Question

$n=25,\sum x=125,\sum x^2=650,\sum y=100,\sum y^2=460,\sum xy=508$. It was observed that two pair of values of $(x,y)$ were copied as $(6,14)$ and $(8,6)$ instead of $(8,12),(6,8).$ The correct correlation coefficient is

• A. $0.667$
• B. $0.87$
• C. $-0.25$
• D. $0.356$

Answer 1

The correct option is A $0.667$

Corrected $\sum x=125-6-8+8+6=125$

Corrected $\sum y=100-14-6+12+8=100$

Corrected $\sum x^2=650-(6{)}^2-(8{)}^2+(6{)}^2+(8{)}^2=650$

Corrected $\sum y^2=460-(14{)}^2-(6{)}^2+(12{)}^2+(8{)}^2=436$

Corrected $\sum xy=508-6\times 18-8\times 6+8\times 12+6\times 8=520$

The formula for Pearson product moment correlation is

$r=\frac{n\sum xy-(\sum x)(\sum y)}{\sqrt{[n\sum x^2-(\sum x{)}^2\left]\right[n\sum y^2-(\sum y{)}^2]}}$

$=\frac{25\times 520-125\times 100}{\sqrt{[25\times 650-(125{)}^2\left]\right[25\times 436-\left(100{)}^2\right]}}$

$=\frac{13000-12500}{\sqrt{[16250-15625][10900-10000]}}$

$=\frac{500}{\sqrt{\left[625\right]\left[900\right]}}$

$=\frac{500}{\left(25\right)\left(30\right)}$

$=\frac{2}{3}$

$=0.667$