Question

Which one of the following can be considered as appropriate pair of values of regression of ${\text{y}}$ on ${\text{x}}$ and regression coefficient of \[{\text{x}}\] on ${\text{y?}}$
${\text{A}}{\text{.}}$$\left( {1,1} \right)$
${\text{B}}{\text{.}}$$\left( { - 1,1} \right)$
${\text{C}}{\text{.}}$$\left( {\dfrac{{ - 1}}{2},2} \right)$
${\text{D}}{\text{.}}$ $\left( {\dfrac{1}{3},\dfrac{{10}}{3}} \right)$

Answer 1

Hint: From the question, here we have to choose one of the correct options as appropriate pair of values of regression of ${\text{y}}$ on ${\text{x}}$ and regression coefficient of ${\text{x}}$ on ${\text{y}}$. We have to find the correct solution by using some properties of regression coefficient in the below mentioned formula.

Complete step-by-step solution:
Let us understand the properties of regression first
Some of the properties of regression coefficient are given below:
$1.$ It is generally denoted by $'{\text{b}}'$ and it can be expressed in the form of an original unit of data.
$2.$ If two variables are they say ${\text{x}}$ and ${\text{y}}$ two values of the regression coefficient are obtained. One will be obtained when ${\text{x}}$ is independent and ${\text{y}}$ is dependent and the other when we consider ${\text{y}}$ as independent and ${\text{x}}$ as a dependent. The regression coefficient of ${\text{y}}$ on ${\text{x}}$ is represented by ${{\text{b}}_{{\text{yx}}}}$ and ${\text{x}}$ on ${\text{y}}$ is represented as ${{\text{b}}_{{\text{xy}}}}$.
$3.$ Both of the regression coefficients must have the same sign. If ${{\text{b}}_{{\text{xy}}}}$ is positive, ${{\text{b}}_{{\text{yx}}}}$ will also be positive and it is true for vice versa.
$4.$ If one regression coefficient is greater than unity, then others will be lesser than unity. The product of regression of ${\text{y}}$ on ${\text{x}}$ and regression coefficient of \[{\text{x}}\] on ${\text{y}}$ is always less than or equal to $1$.
$5.$ Linear regression is one of the most popular techniques in statistics. Despite its popularity, it is sometimes difficult to interpret regression coefficients.
By using the property of regression coefficient, we have to check the given option which is suitable to the given conditions of the property.
Option ${\text{A}}$$ - \left( {1,1} \right)$ be the correct answer. Because, the terms $1$ and $1$ contain the same positive sign and their product is $1$ then it is equal to $1$.
Option ${\text{B}}$ and ${\text{C}}$ be wrong. Because, the given both terms are containing the opposite signs and their product is less than $1$.
Option ${\text{D}}$ also wrong one. Because, the terms contain the same sign but its product is greater than $1$.

$\therefore $ The option ${\text{A}}$ is the correct option.

Note: The regression coefficients are a statistical measure which is used to measure the average functional relationship between variables. In regression analysis, one variable is dependent and other is independent. Also, it measures the degree of dependence of one variable on the others.
The given question is easy to choose the correct one. The students should concentrate on the properties of regression coefficients and on checking the given terms with comparison on the way of properties of regression coefficients.