1. In developing patient appointment schedules, a medical center wants to estimate the mean time that a staff member spends with each patient. Using a value for the population standard deviation of eight minutes, determine:
a. How large a sample should be taken if the desired margin of error is two minutes at a 95% level of confidence?
b. How large a sample should be taken if the desired margin of error is two minutes at a 99% level of confidence?
To determine the sample size needed to estimate the mean time a staff member spends with each patient, we can use the formula:
n = (Z * σ / E)²
Where:
n = sample size
Z = z-score (corresponding to the desired level of confidence)
σ = population standard deviation
E = desired margin of error
a. For a 95% level of confidence:
Z = 1.96 (corresponding to a 95% confidence level)
σ = 8 minutes (population standard deviation)
E = 2 minutes (desired margin of error)
Using the formula:
n = (1.96 * 8 / 2)²
n = 6.24²
n ≈ 38.7
Rounding up to the nearest whole number, the sample size should be at least 39.
b. For a 99% level of confidence:
Z = 2.58 (corresponding to a 99% confidence level)
Using the formula:
n = (2.58 * 8 / 2)²
n = 16.32²
n ≈ 266.43
Rounding up to the nearest whole number, the sample size should be at least 267.
To calculate the required sample size, we will use the formula:
n = (Z * σ / E)²
Where:
n = sample size
Z = Z-score (a critical value that corresponds to the desired level of confidence)
σ = population standard deviation
E = desired margin of error
a. For a 95% level of confidence:
Z-score for a 95% confidence level is approximately 1.96 (obtained from a Z-score table or calculator).
σ = 8 minutes
E = 2 minutes
Plugging these values into the formula:
n = (1.96 * 8 / 2)²
n = (15.68 / 2)²
n = 7.84²
n ≈ 61.45
Therefore, a sample size of approximately 61.45 should be taken.
b. For a 99% level of confidence:
Z-score for a 99% confidence level is approximately 2.58 (obtained from a Z-score table or calculator).
σ = 8 minutes
E = 2 minutes
Plugging these values into the formula:
n = (2.58 * 8 / 2)²
n = (20.64 / 2)²
n = 10.32²
n ≈ 106.50
Therefore, a sample size of approximately 106.50 should be taken.
Note: Since sample sizes must be whole numbers, rounding up to the nearest whole number is recommended. So, the final sample sizes would be 62 for a 95% confidence level and 107 for a 99% confidence level.