Answer: \(\boxed{y=-0.3x+7.2}\), \(\boxed{r=-0.520}\), and there is no evidence of non-zero correlation at the \(10\%\) level.

(i) First calculate the totals:

\[\sum x=20,\qquad \sum y=30,\qquad \sum xy=111,\qquad \sum x^2=110,\qquad \sum y^2=190.\]

The means are

\[\bar{x}=\frac{20}{5}=4,\qquad \bar{y}=\frac{30}{5}=6.\]

Now

\[S_{xy}=\sum xy-\frac{(\sum x)(\sum y)}{n}=111-\frac{20(30)}5=-9,\]

and

\[S_{xx}=\sum x^2-\frac{(\sum x)^2}{n}=110-\frac{20^2}{5}=30.\]

The gradient of the regression line of \(y\) on \(x\) is

\[b=\frac{S_{xy}}{S_{xx}}=\frac{-9}{30}=-0.3.\]

Use

\[y-\bar{y}=b(x-\bar{x}).\]

Hence

\[y-6=-0.3(x-4).\]

So

\[\boxed{y=-0.3x+7.2}.\]

(ii) Also

\[S_{yy}=\sum y^2-\frac{(\sum y)^2}{n}=190-\frac{30^2}{5}=10.\]

The product moment correlation coefficient is

\[r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}.\]

Therefore

\[r=\frac{-9}{\sqrt{30\cdot10}}=-0.520\text{ approximately}.\]

(iii) Test

\[H_0:\rho=0,\qquad H_1:\rho\ne0.\]

For \(n=5\), at the \(10\%\) significance level for a two-tailed test, the critical value is

\[0.805.\]

Since

\[|r|=0.520<0.805,\]

we do not reject \(H_0\).

Therefore there is no evidence, at the \(10\%\) significance level, of non-zero correlation between the variables.