Two researchers separately testing the same research hypothesis end up with the same sample size ( \(N_{i}=12\) ) and standard deviations for the two groups ( \(s_{1}=.50\) and \(s_{2}=.75\) ). However, the experimental manipulation is more effective for the second researcher, which results in a bigger difference between the means of the two groups: \(\bar{X}_{1}=2.50\) and \(\bar{X}_{2}=2.15\) (first researcher), as well as \(\bar{X}_{1}=2.70\) and \(\bar{X}_{2}=1.95\) (second researcher).

a. Calculate the standard error of the difference \(\left(s_{\bar{X}_{1}}-\bar{X}_{2}\right)\) and the \(t\)-test for independent means for each study.

b. How does the increased between-group variability in the second study affect the calculated value of the \(t\)-statistic, as well as the probability of making a Type II error?