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 $\left( x,y \right)$ were copied as $\left( 6,14 \right)$ and $\left( 8,6 \right)$ instead of $\left( 8,12 \right)$, $\left( 6,8 \right)$. The correct correlation coefficient is
A. $0.667$
B. $0.87$
C. $-0.25$
D. $0.356$

Answer 1

In this problem we need to calculate the value of correct correlation coefficient for the given set of data with the given conditions. We can observe that some of the data is corrupted or copied wrong. So we will calculate the actual values of $\sum{x}$, $\sum{{{x}^{2}}}$, $\sum{y}$, $\sum{{{y}^{2}}}$, $\sum{xy}$ by subtracting the respective values which are mistakenly copied and adding the actual values at the same time. After having the actual values of $\sum{x}$, $\sum{{{x}^{2}}}$, $\sum{y}$, $\sum{{{y}^{2}}}$, $\sum{xy}$ for the true data set we will use the formula $r=\dfrac{n\sum{xy}-\left( \sum{x} \right)\times \left( \sum{y} \right)}{\sqrt{\left[ n\sum{{{x}^{2}}}-{{\left( \sum{x} \right)}^{2}} \right]\left[ n\sum{{{y}^{2}}}-{{\left( \sum{y} \right)}^{2}} \right]}}$ to find the correlation coefficient.

Complete step by step answer:
Given that, $n=25$, $\sum{x}=125$, $\sum{{{x}^{2}}}=650$, $\sum{y}=100$, $\sum{{{y}^{2}}}=460$, $\sum{xy}=508$ and
two pair of values of $\left( x,y \right)$ were copied as $\left( 6,14 \right)$ and $\left( 8,6 \right)$ instead of $\left( 8,12 \right)$, $\left( 6,8 \right)$.
If the data set is copied as $\left( {{x}_{1}},{{y}_{1}} \right)$ instead of $\left( {{x}_{2}},{{y}_{2}} \right)$, then the respect values of $\sum{x}$ changed as
$\sum{{{x}_{act}}}=\sum{x}-{{x}_{1}}+{{x}_{2}}$
So, the data set is copied $\left( 6,14 \right)$ and $\left( 8,6 \right)$ instead of $\left( 8,12 \right)$, $\left( 6,8 \right)$, then the values are modified as
$\sum{{{x}_{act}}}=\sum{x}-6-8+8+6$
Substituting the value $\sum{x}=125$ in the above equation, then we will get
$\begin{aligned} & \sum{{{x}_{act}}}=125+0 \\\ & \Rightarrow \sum{{{x}_{act}}}=125 \\\ \end{aligned}$
Now the actual or correct value of $\sum{y}$ is given by
$\begin{aligned} & \sum{{{y}_{act}}}=\sum{y}-14-6+12+8 \\\ & \Rightarrow \sum{{{y}_{act}}}=\sum{y} \\\ & \Rightarrow \sum{{{y}_{act}}}=100 \\\ \end{aligned}$
Now the actual or correct value of $\sum{{{x}^{2}}}$ is given by
$\begin{aligned} & \sum{{{x}^{2}}_{act}}=\sum{{{x}^{2}}}-{{6}^{2}}-{{8}^{2}}+{{6}^{2}}+{{8}^{2}} \\\ & \Rightarrow \sum{{{x}^{2}}_{act}}=\sum{{{x}^{2}}} \\\ & \Rightarrow \sum{{{x}^{2}}_{act}}=650 \\\ \end{aligned}$
Now the actual or correct value of $\sum{{{y}^{2}}}$ is given by
$\begin{aligned} & \sum{{{y}^{2}}_{act}}=\sum{{{y}^{2}}}-{{14}^{2}}-{{6}^{2}}+{{12}^{2}}+{{8}^{2}} \\\ & \Rightarrow \sum{{{y}^{2}}_{act}}=460-24 \\\ & \Rightarrow \sum{{{y}^{2}}_{act}}=436 \\\ \end{aligned}$
Now the actual or correct value of $\sum{xy}$ is given by
$\begin{aligned} & \sum{x{{y}_{act}}}=\sum{xy}-6\times 8-8\times 6+8\times 12+6\times 8 \\\ & \Rightarrow \sum{x{{y}_{act}}}=508+12 \\\ & \Rightarrow \sum{x{{y}_{act}}}=520 \\\ \end{aligned}$
From the all the above values the correlation coefficient will be calculated as
$r=\dfrac{n\sum{xy}-\left( \sum{x} \right)\times \left( \sum{y} \right)}{\sqrt{\left[ n\sum{{{x}^{2}}}-{{\left( \sum{x} \right)}^{2}} \right]\left[ n\sum{{{y}^{2}}}-{{\left( \sum{y} \right)}^{2}} \right]}}$
Substituting all the actual or corrected values in the above equation, then we will get
$r=\dfrac{25\times 520-125\times 100}{\sqrt{\left( 25\times 650-{{125}^{2}} \right)\left( 25\times 436-{{100}^{2}} \right)}}$
Simplifying the above equation, then we will get
$\begin{aligned} & r=\dfrac{13000-12500}{\sqrt{\left( 16250-15625 \right)\left( 10900-10000 \right)}} \\\ & \Rightarrow r=\dfrac{500}{\sqrt{625\times 900}} \\\ & \Rightarrow r=\dfrac{500}{25\times 30} \\\ & \Rightarrow r=0.667 \\\ \end{aligned}$

So, the correct answer is “Option A”.

Note: In type of problem needs the student attention. While calculating the corrected values one may do a lot of mistakes like substituting the wrong values or use $x$ values instead of $y$ values or vice versa. So, one should be careful about substitution while calculating the corrected values.