\( X \) is a normally distributed random variable with mean 19 and standard deviation 2 . What is the probability that \( X \) is between 15 and 17 ? 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 15 and 17 using the \( 0.68-0.95-0.997 \) rule (also known as the empirical rule), follow these steps: 1. **Identify the Parameters:** - Mean (\( \mu \)) = 19 - Standard Deviation (\( \sigma \)) = 2 2. **Standardize the Values:** Convert the \( X \) values to \( Z \)-scores using the formula: \[ Z = \frac{X - \mu}{\sigma} \] - For \( X = 15 \): \[ Z = \frac{15 - 19}{2} = -2 \] - For \( X = 17 \): \[ Z = \frac{17 - 19}{2} = -1 \] 3. **Apply the Empirical Rule:** - **Within 1 Standard Deviation (\( Z = -1 \) to \( Z = 1 \)):** Approximately 68% of the data. - **Within 2 Standard Deviations (\( Z = -2 \) to \( Z = 2 \)):** Approximately 95% of the data. - **Within 3 Standard Deviations (\( Z = -3 \) to \( Z = 3 \)):** Approximately 99.7% of the data. However, we are interested in the probability between \( Z = -2 \) and \( Z = -1 \). 4. **Calculate the Probability:** - The probability from \( Z = -1 \) to \( Z = 1 \) is 68%, so each tail beyond this range is \( \frac{100\% - 68\%}{2} = 16\% \). - The probability beyond \( Z = -2 \) is half of the remaining 5% (since 95% is within \( \pm2\sigma \)), which is 2.5%. - Therefore, the probability between \( Z = -2 \) and \( Z = -1 \) is: \[ 16\% - 2.5\% = 13.5\% \] 5. **Express as a Decimal:** \[ 13.5\% = 0.135 \] **Final Answer:** The probability that \( X \) is between 15 and 17 is **0.135**.