A biologist will test, using a random sample of 20 pairs of observations, whether two variables are correlated in a population whose correlation coefficient is ρ\rhoρ.
For n=20n = 20n=20, the critical value for a one-tailed test for the population correlation coefficient is 0.3783 at the 5% level and 0.4438 at the 2.5% level.
The biologist tests, at the 5% significance level, whether there is any correlation between the two variables. Find, in set notation, the set of values of the sample correlation coefficient r r\,r for which the null hypothesis would be rejected.
The biologist instead tests, at the 5% significance level, whether there is positive correlation between the two variables. Find, in set notation, the set of values of r r\,r for which the null hypothesis would be rejected, and state, giving a reason, whether this test is more or less likely than the test in part (a) to reject the null hypothesis when ρ \rho\,ρ is positive.