Question

In a bivariate data, \[\sum{x=30,\sum{y=400,\sum{{{x}^{2}}=196,\sum{xy=850}}}}\] and \[n=10\]. The regression coefficient of \[y\] on \[x\] is
A). \[-3.1\]
B). \[-3.2\]
C). \[-3.3\]
D). \[-3.4\]

Answer 1

Hint: To solve this question you must have the basic knowledge about the regression coefficients, dependent and independent variables. In the question we have to calculate the regression coefficient of \[y\] on \[x\] i.e. \[{{b}_{yx}}\] use the formula for the same and substitute the values given in the question.

Complete step-by-step solution:
There are three types of analysis on the statistical data. First one is univariate analysis i.e. for only one variable. Second is bivariate analysis i.e. analysis for the two variables. And the last one is multivariate analysis i.e. the analysis for the multiple variables.
The given question is of bivariate data, there are two variables \[x\] and \[y\]. In this statistical analysis of the bivariate data one is dependent while the other is independent.
Regression analysis is used to make the predictions after knowing some set of data. These predictions are very helpful in order to guess the sales of the next month by knowing the sales of the current month or by previous month. These predictions are helpful in the research work.
Basically, the regression coefficient is used to measure the degree of dependency of one variable to the other.
For the two variables \[x\] and \[y\], there are two values for the regression coefficient. One is obtained when \[y\] is independent and \[x\] is dependent i.e. we can say that \[x\] on \[y\]. And another is obtained when \[x\] is independent and \[y\] is dependent or we can say \[y\] on x. The notation used for the regression coefficient is \[b\]. So, the regression coefficient of \[x\] on \[y\] is denoted by \[{{b}_{xy}}\] whereas the regression coefficient for \[y\]on x is denoted by \[{{b}_{yx}}\].
In the given question we have to find \[{{b}_{yx}}\].
The formula used to determine \[{{b}_{yx}}\] is,
\[{{b}_{yx}}=\dfrac{(n\sum{xy-\sum{x\sum{y}}})}{(n\sum{{{x}^{2}}-({{\sum{x)}}^{2}}})}\]
In the question it is given that, \[\sum{x=30,\sum{y=400,\sum{{{x}^{2}}=196,\sum{xy=850}}}}\] and \[n=10\]
Substituting all the values in the formula, we get
\[\Rightarrow {{b}_{yx}}=\dfrac{(10\times 850-30\times 400)}{(10\times 196-{{(30)}^{2}})}\]
Simplifying the above expression, we get
\[\begin{align}
  & \Rightarrow {{b}_{yx}}=\dfrac{(8500-12000)}{(1960-900)} \\
 & \Rightarrow {{b}_{yx}}=-\dfrac{350}{106} \\
 & \Rightarrow {{b}_{yx}}=-3.3 \\
\end{align}\]
So, the regression coefficient of \[y\] on x is\[-3.3\]. Hence we can conclude that option \[(3)\] is correct.

Note: Both the regression coefficients i.e. \[{{b}_{xy}}\] and \[{{b}_{yx}}\] must have the same sign. If \[{{b}_{yx}}\] is negative in nature then \[{{b}_{xy}}\] should be negative in nature or vice-versa. There will be no effect on the regression coefficients if some constant is subtracted from both the variables whereas multiplication by some constant will change the coefficients.