If you have a baby in a private hospital, you stay about a day longer than if you had the baby in a public hospital. The same is true for a hip replacement. In fact, the same is true for any procedure requiring an overnight stay.
If this was simply the private hospital ‘value proposition’ and was simply a matter of private choice and private cost, it would not be a matter of public policy and taxpayer interest. But subsidies to private health insurance cost taxpayers $6 billion dollars a year, and a further $3 billion of annual subsidies are provided for medical costs in hospitals.