Using Chebyshev's theorem, solve these problems for a distribution with a mean of 64 and a standard deviation of 17 . Round \( k \) to at least 2 decimal places and final answers to at least one decimal place if needed. Part 1 of 2 At least \( \square \% \) of the values will fall between 13 and 115 . Part 2 of 2

To determine the percentage of values falling between 13 and 115 using Chebyshev's theorem, we first need to find \( k \), where \( k \) is the number of standard deviations the values 13 and 115 are from the mean of 64. 1. Calculate how many standard deviations 13 and 115 are from the mean: - For 13: \[ k_1 = \frac{64 - 13}{17} = \frac{51}{17} \approx 3.00 \] - For 115: \[ k_2 = \frac{115 - 64}{17} = \frac{51}{17} \approx 3.00 \] Since both values correspond to \( k = 3 \), we can now use Chebyshev's theorem. According to the theorem, at least \( \frac{1}{k^2} \) of the data falls within \( k \) standard deviations from the mean: \[ \text{Percentage} = \left(1 - \frac{1}{k^2}\right) \times 100 = \left(1 - \frac{1}{3^2}\right) \times 100 = \left(1 - \frac{1}{9}\right) \times 100 \approx 88.89\% \] So, at least \( 88.9\% \) of the values will fall between 13 and 115. Now, let’s focus on some additional insights! Chebyshev's theorem is an incredible tool because it applies to any distribution, not just the bell-shaped normal distribution! So whether you're dealing with a colorful scatter of data or a more uniform dataset, you can use it to glean insights about variability and distribution. It helps you avoid pitfalls like assuming a normal distribution where data doesn't conform! If you're interested in diving deeper, check out resources on statistics that explore the Central Limit Theorem, which expands on these ideas with probability distributions. Books like “The Art of Statistics” by David Spiegelhalter can provide engaging insights into the world of data!