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The normal cabin pressure during flight is basically a function of the aircraft type.
Maintaining a lower pressure altitude in the cabin creates a higher pressure difference between the cabin and the air outside of the plane, which the aircraft needs to be able to sustain.
The vast majority of modern aircraft are pressurized to around 8,000 feet during normal flight, although there are 3 (or 4) exceptions.
The Airbus A380, Airbus A350, and Boeing 787 are all designed to be able to maintain a cabin pressure of 6,000 feet, and would normally be set to that for normal flight.
The 4th aircraft that (use to) support a cabin pressure of 6,000 feet was Concorde. Obviously not an option for flying today, but interesting given how long ago they started flying relative to the other models above.
So as far as knowing if an upcoming flight is going to be pressurized at 6,000 or 8,000 feet, simply look at the aircraft type flying it. If it's an A380, A350 or B787 then it'll almost certainly be 6,000 feet. If it's a Concorde, you need to have a word to your travel agent! If it's anything else, it'll be 8,000 feet.
(Note that cabin pressure isn't actually measured in 'feet', but it's a common way to refer to aircraft pressurization, referring to the normal atmospheric pressure at that elevation above sea level).