Yes, but I don’t understand why we don’t touch the mean but divide the variance by 50

You see another photo I attached...

Hi, I revisited this qn tdy and but I want to check on my working and understanding:

For part(iii), I understand it as
P(Ā - B̅ < 20) or P(B̅ - Ā < 20)

Ā - B̅ ~ N( -19, 0.3413)
P(Ā - B̅ < 20)
= 1

B̅ - Ā ~ N(19, 0.3413)
P(B̅ - Ā < 20)
= 0.9565255
= 0.957 (3s.f)

Initially I wanted to add both probabilities because or(+), but adding 1 is definitely wrong and the answer is 0.957

So why can’t I use P(Ā - B̅ < 20)? The question asked for difference of mean weight of both A and B which can go either Ā - B̅ or B̅ - Ā?

Or should I reject Ā - B̅ because the probability is 1?

You may see it in this way:
since P(A bar- B bar < 20)=1, it is for *sure* A bar is more than B bar by less than 20, no need to doubt about it. So the only way to have a difference less than 20 between them lies in the hand of P(B bar - A bar < 20).
The way you listed there, is really not a recommended way to present the difference (the modulus). Firstly, you need to calculate two distribution (though simple, but not needed). Secondly, it gives you wrong interpretation as to + the two probabilities, as it looks like an "OR" cases to add them.
So, use the modulus method.

Is there another to way to look at this beside modulus? I learnt modulus online so I’m not 100% familiar with it

If I have to use modulus, what situations can I use it on? If the question ask for the diff between two variables is more than or less than a number? What do I need to type on my calculator or I just solve it manually?

See the new photo...

Thanks for the explanation

Thanks! Really appreciated!

Would it be the same if I modulus P(|B̅ - A̅|< 20) ?

Yes.