This Bible contradiction is from the Skeptic's Annotated Bible.
Yes, his whole family is saved by his belief.
1 Corinthians 7:14 View context
For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband: else were your children unclean; but now are they holy.
Acts 16:31 View context
And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.
1 Corinthians 7:16 View context
For what knowest thou, O wife, whether thou shalt save [thy] husband? or how knowest thou, O man, whether thou shalt save [thy] wife?