Question

A)  Volume of production (X) and manufacturing expenses (Y) are recorded for 8 randomly selected firms, the data is as under:

Volume of production (X)

46

44

48

55

50

79

88

100

Manufacturing expenses (Y)

150

140

160

170

160

172

185

165

 

1)   Determine the single coefficient correlation (r)?

2)   Construct the regression equation of manufacturing expenses on volume of production. Also estimate manufacturing expenses (Y) when volume of production is 120.

3)   Interpret the findings.

Answer 1

1)

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} X & Y & XY & X^2 & Y^2 \\ \hline 46 & 150 & 6900 & 2116 & 22500 \\ 44 & 140 & 6160 & 1936 & 19600 \\ 48 & 160 & 7680 & 2304 & 25600 \\ 55 & 170 & 9350 & 3025 & 28900 \\ 50 & 160 & 8000 & 2500 & 25600 \\ 79 & 172 & 13588 & 6241 & 29584 \\ 88 & 185 & 16280 & 7744 & 34225 \\ 100 & 165 & 16500 & 10000 & 27225 \\ \hline 510 & 1302 & 84458 & 35866 & 213234 \\ \end{array}$$

$$\bar{x}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nX_i=\dfrac{510}{8}$$

$$\bar{y}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nY_i=\dfrac{1302}{8}$$

$$SS_{XX}=\displaystyle\sum_{i=1}^nX_i^2-\dfrac{1}{n}\bigg(\displaystyle\sum_{i=1}^nX_i\bigg)^2$$

$$=\dfrac{26828}{8}$$

$$SS_{YY}=\displaystyle\sum_{i=1}^nY_i^2-\dfrac{1}{n}\bigg(\displaystyle\sum_{i=1}^nY_i\bigg)^2$$

$$=\dfrac{10668}{8}$$

$$SS_{XY}=\displaystyle\sum_{i=1}^nX_iY_i-\dfrac{1}{n}\bigg(\displaystyle\sum_{i=1}^nX_i\bigg)\bigg(\displaystyle\sum_{i=1}^nY_i\bigg)$$

$$=\dfrac{11644}{8}$$

Correlation coefficient

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}$$

$$=\dfrac{\dfrac{11644}{8}}{\sqrt{\dfrac{26828}{8}}\sqrt{\dfrac{10668}{8}}}\approx0.68828221$$

b) The slope

$$m=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{\dfrac{11644}{8}}{\dfrac{26828}{8}}\approx0.43402415$$

The y-intercept

$$n=\bar{y}-\bar{x}\cdot m=\dfrac{1302}{8}-\dfrac{510}{8}\cdot\dfrac{11644}{26828}$$

$$\approx135.08096019$$

The regression equation of manufacturing expenses on volume of production

$$Y=135.08096019+0.43402415X$$

Estimate manufacturing expenses (Y) when volume of production is 120

$$Y=135.08096019+0.43402415(120)$$

$$Y=187$$

3)

$$r=0.68828221$$

$$0.4<|r|<0.7, r>0$$

There is moderate positive correlation.