Question

Translating Information to Other Significance Levels Suppose in a two-tailed test of \(H_{0}: \rho=0\) vs \(H_{a}: \rho \neq 0,\) we reject \(H_{0}\) when using a \(5 \%\) significance level. Which of the conclusions below (if any) would also definitely be valid for the same data? Explain your reasoning in each case. (a) Reject \(H_{0}: \rho=0\) in favor of \(H_{a}: \rho \neq 0\) at a \(1 \%\) significance level. (b) Reject \(H_{0}: \rho=0\) in favor of \(H_{a}: \rho \neq 0\) at a \(10 \%\) significance level. (c) Reject \(H_{0}: \rho=0\) in favor of the one-tail alternative, \(H_{a}: \rho>0,\) at a \(5 \%\) significance level, assuming the sample correlation is positive.

Short answer

The conclusions for Scenarios (b) and (c) would definitely be valid for the same data while the conclusion for Scenario (a) would not necessarily hold true.

Step-by-step solution

Scenario (a)

In this case, we're asked to consider if we would reject the null hypothesis at a 1% significance level. Recall that a lower significance level means the test is more stringent – there is a smaller probability of rejecting the null hypothesis when it is true. This means that if we rejected the null hypothesis at a 5% significance level, we cannot automatically assume the same conclusion at a 1% significance level. The sample data may not provide enough evidence to reject the null hypothesis at this more stringent level.

Scenario (b)

Here, we're looking at a 10% significance level, which is less stringent than a 5% level. Thus, if we rejected the null hypothesis at a 5% level, we would definitely also reject it at a 10% level.

Scenario (c)

The final part involves a one-tail alternative hypothesis, which means we're only concerned about one direction (here, rho>0). This is a different framework than the original two-tail test, where we considered both rho<0 and rho>0. If the null hypothesis was rejected at a 5% level for the two-tail test and the sample correlation is positive, we can say that we would also reject the null hypothesis at a 5% level for a one-tail test. The evidence for this lies in the fact that in a one-tail test, the full 5% significance level is allocated to testing the observed difference in one direction (in this case, rho>0), whereas in a two-tail test, the 5% significance level is split evenly between the two tails, making it a more stringent test.