a lot better or much better?

a lot better or much better? 

which one right?

Just Googled for a while. It seems that 'much' can be a modifier while 'a lot' cannot.

For the quiz, 
He speaks ____ better than Mary . 
A.a lot      B.much

You have to choose A, rather than B. 

However, when there is no 'then', it seems that 'a lot better' is common.

Examples: 

Much

Mary sings much better than Helen.
Her handwriting is much better than mine.

a lot better: 

I feel a lot better, thanks for asking.
"We can use a lot better science," he said."
I could have done a lot better.

If I am wrong, please leave a message and correct me. Thanks.