How do the banks know whether or not to apply the Higher Rate of Tax (HRT) to your savings accounts? Do they ask you? Ask Revenues and Customs?

I’m thinking of putting a monthly amount into a HSBC regular saver instead of my ISA (at Yorkshire BS). I’m thinking that the regular saver account will pay me a higher rate of interest even after the standard tax rate.

(I don’t earn enough to pay the HRT on my wages).