A majority of Americans believe health insurance providers should be required to cover birth control, according to a new survey, a demonstration of strong support for the provision of President Obama’s signature health care law that has incensed some of its loudest critics.

A survey out Tuesday by researchers at the University of Michigan Health System found that 69% of respondents nationwide support mandated coverage of contraception, with even higher support among women, African-Americans, Hispanics, and parents of children under the age of 18.

Between 85% and 75% of respondents favored mandated coverage of mammograms, colonoscopies, recommended vaccinations, diabetes and high cholesterol screenings, mental health care and dental care. Less than 10% of respondents (chief among them men, adults older than 60, and adults without children at home) favored mandated coverage of all these services except birth control.

Insurers have been required to provide birth control to policy holders at no out-of-pocket cost since 2012, as a key tenet of the 2010 Affordable Care Act. The health care law offers an exemption for religious groups.

More than 90 legal challenges have been brought nationwide. The Supreme Court is expected to rule this spring on an appeal by Hobby Lobby and Consestoga, two employers who have objected to the provision on religious grounds.