\( X \) is a normally distributed random variable with mean 74 and standard deviation 17. What is the probability that \( X \) is between 45 and 85 ? Write your answer as a decimal rounded to the nearest thousandth.

To find the probability that \( X \) is between 45 and 85 for a normally distributed variable with mean 74 and standard deviation 17, we need to standardize the values using the Z-score formula: \[ Z = \frac{X - \mu}{\sigma} \] 1. Calculate the Z-scores for 45 and 85: For \( X = 45 \): \[ Z_{45} = \frac{45 - 74}{17} \approx -1.706 \] For \( X = 85 \): \[ Z_{85} = \frac{85 - 74}{17} \approx 0.647 \] 2. Now, look up these Z-scores in the standard normal distribution table or use a calculator: - The probability associated with \( Z_{45} \approx -1.706 \) is approximately 0.044. - The probability associated with \( Z_{85} \approx 0.647 \) is approximately 0.738. 3. To find the probability that \( X \) is between 45 and 85, we subtract the probability at \( Z_{45} \) from the probability at \( Z_{85} \): \[ P(45 < X < 85) = P(Z < 0.647) - P(Z < -1.706) \\ \approx 0.738 - 0.044 = 0.694 \] Thus, the probability that \( X \) is between 45 and 85 is approximately **0.694** when rounded to the nearest thousandth.