Below are two hypothetical situations:

Study A: \(N_{1}=20, \bar{X}_{1}=14.50, s_{1}=2.50\)

\(N_{2}=20, \bar{X}_{2}=12.75, s_{2}=3.25\)

Study B: \(N_{1}=65, \bar{X}_{1}=14.50, s_{1}=2.50\)

\(N_{2}=65, \bar{X}_{2}=12.75, s_{2}=3.25\)

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 \(N_{i}=20\) to \(N_{i}=65\) affect the calculated values of the standard error of the difference and the \(t\)-statistic?

c. How does increasing the sample size affect the probability of making a Type II error?