A health services provider has been reimbursed by a national health insurance organization for a large number of services. It appears that overpayments may have occurred on many of the reimbursements owing to errors made during the billing process or to up-coding (billing for a more complicated service than was actually provided). Overpayments in the past have been skewed, with a standard deviation of $12.00 per service provided. The health services provider claims that no overpayments have occurred on average, that is, that the actual mean overpayment is $0. An audit of a random sample of 36 reimbursements was conducted, and the sample mean overpayment was $4.00.
a. Determine the probability that a sample mean overpayment is greater than or equal to that found in the audit if the provider's claim is true.
b. Determine the probability that the sample mean overpayment for the audited reimbursements will be between $2.00 and $4.00 if the provider's claim is correct.
c. Determine the probability that the total overpayment to the provider for the sampled reimbursements will be greater than or equal to $144.00 if the provider's claim is true.
This solution finds the probability of the sample mean for varying scenarios if the provider's claim is correct. All steps and formulas are shown in a clear, structure manner.