So this questioned was sent to me and needs to be answered in a more open setting.Do stats affect rolls, besides combat and noble standing?
RULE #2 "Do not break the 2d6 curve."
Those into statistics will know that it is technically not a curve ... the lines are straight to a peak ... but the point is still the same. Here are your chances of success:
2d6, roll 2+ = 100%
2d6, roll 3+ = 97%
2d6, roll 4+ = 92%
2d6, roll 5+ = 83%
2d6, roll 6+ = 72%
2d6, roll 7+ = 58%
2d6, roll 8+ = 42%
2d6, roll 9+ = 28%
2d6, roll 10+ = 17%
2d6, roll 11+ = 8%
2d6, roll 12+ = 3%
I use Rule 68A for most tasks because it is 'stupid' simple.
RULE 68A:
The target number to accomplish an EASY task is 6+ on 2d6.
The target number to accomplish an AVERAGE task is 8+ on 2d6.
The target number to accomplish an HARD task is 10+ on 2d6. [the hexidecimal notation for 10 is 'A']
That's it. You roll 2d6, add your special modifiers (like any appropriate skill) and compare it to the target number.
So the thing about a 2d6 curve is that modifiers matter a LOT!
An average task (like a Medic patching a bullet wound in the field with a first aid kit) is 8+, so with a skill-0 the chance of success is 42%.
With any +1 modifier (like skill-1), the chance of success is 58%.
With any +2 modifier (like skill-2), the chance of success is 72%.
With any +3 modifier (like skill-3), the chance of success is 83%.
If we pile attribute bonuses on skill bonuses, then the odds quickly become so high that rolling is a waste of time ...
Medical skill with +2 attribute bonus (+1 Edu & +1 Dex) = 72% success for Medic-0 to 97% success for Medic-3.
Maybe we should add another +1 for a Sick Bay on a ship = 83% success for Medic-0 to 100% success for Medic-3.
With the modifiers for a good (not great) education and dexterity, the person with a little First Aid training (medical-0) is as good as an average Doctor (Medical-3) at treating a bullet wound!
The 2d6 curve has been broken.
So I do not add ability modifiers to skills unless they are exceptional ... ANY ABILITY OF 10+ may add +1 to an appropriate skill check.
The question then becomes "What do abilities do?" ... Good question.

Anything you can think of.
Seriously. Traveller is not like other games where the assumption is that I must have THE SKILL to do THE TASK. "How can I swim if I don't have Swimming-1?" is not how Traveller works. Assume that you can do anything and everything you want to attempt. Now common sense will tell you that some things have very poor odds ... attempting brain surgery to alter someone's personality using only your hunting knife and having no Medical training is not likely to end well, but you can try it with an actual chance of success (about 3%).
So a skill (like Medical-1 or Pistol Combat-1) indicates that you spent 2 years of your life engaged in that task on what approaches full-time employment. That makes a Skill-1 closer to a mini "prestige class" in D&D terms. You are assumed to know EVERYTHING that comes with 2 years of life experience as a Medic or Pistol shooter.
But what about all those things in life that are not about WORK ... like "Can I swim?" That is where your ATTRIBUTES come into play. Treat them just like skill as follows:
RULE 68A:
Easy Task = roll 6+
Average Task = roll 8+
Hard Task = roll 10+
Attribute = 0-2 = -1
Attribute = 3-5 = 0
Attribute = 6-8 = +1
Attribute = 9-11 = +2
Attribute = 12+ = +3
So swimming a short distance across calm water is EASY with a DEX of 7 grants +1 so roll 6+ on 2d6+1 to swim from the boat to shore (83% chance).
Now let's say that the water is rough and you are attempting to swim with someone else. The rough water makes it an Average Task (8+) so you roll 8+ on 2d6+1 (58% chance). The person with you is a bad swimmer (DEX of 2) and clinging to a life preserver. They must roll 8+ on 2d6-1 (28% chance).
They fail. so you attempt to pull them along with you. This is not a Dexterity issue, but more of an easy Endurance (END of 5) issue, so you roll 6+ on 2d6+0 to pull them with you (72% chance).
Any activity that is not DIRECTLY a skill can probably be attempted as an ability check.