\( X \) is a normally distributed random variable with mean 29 and standard deviation 4 . What is the probability that \( X \) is between 34 and 36 ? Write your answer as a decimal rounded to the nearest thousandth.

To find the probability that \( X \) is between 34 and 36, where \( X \) is normally distributed with a mean (\( \mu \)) of 29 and a standard deviation (\( \sigma \)) of 4, follow these steps: 1. **Standardize the Values:** Convert the \( X \)-values to \( Z \)-scores using the formula: \[ Z = \frac{X - \mu}{\sigma} \] - For \( X = 34 \): \[ Z_1 = \frac{34 - 29}{4} = \frac{5}{4} = 1.25 \] - For \( X = 36 \): \[ Z_2 = \frac{36 - 29}{4} = \frac{7}{4} = 1.75 \] 2. **Use the Standard Normal Distribution Table:** - Find the cumulative probability for \( Z = 1.25 \) and \( Z = 1.75 \). - \( \Phi(1.25) \approx 0.8944 \) - \( \Phi(1.75) \approx 0.95994 \) 3. **Calculate the Probability:** Subtract the smaller cumulative probability from the larger one: \[ P(34 \leq X \leq 36) = \Phi(1.75) - \Phi(1.25) = 0.95994 - 0.8944 = 0.06554 \] 4. **Round to the Nearest Thousandth:** \[ 0.06554 \approx 0.066 \] **Final Answer:** The probability that \( X \) is between 34 and 36 is **0.066**.