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If the relation between $x$ and $u$ is $3x+4u+7=0$ and the correlation coefficient between $x$ and $y$ is -0.6, then what is the correlation coefficient between $u$ and $y$?

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Last updated date: 21st Jul 2024
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Answer
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Hint: For solving this question you should know about the correlation coefficient. The correlation coefficient between variables is determined by solving the equation in which they are given. And then determine the value of correlation coefficient for them.

Complete step by step answer:
According to our question it is asked of us that if the relation between $x$ and $u$ is $3x+4u+7=0$ and the correlation coefficient between $x$ and $y$ is -0.6, then what is the correlation coefficient between $u$ and $y$. Now as we know that if we want to solve any equation or any expression, then we can add any term as a form of zero. It means that we will be doing addition and subtraction of the same term with the same signs, if we are adding or subtracting that from both sides of an equation. And if we add in only one side of the equation, then we add and subtract the same term with one negative and one positive sign. In this type of problem we will find one value from the equation, so we will take one side to our required value variable and remaining terms on the opposite side. So,
$\begin{align}
  & 3x+4u+7=0 \\
 & \therefore u=\dfrac{-3x-7}{4} \\
\end{align}$
We can write this as,
$u=\left( -\dfrac{3}{4} \right)x-\left( \dfrac{7}{4} \right)$
Therefore, perfect negative correlation between $x$ and $y$ and that is -0.6.
Therefore the correlation between $u$ and $y$ is,
$\begin{align}
  & =\dfrac{\left( -0.6 \right)\times \left( -\dfrac{3}{4} \right)}{\left( \dfrac{3}{4} \right)} \\
 & =0.6 \\
\end{align}$
So, the correlation between $u$ and $y$ is 0.6.

Note: While solving this type of questions we have to always add or subtract the term which will make it easy for the question. And if the variables are of higher order, then too we will make the higher ordered term to add or subtract in the equation.