\( X \) is a normally distributed random variable with mean 20 and standard deviation 20 . What is the probability that \( X \) is greater than 22 ? Write your answer as a decimal rounded to the nearest thousandth.

To find the probability that \( X \) is greater than 22 for a normally distributed random variable with a mean (\( \mu \)) of 20 and a standard deviation (\( \sigma \)) of 20, we first need to calculate the Z-score. The Z-score is given by the formula: \[ Z = \frac{X - \mu}{\sigma} \] Substituting the values: \[ Z = \frac{22 - 20}{20} = \frac{2}{20} = 0.1 \] Now, we look up the Z-score of 0.1 in the standard normal distribution table, or we can use a calculator. The table typically gives the probability that \( Z \) is less than a certain value. Looking up Z = 0.1, we find: \[ P(Z < 0.1) \approx 0.5398 \] To find \( P(X > 22) \), we need the complement of this probability: \[ P(X > 22) = 1 - P(Z < 0.1) = 1 - 0.5398 = 0.4602 \] Thus, the probability that \( X \) is greater than 22 is: \[ \boxed{0.460} \]