\( X \) is a normally distributed random variable with mean 50 and standard deviation 12 . What is the probability that \( X \) is between 14 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 determine the probability that \( X \) is between 14 and 86, we'll use the \( 68\%-95\%-99.7\% \) rule, also known as the empirical rule, which applies to normally distributed data. 1. **Identify the Mean and Standard Deviation:** - Mean (\( \mu \)) = 50 - Standard Deviation (\( \sigma \)) = 12 2. **Determine How Many Standard Deviations Away 14 and 86 Are:** - For 14: \[ \frac{14 - 50}{12} = -3 \sigma \] - For 86: \[ \frac{86 - 50}{12} = +3 \sigma \] 3. **Apply the Empirical Rule:** - The rule states that approximately: - 68% of the data falls within \( \mu \pm 1\sigma \) - 95% within \( \mu \pm 2\sigma \) - **99.7% within \( \mu \pm 3\sigma \)** Since both 14 and 86 are \( 3\sigma \) away from the mean, the probability that \( X \) lies between 14 and 86 is approximately **99.7%**. 4. **Express as a Decimal:** - 99.7% as a decimal is **0.997**. **Answer:** 0.997