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

Hint: 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.