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The woman was home alone inside her Queenstown property when she had a medical emergency and was unable to get through to Triple Zero on Thursday.
Paramedics and police eventually arrived at the home but the woman could not be saved.
She is one of four people who died during the outage after the fault also led to the deaths of an eight-week-old boy from Gawler West in Adelaide, a 74-year-old man from Willetton in Perth and a 49-year-old man from Kensington in Perth.
The family of the 68-year-old woman said the telecommunications giant has not reached out to them personally.
“On behalf of our family we mourn the loss of our wife/sister, we hope that authorities are able to investigate the terrible circumstances so other families don’t experience a tragedy like this,” they said in a statement.
“We are saddened to hear of the loss of the young child and our hearts go out to the family and others that have been impacted by this event.”
The family told 9News while are currently focused on supporting one another in their grief, they intend to seek answers in time.
The parents of the newborn baby who died during the outage tried to call Triple Zero but their calls wouldn’t connect.
The news has devastated residents of the Gawler West community.

“I just couldn’t believe it, I’ve got kids and I couldn’t imagine that,” neighbour Tasha Delaine said.
The family’s call eventually got through and several ambulances were dispatched to the home at 5.20am on Thursday morning to reports of a cardiac arrest.
Paramedics called specialised medical back-up but the infant was unable to be saved.
Since the tragedies, South Australian police have conducted welfare checks on more than 150 callers in the state.
Adelaide resident Chris Tyndall received a follow-up call after he spent Thursday morning repeatedly trying to call for an ambulance with two phones.
The 70-year-old, who lives with heart disease and diabetes, had spent a week in bed with the flu.
“I was struggling,” Tyndall said.
“When they got here, I went to open the door and nearly fell, that’s when she knew something was dreadfully wrong.
“She took my blood pressure standing up and it dropped dramatically.
“She said that could cause a stroke.”
“South Australian police didn’t know. No one in SAS or ambulance service knew; my office didn’t know. But then they conducted a press conference,” said Malinauskas.
“I have not witnessed such incompetence from an Australian corporation in respect to communications worse than this.”
Western Australia’s Premier, Roger Cook, also spoke out against the telco, deeming the outage “completely unacceptable”.
Optus said a botched firewall upgrade was to blame for the failure of Triple Zero calls.
The outage stretched on for 13 hours and impacted about 600 emergency services calls across South Australia, Western Australia and the Northern Territory.
Two complaints have since been raised with the Telecommunications Industry Ombudsman.