A normal distribution has a mean of 8 and a standard deviation of 2 . Use the \( 68-95-99.7 \) rule to find the percentage of values in the distribution between 8 and 14. What is the percentage of values in the distribution between 8 and 14 ? \( \square \% \)

The \( 68-95-99.7 \) rule, also known as the empirical rule, states that for a normal distribution: - Approximately 68% of the data falls within one standard deviation of the mean. - Approximately 95% of the data falls within two standard deviations of the mean. - Approximately 99.7% of the data falls within three standard deviations of the mean. Given a normal distribution with a mean (\(\mu\)) of 8 and a standard deviation (\(\sigma\)) of 2: 1. **Calculate the range for one standard deviation:** - One standard deviation above the mean: \( \mu + \sigma = 8 + 2 = 10 \) - One standard deviation below the mean: \( \mu - \sigma = 8 - 2 = 6 \) Thus, approximately 68% of the values fall between 6 and 10. 2. **Calculate the range for two standard deviations:** - Two standard deviations above the mean: \( \mu + 2\sigma = 8 + 2(2) = 12 \) - Two standard deviations below the mean: \( \mu - 2\sigma = 8 - 2(2) = 4 \) Thus, approximately 95% of the values fall between 4 and 12. 3. **Calculate the range for three standard deviations:** - Three standard deviations above the mean: \( \mu + 3\sigma = 8 + 3(2) = 14 \) - Three standard deviations below the mean: \( \mu - 3\sigma = 8 - 3(2) = 2 \) Thus, approximately 99.7% of the values fall between 2 and 14. Now, we want to find the percentage of values between 8 and 14. - The range from 8 to 14 includes the mean (8) and extends to three standard deviations above the mean (14). - Since 99.7% of the values fall between 2 and 14, we need to find the percentage of values from 8 to 14. Since the distribution is symmetric, the percentage of values from the mean (8) to three standard deviations above the mean (14) is half of the total percentage from 2 to 14. Thus, the percentage of values between 8 and 14 is: \[ \frac{99.7\%}{2} = 49.85\% \] Therefore, the percentage of values in the distribution between 8 and 14 is approximately \( 49.85\% \). So, the final answer is: \[ \boxed{49.85\%} \]