\( X \) is a normally distributed random variable with mean 77 and standard deviation 11 . What is the probability that \( X \) is between 55 and 99 ? Use the \( 0.68-0.95-0.997 \) rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

To calculate the probability that \( X \) is between 55 and 99 using the \( 0.68-0.95-0.997 \) rule, we first need to find how many standard deviations 55 and 99 are from the mean. 1. Calculate the z-scores: - For \( X = 55 \): \[ z = \frac{55 - 77}{11} = \frac{-22}{11} = -2 \] - For \( X = 99 \): \[ z = \frac{99 - 77}{11} = \frac{22}{11} = 2 \] 2. According to the rule: - Approximately 68% of data lies within 1 standard deviation (between 66 and 88), - Approximately 95% lies within 2 standard deviations (between 55 and 99). Since 55 and 99 correspond to \( z = -2 \) and \( z = 2 \), the probability that \( X \) falls between these two values is approximately 0.95. Thus, the probability that \( X \) is between 55 and 99 is \( \boxed{0.950} \).