Answer: \(\boxed{r=0.733}\); there is no evidence of positive correlation at the \(1\%\) level; \(\boxed{n=15}\).

(i) First calculate the corrected sums.

\[S_{xy}=\sum xy-\frac{\sum x\sum y}{n}.\]

So

\[S_{xy}=212.62-\frac{35.9(36.8)}{8}=47.48.\]

Also

\[S_{xx}=\sum x^2-\frac{(\sum x)^2}{n}=216.47-\frac{35.9^2}{8}=55.37,\]

and

\[S_{yy}=\sum y^2-\frac{(\sum y)^2}{n}=244.96-\frac{36.8^2}{8}=75.68.\]

The product moment correlation coefficient is

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

Therefore

\[r=\frac{47.48}{\sqrt{55.37(75.68)}}.\]

Hence

\[\boxed{r=0.733}\quad\text{to 3 significant figures}.\]

(ii) We test

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

For \(n=8\), the \(1\%\) one-tailed critical value for \(r\) is

\[0.789.\]

Since

\[0.733<0.789,\]

we do not reject \(H_0\).

Therefore there is not enough evidence, at the \(1\%\) level, to conclude that there is positive correlation between land area and population.

(iii) Now \(r=0.651\), and the test is for non-zero correlation, so it is a two-tailed test.

We need the smallest sample size for which \(0.651\) is greater than the \(1\%\) two-tailed critical value.

From the tables, for \(n=14\), the critical value is

\[0.661,\]

which is still larger than \(0.651\), so this is not significant.

For \(n=15\), the critical value is

\[0.641.\]

Now

\[0.651>0.641,\]

so the result is significant.

Therefore the least possible value is

\[\boxed{n=15}.\]