Question

If $Var(x) = 8.25$, $Var(y) = 33.96$ and $Cov(x,y) = 10.2$ then the correlation coefficient is
$A)0.89$
$B) - 0.98$
$C)0.61$
$D) - 0.16$

Answer 1

Hint: First, we will need to know about the concept of the correlation coefficient.
The coefficient of the correlation is used to measure the relationship extent between $2$ separate intervals or variables.
Denoted by the symbol $r$. Where r is the value of positive or negative. Thus, this will further be generalized into the form of Pearson’s correlation coefficient.
We will simply use the formula of standard deviation and variance to get the correlation coefficient of the problem.
Formula used:
$r = \dfrac{{Cov(x,y)}}{{{\sigma _X}{\sigma _Y}}}$, where ${\sigma _X}$is the standard deviation of X and ${\sigma _Y}$ is the standard deviation of Y.

Complete step-by-step solution:
Since from the given that we have, $Var(x) = 8.25$, $Var(y) = 33.96$ and $Cov(x,y) = 10.2$
Now let us find the standard deviation for the X, which is ${\sigma _X} = \sqrt {VarX} = \sqrt {8.25} $
Similarly, we can also able to find the standard deviation for Y, which is ${\sigma _Y} = \sqrt {VarY} = \sqrt {33.96} $
Also, from the given that, we know $Cov(x,y) = 10.2$
Now substituting all the know values into the given correlation formula we get $r = \dfrac{{Cov(x,y)}}{{{\sigma _X}{\sigma _Y}}} \Rightarrow \dfrac{{10.2}}{{\sqrt {8.25} \sqrt {33.96} }}$
Since $\sqrt {8.25} = 2.872$ and $\sqrt {33.96} = 5.827$ , the values of the root
Thus, we get $r = \dfrac{{Cov(x,y)}}{{{\sigma _X}{\sigma _Y}}} = \dfrac{{10.2}}{{\sqrt {8.25} \sqrt {33.96} }} = \dfrac{{10.2}}{{2.872 \times 5.827}}$
With the help of multiplication operation, we get $r = \dfrac{{Cov(x,y)}}{{{\sigma _X}{\sigma _Y}}} = \dfrac{{10.2}}{{2.872 \times 5.827}} = \dfrac{{10.2}}{{16.735}}$
With the help of division operation, we get $r = \dfrac{{Cov(x,y)}}{{{\sigma _X}{\sigma _Y}}} = \dfrac{{10.2}}{{16.735}} = 0.609$ which is approximately $0.61$
Hence, the option $C)0.61$ is correct.

Note: The standard formula for the correlation coefficient:
Let us consider two different variables x and y that are related commonly, to find the extent of the link between the given numbers x and y, we will choose Pearson's coefficient r method.
In that process, the formula given is used to identify the extent or range of the two variables' equality.
Which is $r = \dfrac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {[n{{\sum {(y)} }^2} - (\sum {x{)^2}} ][n{{\sum {(y)} }^2} - (\sum {y{)^2}} ]} }}$
In this formula $r = \dfrac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {[n{{\sum {(y)} }^2} - (\sum {x{)^2}} ][n{{\sum {(y)} }^2} - (\sum {y{)^2}} ]} }}$
$\sum x $denotes the number of first variable values.
\[\sum y \] denotes the count of the second variable values.
${\sum x ^2}$ denotes the addition of a square for the first value.
\[\;{\sum y ^2}\] denotes the sum of the second values. And n denotes the total count data quantity.