Hi Debra
I suspect the family did pay their own way out given their absence from the online lists. Donald Matheson was a surveyor so they probably weren't a poor family. At the time of the Chatsworth's arrival in Queensland Donald would have been about 35, his wife Elizabeth 27 and their daughter Catherine/Katie 4. There was a son Matthew less than 2, and I think he may have died on the voyage.
Would appreciate any further info you can provide.
Regards,
Andrew