\( X \) is a normally distributed random variable with mean 52 and standard deviation 14 . What is the probability that \( X \) is between 10 and 80 ? 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 the normally distributed random variable \( X \) with mean \( \mu = 52 \) and standard deviation \( \sigma = 14 \) lies between 10 and 80, we can use the **68-95-99.7 rule** (also known as the empirical rule). This rule approximates the distribution of data in a normal distribution: - **68%** of the data falls within **±1σ** of the mean. - **95%** of the data falls within **±2σ** of the mean. - **99.7%** of the data falls within **±3σ** of the mean. ### Step-by-Step Calculation: 1. **Determine How Many Standard Deviations 10 and 80 Are from the Mean:** - **For 10:** \[ \text{Number of } \sigma \text{ from mean} = \frac{\mu - 10}{\sigma} = \frac{52 - 10}{14} = \frac{42}{14} = 3σ \] So, 10 is **3σ below** the mean. - **For 80:** \[ \text{Number of } \sigma \text{ from mean} = \frac{80 - \mu}{\sigma} = \frac{80 - 52}{14} = \frac{28}{14} = 2σ \] So, 80 is **2σ above** the mean. 2. **Apply the Empirical Rule:** - The interval from **\( -3σ \) to \( +3σ \)** encompasses **99.7%** of the data. - The interval from **\( -2σ \) to \( +2σ \)** encompasses **95%** of the data. - However, our interval of interest is from **\( -3σ \)** (which is 10) to **\( +2σ \)** (which is 80). 3. **Calculate the Probability:** - The total probability within \( -3σ \) to \( +3σ \) is **99.7%**. - The probability beyond \( +2σ \) to \( +3σ \) is approximately **2.5%** (since the empirical rule is symmetric). - Therefore, the probability from \( -3σ \) to \( +2σ \) is: \[ 99.7\% - 2.5\% = 97.2\% \] 4. **Express as a Decimal:** \[ 97.2\% = 0.972 \] ### Final Answer: The probability that \( X \) is between 10 and 80 is **0.972**.