Nationwide, the amount charged by doctors for performing a particular minor surgical procedure averages $1220 and varies with a standard deviation of $270. We randomly select 140 bills from the population of all bills charged for this surgery. Let represent the average amount charged for these 140 surgical procedures.
To find the average amount charged for these 140 surgical procedures, we need to use the concept of the sampling distribution of the mean. The sampling distribution of the mean is a theoretical distribution that shows all possible sample means that could be obtained from a population.
In this case, we know that the average amount charged for the surgical procedure is $1220 with a standard deviation of $270. Since we are randomly selecting 140 bills from the population, we can assume that the sample mean will follow a normal distribution (based on the Central Limit theorem).
The standard deviation of the sampling distribution of the mean (also known as the standard error) can be calculated using the formula:
Standard Error (SE) = Standard Deviation (SD) / √(sample size)
In this case, the standard error would be:
SE = 270 / √140 ≈ $22.83
Now, with the average amount charged for the surgical procedures ($1220) and the standard error ($22.83), we can find the range within which the sample mean is likely to fall.
Using the concept of confidence intervals, let's say we want to find the 95% confidence interval. This means that we want to find the range within which we can be 95% confident that the true population mean lies.
To calculate the confidence interval, we need to use a t-score or z-score, depending on whether the sample size is large (typically more than 30) or small (less than 30). In this case, since we have a sample size of 140, we can use a z-score.
For a 95% confidence interval, we would use a z-score of approximately 1.96 (which corresponds to an area of 0.025 on each tail of the normal distribution).
The formula for the confidence interval is as follows:
Confidence Interval = sample mean ± (z-score * standard error)
For example, the lower bound of the confidence interval would be:
Lower Bound = sample mean - (1.96 * standard error)
And the upper bound would be:
Upper Bound = sample mean + (1.96 * standard error)
Plugging in the values, the confidence interval would be:
Confidence Interval = $1220 ± (1.96 * $22.83)
After calculating the confidence interval, we would have an estimate of the range within which we can be 95% confident that the true population mean lies based on the sample of 140 bills selected.