Imagine two studies examining the same research question find the same means and standard deviations for two groups: \(\bar{X}_{1}=10.00, s_{1}=3.00\), and \(\bar{X}_{2}=12.00, s_{2}=2.00\). However, the two studies differ in sample sizes: \(N_{i}=9\) in the first study and \(N_{i}=16\) in the second study.

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 increasing the sample size from \(N_{i}=9\) to \(N_{i}=16\) affect the calculated values of the standard error of the difference and the \(t\)-statistic, as well as the probability of making a Type II error?