\( X \) is a normally distributed random variable with mean 44 and standard deviation 5 . What is the probability that \( X \) is between 34 and 49 ? 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 34 and 49, we will use the \( 0.68-0.95-0.997 \) rule, also known as the empirical rule. 1. First, we calculate how many standard deviations 34 and 49 are from the mean \( \mu = 44 \). - For 34: \[ z_1 = \frac{34 - 44}{5} = \frac{-10}{5} = -2 \] - For 49: \[ z_2 = \frac{49 - 44}{5} = \frac{5}{5} = 1 \] 2. Now, using the empirical rule, we know: - About 68% of the data lies within 1 standard deviation (from \( \mu - 1\sigma \) to \( \mu + 1\sigma \)), which is between 39 and 49. - About 95% of the data lies within 2 standard deviations (from \( \mu - 2\sigma \) to \( \mu + 2\sigma \)), which is between 34 and 54. 3. Therefore, to find the probability that \( X \) is between 34 and 49, we find the area from 34 to 44 and from 44 to 49: - From 34 to 44 (which is half of the 95%): \( \frac{95\%}{2} = 47.5\% \) - From 44 to 49 (from 44 to 49 is half of the 68%): \( \frac{68\%}{2} = 34\% \) Combining these, we get \( 47.5\% + 34\% = 81.5\% \). Therefore, rounding to the nearest thousandth, the probability that \( X \) is between 34 and 49 is: \[ \text{Probability} = 0.815 \]