

The table below shows the value of three variables age of injury (X), level of treatment (Y), 12-month treatment score (Z),

Age at Injury Level of Treatment 12-month Treatment score
X Y Z
25 1 78
16 2 66
8 2 78
23 3 89
31 4 87
19 4 90
15 4 98
31 5 76
21 1 56
26 1 72
24 5 84
25 5 87
36 4 69
45 4 87
16 4 88
23 1 92
31 2 97
53 2 69
11 3 79
33 2 69

Calculation:

The correlation between the variables X, Y and Z can be calculated with formulas given below,

rXY=n(XY)(X)(Y)[nX2(X)2][nY2(Y)2] (1)

rXZ=n(XZ)(X)(Z)[nX2(X)2][nZ2(Z)2] (2)

rZY=n(ZY)-(Z)(Y)[nZ2-(Z)2][nY2-(Y)2] (3)

Where,

  • r is the coefficient correlation.
  • rXY is the correlation between variable X and Y.
  • rXZ is the correlation between variable X and Z.
  • rYZ is the correlation between variable Y and Z.
  • X is variable age of injury.
  • Y is variable level of treatment.
  • Z is variable 12-month treatment score.

Calculate the values of all the variables as shown in figure below,

X Y Z XY YZ ZX X2 Y2 Z2
25 1 78 25 78 1950 625 1 6084
16 2 66 32 132 1056 256 4 4356
8 2 78 16 156 624 64 4 6084
23 3 89 69 267 2047 529 9 7921
31 4 87 124 348 2697 961 16 7569
19 4 90 76 360 1710 361 16 8100
15 4 98 60 392 1470 225 16 9604
31 5 76 155 380 2356 961 25 5776
21 1 56 21 56 1176 441 1 3136
26 1 72 26 72 1872 676 1 5184
24 5 84 120 420 2016 576 25 7056
25 5 87 125 435 2175 625 25 7569
36 4 69 144 276 2484 1296 16 4761
45 4 87 180 348 3915 2025 16 7569
16 4 88 64 352 1408 256 16 7744
23 1 92 23 92 2116 529 1 8464
31 2 97 62 194 3007 961 4 9409
53 2 69 106 138 3657 2809 4 4761
11 3 79 33 237 869 121 9 6241
33 2 69 66 138 2277 1089 4 4761
Total 512 59 1611 1527 4871 40882 15386 213 132149

Substitute 1527 for XY , 512 for X , 15386 for X2 , 213 for Y2 and 59 for (Y)2 in the equation 1 to get the correlation between X and Y.

rXY=20(1527)(512)(59)[20(15386)(512)2][20(213)(59)2]=3054030208307720262144×42603481=332213.4853×27.9105=0.055719

Thus, the correlation between X and Y is 0.055719.

rXZ=20(40882)(512)(1611)[20(15386)(512)2][20(132149)(1611)2]=0.15432

Thus the correlation between X and Z is -0.15432.

rZY=20(4871)-(1611)(59)[20(132149)(1611)2][20(213)(59)2]=0.389126

Thus, the correlation between Z and Y is 0.389126.

Conclusion:

The correlation between variables age of injury and level of treatment and level of treatment and 12-month treatment score is positive, but the correlation between level of treatment and 12-month treatment score is negative. This indicates that with increase in the age of injury, the level of treatment done also increases. Whereas, with increase in level of treatment, the 12-months score of treatment do not increase.

The matrix of correlation can thus be formulated as given below:

X Y Z
1
Y 0.055719 1
Z –0.15432 0.389126 1

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