How to test my hypothesis? With a paired sample t-test I found out that participants communicated more pride about a team achievement than about an individual achievement.
What I would like to examine now, is if this difference between communicated pride about a team achievement and an individual achievement is larger for Indians compared to Americans. 
I have a sample consisting of 148 Indians and 268 Americans. All participants had to indicate their communicated pride over both a team achievement and an individual achievement (so it is a within subjects design). Which test should I run? 
I thought about computing a new variable which is computed as communicated pride over a team achievement - comunnicated pride over an individual achievement. Then I have to determine if these differ from each other, but I don't know how!?
 A: The "orthodox" solution would be a 2x2 ANOVA, with team vs. individual as within (repeated-measures) factor, and nationality as a between factor. You would be interested in the interaction between these factors. Your idea is that the behaviour on one factor - team/individual - depends on the level of the other factor - nationality.
Conceptually, the ANOVA can be seen as the generalisation of the two-sample t-test to three or more (in your case, four) groups. It checks if all group means are identical by comparing within- and between-group variance, and typically, all main effects and their interactions are evaluated at the same time.
Traditionally, the interaction would be followed up by a test by factor - here, an unpaired t-test of team minus individual scores for the two nationalities. Pragmatically speaking, your readers will probably expect to see the ANOVA in this case, you'll probably hardly get around it.
Either way, don't forget to provide an inference for a measure of effect size, such as a point estimate and confidence interval for $\eta^2$ (eta squared) for your main effects and interaction.
