Question

he following table shows the incomes of super 10 taxis by year

YEAR x-CODE

2001 1

2002 2

2003 3

2004 4

2005 5

Total 15

a) Plot the data on a scatter plot diagram

b) Compute Pearson’s correlation coefficient (r)

c) Compute Spearman’s ranking (rho)

        

Answer 1

(a) Scatter Plot of the data is-

(b)

$E(XY)=\dfrac{\sum XY}{n}=\dfrac{30055}{5}=6011 \\[9pt] E(X)=\dfrac{\sum X}{n}=\dfrac{10015}{5}=2003 \\[9pt] E(Y)=\dfrac{\sum Y}{n}=\dfrac{15}{5}=3 \\[9pt] \sigma_X=\sqrt{\dfrac{ (x-\bar{x})^2}{n}}=\sqrt{\dfrac{10}{5}}=\sqrt{2} \\[9pt] \sigma_Y=\sqrt{\dfrac{(y-\bar{y})^2}{n}}=\sqrt{\dfrac{10}{5}}=\sqrt{2}$

Pearson correlation coefficient $r=\dfrac{Cov(X,Y)}{\sigma_X,\sigma_Y}$

$=\dfrac{E(XY)-E(X)E(Y)}{\sigma_X.\sigma_Y} \\[9pt] =\dfrac{6011-2003(3)}{\sqrt{2}.\sqrt{2}}\\[9pt] =\dfrac{2}{2}=1$

(c) spearmen rank correlation coefficient-

$=1-\dfrac{6d^2}{n^3-n}$

where, d= Difference between x and Y

$=1-\dfrac{6\times 20000000}{(5)^3-5}$

$=1-\dfrac{1200000000}{120}=1-10^6$