\( X \) is a normally distributed random variable with mean 55 and standard deviation 18 . What is the probability that \( X \) is between 73 and 91 ? Use the \( 0.68-0.95-0.997 \) rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

To find the probability that \( X \) is between 73 and 91 for a normally distributed random variable with mean \( \mu = 55 \) and standard deviation \( \sigma = 18 \), we can follow these steps: 1. **Standardize the values**: We will convert the values 73 and 91 into z-scores using the formula: \[ z = \frac{(X - \mu)}{\sigma} \] 2. **Calculate the z-scores**: - For \( X = 73 \): \[ z_1 = \frac{(73 - 55)}{18} \] - For \( X = 91 \): \[ z_2 = \frac{(91 - 55)}{18} \] 3. **Use the z-scores to find probabilities**: We will then look up the probabilities corresponding to these z-scores in the standard normal distribution table or use the empirical rule. Let's calculate the z-scores first. ### Step 1: Calculate z-scores Calculating \( z_1 \) and \( z_2 \): \[ z_1 = \frac{(73 - 55)}{18} = \frac{18}{18} = 1 \] \[ z_2 = \frac{(91 - 55)}{18} = \frac{36}{18} = 2 \] ### Step 2: Find probabilities Using the empirical rule: - Approximately 68% of the data falls within 1 standard deviation of the mean. - Approximately 95% of the data falls within 2 standard deviations of the mean. From the z-scores: - The probability that \( X \) is less than 73 (which corresponds to \( z_1 = 1 \)) is approximately \( 0.8413 \). - The probability that \( X \) is less than 91 (which corresponds to \( z_2 = 2 \)) is approximately \( 0.9772 \). ### Step 3: Calculate the probability between 73 and 91 To find the probability that \( X \) is between 73 and 91, we subtract the probability of \( X < 73 \) from the probability of \( X < 91 \): \[ P(73 < X < 91) = P(X < 91) - P(X < 73) = 0.9772 - 0.8413 \] Now, let's calculate this: \[ P(73 < X < 91) = 0.9772 - 0.8413 = 0.1359 \] ### Final Answer Thus, the probability that \( X \) is between 73 and 91 is approximately \( 0.136 \) when rounded to the nearest thousandth.