How do we decide if the correlation coefficient is close enough to ±1 to declare that there is correlation?

Answer 1

You can compare the computed correlation coefficient to a table of critical r-values. if |correlation coefficient| > critical value, then there is a significant linear correlation.

You can also test null hypothesis that correlation coefficient = 0 vs alternate hypothesis that correlation coefficient is non-zero using t-statistic r * sqrt(n-2)/sqrt(1-r^2) with n-2 degrees of freedom.