Final answer:
To achieve a margin of error of two minutes at a 95% level of confidence with a population standard deviation of eight minutes, the medical center should take a sample of at least 62 patients.
Step-by-step explanation:
When developing patient appointment schedules, a medical center needs to estimate the mean time that a staff member spends with each patient. To determine an appropriate sample size with a desired margin of error of two minutes at a 95% level of confidence, and using a planning value for the population standard deviation (σ) of eight minutes, we can apply the formula for sample size in estimating a population mean:
n = (Z*σ/E)^2
where Z is the Z-value from the standard normal distribution for the desired confidence level (1.96 for 95%), σ is the population standard deviation, and E is the desired margin of error.
Plugging in the values:
n = (1.96*8/2)^2
n = (15.68/2)^2
n = (7.84)^2
n = 61.47
Since we cannot collect a fraction of a sample, we round up to the next whole number. Therefore, the medical center should take a sample of at least 62 patients to achieve the desired precision in their estimate of the mean time a staff member spends with each patient.