\( X \) is a normally distributed random variable with mean 39 and standard deviation 16 . What is the probability that \( X \) is between 7 and 87 ? 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 7 and 87, we can apply the empirical rule, often referred to as the \( 0.68-0.95-0.997 \) rule, which relates to how data is distributed around the mean in a normal distribution. 1. **Calculate Z-scores**: - For \( X = 7 \): \[ Z = \frac{7 - 39}{16} = \frac{-32}{16} = -2 \] - For \( X = 87 \): \[ Z = \frac{87 - 39}{16} = \frac{48}{16} = 3 \] 2. **Use the empirical rule**: - The range \( -2 \) to \( 3 \) involves: - From \( -2 \) to \( 0 \) covers 50% of the area below the mean (which is 68%). - The area from \( 0 \) to \( +3 \) can be approximated by knowing that from the mean to \( +3 \) approximately covers 49.85% of the total area (since 95% is within -2 to +2, we add the area in the tail). So, we can combine the probabilities: - The area from \( Z = -2 \) to \( Z = 0 \) (which is 34%) plus the area from \( Z = 0 \) to \( Z = 3 \) (which is approximately 49.85%) totals to: \[ P(7 < X < 87) \approx 0.34 + 0.4975 \approx 0.8375 \] Thus, rounding to the nearest thousandth, the probability that \( X \) is between 7 and 87 is approximately \( 0.838 \).