Final answer:
Using the normal distribution and standard error, it is calculated that 17% of the time, the mean hours worked by a sample of 22 ICU nurses will be greater than approximately 55.8 hours.
Step-by-step explanation:
To determine 17% of the time their mean hours worked will be greater than how many hours, we need to use the concept of the normal distribution and standard error of the mean. Since the distribution of hours worked weekly by ICU nurses is normally distributed, with a population mean (μ) of 55 hours and a population standard deviation (σ) of 3.9 hours, we can calculate the standard error of the mean (SEM) for a sample size (n) of 22 nurses using the formula SEM = σ / √n.
SEM = 3.9 / √22 = 3.9 / 4.69 ≈ 0.83 hours
Next, we need to determine the z-score that corresponds to the top 17% of the normal distribution curve. Using a z-score table or a calculator, we find that the z-score that leaves 17% in the upper tail is approximately 0.93. We then use this z-score to find the cut-off point for the sample mean.
Cut-off point for sample mean = μ + (z * SEM) = 55 + (0.93 * 0.83) ≈ 55 + 0.77 = 55.77 hours
Therefore, 17% of the time, the mean hours worked by a sample of 22 nurses will be greater than approximately 55.8 hours.