"a higher majority" is a grammatically correct and usable phrase in written English.
It is often used to refer to a much larger number or proportion of people than what is commonly accepted or expected. For example, "Although the senator received a lot of negative feedback, a higher majority of the population still agreed with her position."
Since a high majority of Nassau voters have to work a lot harder and for less money, now is the time to say no to another increase.
The party, which also opposes Heathrow expansion, hopes to put the focus on Brexit, after Goldsmith campaigned for leave despite a high majority of his constituents backing remain.
Families with school-age children could get cheaper holidays thanks to a high majority of headteachers planning to immediately use new powers to change term dates to allow off-peak travel.
"He's got a high majority because of the Asian vote.
A high majority of the survey respondents - 79%, in fact, - indicated that businesses need to do whatever it takes to protect the financial data of its customers.
My rent is absurdly high, and I'm pretty sure a high majority of us can relate to that feeling of writing your rent check thinking of the property you could own out in the suburbs for the same price.
Decades of research support that a high majority of people have implicit biases and recent studies find that most Filipino Americans have symptoms of colonial mentality, which negatively impact the ways they feel about themselves.
Regarding their attitude to whether organ donation should be mandatory by law or only through personal choice, a high majority 127 (80.4%) were in favor of the latter with only 24 (15.2%) favoring the former.
Thus from July 2009 a selected subgroup of residents was approached for the study in the intervention region, and because of the preceding close personal encounter, a high majority of these residents consented.
