Question

Problem 1

Calculate and interpret the correlation coefficient of the two variables below.

Person

Hand

Height

A

17

150

B

15

154

C

19

169

D

17

172

E

21

175

Answer 1

Pearson Correlation coefficient

$$r = \frac{\sum(x_i - \bar x) (y_i - \bar y)}{\sum(x_i - \bar x)^2 \sum(y_i - \bar y)^2}$$

Let $x\ be\ hand \ and \ y\ be\ height$

$$\bar x =\frac{17+15+19+17+21}{5}= 17.8$$

$$\bar y = \frac{150+154+169+172+175}{5}=164$$

$$\sum(x_i - \bar x) (y_i - \bar y)=(17-17.8)(150-164)+(25-17.8) (154-164)+(19-17.8)(169-164)+(17-17.8)(172-164)+(21-17.8)(175-164)$$ $$=-26$$

$$\sum(x_i - \bar x)^2=(17-17.8)^2+(25-17.8)^2+(19-17.8)^2+(17-17.8)^2+(21-17.8)^2$$ $$=64.8$$

$$\sum (y_i - \bar y)^2=(150-164)^2+ (154-164)^2+(169-164)^2+(172-164)^2+(175-164)^2$$ $$=506$$

Therefore

$$r=\frac{-26}{506*64.8}=-0.000793$$

Since we have 10 data points, the degrees of freedom are

$$10-2=8$$

The critical value for the correlation coefficient with 8 degrees of freedom is

$\underset{-}{+} 0.632.$ We conclude that the correlation is significant if it is greater than the critical value. In our case, $-0.000793 > - 0.632$ therefore we can conclude that there is a significant correlation between hand and height