Sierra is analyzing patient wait times at a clinic. The average wait time is 22 minutes with a standard deviation of 5 minutes. Assuming a normal distribution, what percentage of patients wait longer than 30 minutes?

["Analyzing Patient Wait Times at Sierra Clinic: Understanding Wait Durations with Statistics", "Waiting longer than expected at a clinic can be frustrating for patients—and for clinics aiming to improve care efficiency. At Sierra Clinic, recent data reveals an average patient wait time of 22 minutes, with a standard deviation of 5 minutes. With wait times following a normal distribution, healthcare administrators can use statistical analysis to better understand patient flow and enhance operational planning.", "### The % of Patients Waiting Longer Than 30 Minutes", "Given a normal distribution with a mean (μ) of 22 minutes and a standard deviation (σ) of 5 minutes, we want to determine what percentage of patients experience wait times exceeding 30 minutes.", "#### Step 1: Calculate the z-score\nThe z-score measures how many standard deviations a data point is from the mean:", "[\nz = \frac{X - \mu}{\sigma} = \frac{30 - 22}{5} = \frac{8}{5} = 1.6\n]", "#### Step 2: Use the z-table or normal distribution calculator\nA z-score of 1.6 corresponds to the cumulative probability up to 30 minutes. Using standard normal distribution tables or statistical software, the cumulative probability P(Z ≤ 1.6) is approximately 0.9452, or 94.52%.", "To find the percentage of patients waiting longer than 30 minutes:", "[\nP(X > 30) = 1 - P(Z \leq 1.6) = 1 - 0.9452 = 0.0548\n]", "#### Step 3: Convert to percentage\n[\n0.0548 \ imes 100 = 5.48%\n]", "---", "### Interpretation and Implications", "Approximately 5.48% of patients at Sierra Clinic wait longer than 30 minutes. This insight is critical for clinic management because it highlights that close to 5% of patients face wait times well above the median—potentially impacting patient satisfaction and trust.", "By identifying this threshold, clinic leadership can investigate causes of extended waits—such as understaffing, scheduling inefficiencies, or bottlenecks in procedures—and implement targeted improvements like adjusting appointment lengths, optimizing staffing, or integrating faster triage protocols.", "Using this type of statistical analysis enables data-driven decision-making to reduce patient wait times, enhance operational efficiency, and ultimately deliver better care.", "---", "In summary: With a normal distribution of patient wait times averaging 22 minutes (±5 minutes), about 5.5% of patients experience waits exceeding 30 minutes. Monitoring and reducing this percentage can significantly improve patient experience and clinic performance."]