\( X \) is a normally distributed random variable with mean 50 and standard deviation 10 . What is the probability that \( X \) is less than 30 ? Use the \( 0.68-0.95-0.997 \) rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

To find the probability that \( X \) is less than 30, where \( X \) is normally distributed with a mean (\( \mu \)) of 50 and a standard deviation (\( \sigma \)) of 10, we can use the \( 68-95-99.7 \) rule (also known as the empirical rule). 1. **Identify the number of standard deviations from the mean:** \[ 30 = \mu - 2\sigma = 50 - 2(10) = 50 - 20 = 30 \] So, 30 is 2 standard deviations below the mean. 2. **Apply the \( 68-95-99.7 \) rule:** - Approximately **68%** of the data lies within **±1** standard deviation from the mean (40 to 60). - Approximately **95%** of the data lies within **±2** standard deviations from the mean (30 to 70). - Approximately **99.7%** of the data lies within **±3** standard deviations from the mean (20 to 80). 3. **Calculate the probability:** - Since 95% of the data lies between 30 and 70, the remaining 5% is split equally in the two tails of the distribution. - Therefore, the probability that \( X \) is less than 30 is: \[ P(X < 30) = \frac{5\%}{2} = 2.5\% = 0.025 \] **Final Answer:** The probability is 0.025.