\( X \) is a normally distributed random variable with mean 84 and standard deviation 2 . What is the probability that \( X \) is between 82 and 86 ? 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 82 and 86, given that \( X \) is normally distributed with a mean (\( \mu \)) of 84 and a standard deviation (\( \sigma \)) of 2, we can use the **68-95-99.7 rule** (also known as the empirical rule). ### Steps: 1. **Determine the number of standard deviations from the mean:** - **Lower bound (82):** \[ \frac{82 - 84}{2} = -1\sigma \] - **Upper bound (86):** \[ \frac{86 - 84}{2} = +1\sigma \] 2. **Apply the 68-95-99.7 rule:** - Approximately **68%** of the data in a normal distribution lies within **±1 standard deviation** from the mean. Therefore, the probability that \( X \) is between 82 and 86 is **68%**. Expressed as a decimal and rounded to the nearest thousandth: \[ 0.680 \] **Final Answer:** The probability is 0.680