Here is the complete text of 13 U.S.C. § 221 (Refusal or neglect to answer census questions; false answers):
- (a) Whoever, being over eighteen years of age, refuses or
willfully neglects, when requested by the Secretary, or by any
other authorized officer or employee of the Department of Commerce
or bureau or agency thereof acting under the instructions of the
Secretary or authorized officer, to answer, to the best of his
knowledge, any of the questions on any schedule submitted to him in
connection with any census or survey provided for by subchapters I,
II, IV, and V of chapter 5 of this title, applying to himself or to
the family to which he belongs or is related, or to the farm or
farms of which he or his family is the occupant, shall be fined not
more than $100 (note: some pamphlets say the fine is up to $5000.00, but I can't find the source for that).
(b) Whoever, when answering questions described in subsection (a)
of this section, and under the conditions or circumstances
described in such subsection, willfully gives any answer that is
false, shall be fined not more than $500.
(c) Notwithstanding any other provision of this title, no person
shall be compelled to disclose information relative to his
religious beliefs or to membership in a religious body.
In addition,
this section, 13 U.S.C. § 222, criminalizes the giving suggestions or information with intent to cause inaccurate enumeration of population
- Whoever, either directly or indirectly, offers or renders to any officer or employee of the Department of Commerce or bureau or agency thereof engaged in making an enumeration of population under subchapter II, IV, or V of chapter 5 of this title, any suggestion, advice, information or assistance of any kind, with the intent or purpose of causing an inaccurate enumeration of population to be made, shall be fined not more than $1,000 or imprisoned not more than one year, or both.
Based on the foregoing, it looks like you have to answer any goddam question they ask unless, it's about your religious beliefs.
As for why this isn't a gross invasion of your privacy, the government will claim it is because of
this section, which supposedly makes census data private:
- Whoever, being or having been an employee or staff member referred to in subchapter II of chapter 1 of this title, having taken and subscribed the oath of office, or having sworn to observe the limitations imposed by section 9 of this title, or whoever, being or having been a census liaison within the meaning of section 16 of this title, publishes or communicates any information, the disclosure of which is prohibited under the provisions of section 9 of this title, and which comes into his possession by reason of his being employed (or otherwise providing services) under the provisions of this title, shall be fined not more than $5,000 or imprisoned not more than 5 years, or both.
(Of course, as we all know, the federal government would never, ever, wrongfully use or disclose private information. 
)So, unfortunately, it would appear that, absent an expensive court challenge, you have to answer truthfully no matter how invasive the questions might be.
Hail Supreme Leader Obama!
