\( X \) is a normally distributed random variable with mean 38 and standard deviation 13 . What is the probability that \( X \) is between 51 and 77 ? 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 51 and 77 for a normally distributed random variable with mean \( \mu = 38 \) and standard deviation \( \sigma = 13 \), we can follow these steps: 1. **Standardize the values**: We will convert the values 51 and 77 into z-scores using the formula: \[ z = \frac{(X - \mu)}{\sigma} \] 2. **Calculate the z-scores**: - For \( X = 51 \): \[ z_1 = \frac{(51 - 38)}{13} \] - For \( X = 77 \): \[ z_2 = \frac{(77 - 38)}{13} \] 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 \( z_2 \): \[ z_1 = \frac{(51 - 38)}{13} = \frac{13}{13} = 1 \] \[ z_2 = \frac{(77 - 38)}{13} = \frac{39}{13} = 3 \] ### Step 2: Find probabilities Using the empirical rule: - Approximately 68% of the data falls within 1 standard deviation of the mean (between \( \mu - \sigma \) and \( \mu + \sigma \)). - Approximately 95% of the data falls within 2 standard deviations of the mean (between \( \mu - 2\sigma \) and \( \mu + 2\sigma \)). - Approximately 99.7% of the data falls within 3 standard deviations of the mean (between \( \mu - 3\sigma \) and \( \mu + 3\sigma \)). Now, we need to find the probability that \( X \) is between 51 and 77, which corresponds to \( z_1 = 1 \) and \( z_2 = 3 \). - The probability that \( Z < 1 \) is approximately 0.8413 (from z-tables). - The probability that \( Z < 3 \) is approximately 0.9987 (from z-tables). ### Step 3: Calculate the probability between z-scores Now, we can find the probability that \( X \) is between 51 and 77: \[ P(51 < X < 77) = P(Z < 3) - P(Z < 1) = 0.9987 - 0.8413 \] Calculating this gives: \[ P(51 < X < 77) = 0.9987 - 0.8413 = 0.1574 \] ### Final Answer Thus, the probability that \( X \) is between 51 and 77 is approximately \( 0.157 \) when rounded to the nearest thousandth. \[ \boxed{0.157} \]