An epidemiologist models recovery time from an illness using a normal distribution with mean 14 days and standard deviation 3 days. What percentage of patients recover in more than 20 days? (Use the 68-95-99.7 rule and appropriate z-score calculation.)

["Understanding Patient Recovery: Using Normal Distribution to Predict Recovery Time", "When tracking illness outcomes, epidemiologists often model recovery time using statistical tools—especially the normal distribution. This powerful model helps estimate how many patients recover within a certain timeframe, enabling better healthcare planning and patient management.", "### Modeling Recovery Time: The Normal Distribution Approach", "In one study, researchers modeled the recovery time of patients using a normal distribution with a mean recovery time of 14 days and a standard deviation of 3 days. This means that most patients recover within a predictable timeframe, centered around the average of 14 days, with variation captured by the standard deviation (3 days).", "Understanding how recovery times spread around the mean helps clinicians set realistic expectations and design efficient recovery protocols. A key calculation in this modeling is: what percentage of patients take more than 20 days to recover?", "---", "### Step-by-Step: Calculating the Percentage of Patients Recovering in More Than 20 Days", "To find this percentage, epidemiologists use z-scores and the properties of the normal distribution.", "#### 1. Compute the z-score\nThe z-score standardizes a given recovery time by subtracting the mean and dividing by the standard deviation:", "[\nz = \frac{X - \mu}{\sigma}\n]", "Where:\n- $ X = 20 $ days (the recovery time of interest),\n- $ \mu = 14 $ days (mean),\n- $ \sigma = 3 $ days (standard deviation).", "[\nz = \frac{20 - 14}{3} = \frac{6}{3} = 2\n]", "A z-score of 2 means 20 days is 2 standard deviations above the mean.", "#### 2. Use the 68-95-99.7 Rule\nThe empirical rule states:", "- Approximately 68% of data falls within ±1 standard deviation (between 11 and 17 days),\n- 95% within ±2 standard deviations (between 8 and 20 days),\n- 99.7% within ±3 standard deviations (between 5 and 23 days).", "With a z-score of 2, we know:\n- About 95% of patients recover in 8 to 20 days.\n- Therefore, only the remaining 5% fall outside this interval — 2.5% below 8 days and 2.5% above 20 days.", "#### 3. Find the percentage recovering in more than 20 days\nSince 95% recover in 8–20 days, the remaining 5% split evenly between below 8 days (2.5%) and above 20 days (2.5%).", "Thus,\nApproximately 2.5% of patients recover in more than 20 days.", "---", "### Real-World Implications", "Knowing that only 2.5% of patients exceed 20 days of recovery allows healthcare providers to:", "- Identify high-risk cases early\n- Allocate additional care resources effectively\n- Communicate realistic recovery timelines to patients and families", "This statistical modeling proves indispensable in public health planning and clinical decision-making.", "---", "Conclusion", "Modeling recovery times with a normal distribution provides clear, actionable insights. Using the z-score for 20 days (z = 2) reveals that 2.5% of patients take more than 20 days to recover, a key figure that supports predictive analytics and patient management. The 68-95-99.7 rule simplifies interpretation, turning complex data into practical understanding—fundamental in modern epidemiology."]