Question

In a biology class, a group of students brought 9 mice for their experiment. They measured the weight and the length of the body of each mouse from the tail to the nose. The findings are recorded below:

Length in cm

4

5

5

5

7

8

9

9

10

Weight in kg

6

7

8

10

10

11

12

12

13

Calculate the Pearson correlation coefficient. Round your answers to the nearest hundredths.

Answer 1

In order to compute the regression coefficients, the following table needs to be used:

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2 \\ \hline & 4 & 6 & 24 & 16 & 36 \\ \hdashline & 5 & 7 & 35 & 25 & 49 \\ \hdashline & 5 & 8 & 40 & 25 & 64 \\ \hdashline & 5 & 10 & 50 & 25 & 100 \\ \hdashline & 7 & 10 & 70 & 49 & 100 \\ \hdashline & 8 & 11 & 88 & 64 & 121 \\ \hdashline & 9 & 12 & 108 & 81 & 144 \\ \hdashline & 9 & 12 & 108 & 81 & 144 \\ \hdashline & 10 & 13 & 130 & 100 & 169 \\ \hdashline Sum= & 62 & 89 & 653 & 466 & 927 \\ \hdashline \end{array}$$ $$\bar{X}=\dfrac{1}{n}\sum _{i}X_i=\dfrac{62}{9}=6.89$$

$$\bar{Y}=\dfrac{1}{n}\sum _{i}Y_i=\dfrac{89}{9}=9.89$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum _{i}X_i)^2$$ $$=466-\dfrac{62^2}{9}=38.89$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum _{i}Y_i)^2$$ $$=927-\dfrac{89^2}{9}=46.89$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum _{i}X_i)(\sum _{i}Y_i)$$ $$=653-\dfrac{62(89)}{9}=39.89$$

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}SS_{YY}}}=\dfrac{653-\dfrac{62(89)}{9}}{\sqrt{(466-\dfrac{62^2}{9})(927-\dfrac{89^2}{9})}}$$

$$=0.93$$