\( X \) is a normally distributed random variable with mean 41 and standard deviation 7 . What is the probability that \( X \) is between 55 and 62 ? Use the \( 0.68-0.95-0.997 \) rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

To determine the probability that \( X \) is between 55 and 62 using the \( 68-95-99.7 \) rule (also known as the empirical rule), follow these steps: 1. **Identify the Mean and Standard Deviation:** - Mean (\( \mu \)) = 41 - Standard Deviation (\( \sigma \)) = 7 2. **Calculate the Z-scores for 55 and 62:** - \( z_1 = \frac{55 - 41}{7} = 2 \) - \( z_2 = \frac{62 - 41}{7} = 3 \) 3. **Apply the Empirical Rule:** - Approximately **95%** of the data lies within **2 standard deviations** from the mean (\( \mu \pm 2\sigma \)). - Approximately **99.7%** of the data lies within **3 standard deviations** from the mean (\( \mu \pm 3\sigma \)). 4. **Determine the Probability Between \( z = 2 \) and \( z = 3 \):** - The probability within \( 2\sigma \) is 95%. - The probability within \( 3\sigma \) is 99.7%. - The probability between \( 2\sigma \) and \( 3\sigma \) is \( 99.7\% - 95\% = 4.7\% \). - Since this range is only on one side of the mean (from \( 2\sigma \) to \( 3\sigma \)), the probability is \( \frac{4.7\%}{2} = 2.35\% \). 5. **Express the Probability as a Decimal:** - \( 2.35\% \) as a decimal is \( 0.0235 \). - Rounded to the nearest thousandth, the probability is **0.024**. **Final Answer:** The probability is **0.024**.