There was a significant difference between mean scores, so what’s next?

#1
What’s the mathematical meaning of a significant difference between mean scores? What should we expect? What we are allowed to say about our two groups? For example, if I have done a survey and I have a significant difference between males and females in intelligence. After this survey what I am allowed to say (or predict) about males and females intelligence?
Thanks,
 
#2
1] "Statistically significant" does not mean "substantial". You need to check whether the estimated difference between the group means is meaningful.

2] Simple tests for exploring the difference in expectation of Y between two groups are not good for predictive purposes, typically. You need to build a bigger model, adjusting for potential effects of various covariates and factors.
 

hlsmith

Not a robit
#3
The design of your study can come into play and whether you had an unbiased random sample of the target population. I would present the two means along with their respective confidence intervals.

Given no biases or systematic errors, you are rejecting the nulls that both samples came from the same underlying distribution.
 
#4
It was my problem: “ … rejecting the nulls that both samples came from the same underlying distribution.” I mean, can it have any other meaning? Or can we say something more?