The family with several young children have lived in a house on the north side of Montagu Street in Campbell Town - between Forster and Bedford streets - since the start of this year, and had not lived in the town before.

While moving in on January 1 they found a shot had been fired through their front window causing damage to the ceiling. They had not noticed the damage prior to moving in, estimating it occurred between Christmas and New Year's.

A .223 calibre casing was also found on their driveway, which they had not noticed before the incident.

Then overnight between last Sunday and Monday, November 7-8, another shot was fired at the house, this time becoming lodged in the front door frame.

Police attended following both incidents and retrieved the chunks of lead - fully on the first occasion, and a partial fragment on the second occasion.

In a statement, Tasmania Police said it believed it was due to hunting in the area.

"At this stage in the investigation it is believed that fragments from an errant air-rifle pellet struck the residence, potentially while a person was hunting wildlife," the statement read.

"Police do not believe the incident was deliberate or targeted."

But the family - who cannot be identified for legal reasons - disputed these findings, and argued that it was unlikely hunting would be occurring close enough to their house, which is in the township, for this level of damage to occur from an air rifle.

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The mother said it was their view that the amount of lead that became lodged in their house was greater than an air rifle pellet. They were dissatisfied with the extent of the police investigation.

"For an air rifle to do that damage, it'd have to be fired from a very close range," she said.

"And legally, no one is allowed to discharge a firearm within 200 metres of a house anyway so they're still committing an offence, even if it's in the paddock across the railway line from us. And that paddock has sheep in it.

"It seems like a bit more than an isolated incident. We'd like police to take it more seriously.

"If they're saying it's just someone having a pot shot at a rabbit, they still could've hurt or killed one of us."

The family has since installed security cameras and have spent several nights away from the house out of fear for their children's safety, given both gunshots could have been at head height during their trajectories.

"We shouldn't be forced to feel like our house is not safe," the mother said.

"It doesn't have that same safe feeling of being at home.

"I do understand police have a lot going on and are careful about wasting resources, but with something like this, you'd think you'd doorknock the vicinity and at least ask if anyone saw anything. None of our neighbours were visited."

Police have encouraged anyone with information about the incident to contact Crime Stoppers on 1800 333 000.