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 calculate the sample mean, we can divide the sum of all the observed values by the number of observations.
In this case, we have randomly selected 140 bills and let x̄ represent the average amount charged for these 140 surgical procedures.
The mean or average of the observed values can be calculated using the formula:
x̄ = (x1 + x2 + x3 + ... + xn) / n
However, since we don't have the specific values of the 140 bills, but only the population average and standard deviation, we need to use a different formula to estimate the sample mean.
Given that the population average is $1220, we expect the sample mean to be around this value. However, the sample mean may vary from the population mean due to random sampling.
To estimate the variability of the sample mean, we can use the standard deviation divided by the square root of the sample size (140 in this case). This is known as the standard error of the mean.
The formula for the standard error of the mean is:
SE = σ / √(n)
Where σ is the standard deviation and n is the sample size.
In our case, the standard deviation is $270, and the sample size is 140:
SE = 270 / √(140)
By evaluating this expression, we can find the standard error of the mean.
Is there anything else you would like to know about estimating the sample mean?