Take a step forward to the 1910 Census, do you think you have the correct family in that census? Is there enough (relatively) same data, (ie, parents and siblings) in the 1910 Census to qualify you have the same family in the 1900?
>> Sould i infer that Kate and Catherine are two different people or that I have the wrong family? And if it is her family, how would I find her birthdate to get a cerificate to be sure? <<
The official enumeration day of the 1900 census was 1 June 1900.
>>There was another name, Kate born May 1900, which I initially assumed was Catherine. However, I found in the NJ birth index that "Kate" died at one year old. <<
The math says the Kate, b May 1900 and died at one year old, and was in the 1900 Census, is not the Catherine born later (abt 1901). It was not unusual for a subsequent born child to be given the same name as a deceased sibling.
If you want to post the two census records, 1900 and 1910 for an opinion as to whether they're the same household, that would help.
>>Also, where do i find Naturalization records for people in NY and NJ that are not online that according to cencus'were either naturalized or got first papers?<<
fold3.com (a fee subscription site)has some of this data on-line. If you want to post the names, I'll have a look.