Question

Find the regression equation showing the regression equation of capacity utilization on production from the following data

| Average| Standard deviation
---|---|---
Production (in lakh units)| 35.6| 10.5
Capacity utilization (in percentage) | 84.8| 8.5

Coefficient correlation $= r = 0.62$
Estimate the production when the capacity utilization is 70 percent.

Answer 1

If $X$ depends on $Y$, then the regression equation is $X$ on $Y$ and is given using the formula,
$X-\overline{X} = rb_{xy}(Y-\overline{Y})$
Here $r$ is the correlation coefficient and $b_{xy}$ is the ratio of the deviation of $X$ to the deviation of $Y$.
And $\overline{X}\,\text{and}\,\overline{Y}$ are the mean of values of $X$ and $Y$.

Complete step-by-step answer:
Let $x$ denote the production (in lakh units) and $y$ denote the capacity utilization (in percentage).
Since the average production is given to be 35.6 lakh units and the average capacity utilization is given to be 84.8%,
$\overline{x} = 35.6$
$\overline{y} = 84.8$
As the deviation from the production and the capacity utilization is 10.5 lakh units and 8.5% respectively, it gives,
$\sigma (x) = 10.5$
$\sigma (y) = 8.5$
Since you have to find the regression equation showing the regression of capacity utilization on production, it implies you have to find the regression equation of $x$ on $y$.
Now, the formula for the $x$ on $y$ regression line is,
$x-\overline{x} = r\dfrac{\sigma (x)}{\sigma (y)}(y-\overline{y})$, where $r$ is the correlation coefficient.
Substituting the values into the formula as,
\begin{align*}x-35.6 &= (0.62)\left(\dfrac{10.5}{8.5}\right)(y-84.8)\\\ x-35.6 &= 0.7658(y-84.8)\\\ x &= 0.7658y-29.34\end{align*}
So, the regression equation is $x = 0.7658y-29.34$.
Now the capacity utilization is given to be 70 percent. So, the production at this capacity utilization is,
\begin{align*}x &= 0.7658(70)-29.34\\\ &= 53.606-29.34\\\ &= 24.266\end{align*}

Therefore, the production is 24.266 lakh units.

Note: The regression equation of $y$ on $x$ is also given using the same formula, i.e.,
$y-\overline{y} = rb_{yx}(x-\overline{x})$
These regression equations are used to determine the value of either independent or dependent variable, when one is known.