\( X \) is a normally distributed random variable with mean 50 and standard deviation 16 . What is the probability that \( X \) is greater than \( 19 ? \) Write your answer as a decimal rounded to the nearest thousandth.

To determine the probability that \( X \) is greater than 19, where \( X \) is normally distributed with a mean (\( \mu \)) of 50 and a standard deviation (\( \sigma \)) of 16, follow these steps: 1. **Standardize the Variable**: Convert \( X = 19 \) to a standard normal variable \( Z \) using the formula: \[ Z = \frac{X - \mu}{\sigma} = \frac{19 - 50}{16} = \frac{-31}{16} \approx -1.9375 \] 2. **Find the Probability**: We need to find \( P(X > 19) \), which is equivalent to \( P(Z > -1.9375) \). Due to the symmetry of the normal distribution: \[ P(Z > -1.9375) = P(Z < 1.9375) \] 3. **Use the Standard Normal Distribution Table**: Look up the value for \( Z = 1.9375 \) in the standard normal distribution table. Since tables typically provide values up to two decimal places: - \( Z = 1.93 \) corresponds to \( \Phi(1.93) \approx 0.9732 \) - \( Z = 1.94 \) corresponds to \( \Phi(1.94) \approx 0.9738 \) For \( Z = 1.9375 \), interpolate between these two values: \[ \Phi(1.9375) \approx 0.9732 + 0.00045 = 0.97365 \] 4. **Round the Result**: Rounding to the nearest thousandth: \[ P(X > 19) \approx 0.974 \] **Final Answer:** The probability that \( X \) is greater than 19 is approximately **0.974**.