1. **Problem Statement:** We are given a widget weight distribution and asked to use the Empirical Rule to find: a) The interval containing 95% of the weights. b) The percentage of weights between 48 and 63 ounces. c) The percentage of weights below 68 ounces. 2. **Empirical Rule:** For a normal distribution: - About 68% of data lies within 1 standard deviation ($\sigma$) of the mean ($\mu$). - About 95% lies within 2 standard deviations. - About 99.7% lies within 3 standard deviations. 3. **Assumptions:** Since the problem does not provide mean or standard deviation explicitly, we infer from the values given: - The interval 48 to 63 likely corresponds to $\mu - \sigma$ to $\mu + \sigma$. - The value 68 is likely $\mu + 2\sigma$. 4. **Step a) Find the 95% interval:** Using the Empirical Rule, 95% of weights lie between $\mu - 2\sigma$ and $\mu + 2\sigma$. Given $\mu + 2\sigma = 68$ and $\mu + \sigma = 63$, we find $\sigma$: $$ \sigma = 68 - 63 = 5 $$ Then $\mu = 63 - \sigma = 63 - 5 = 58$ So the 95% interval is: $$ [\mu - 2\sigma, \mu + 2\sigma] = [58 - 2(5), 58 + 2(5)] = [48, 68] $$ 5. **Step b) Percentage between 48 and 63:** 48 is $\mu - 2\sigma$ and 63 is $\mu + \sigma$. From the Empirical Rule: - Between $\mu - 2\sigma$ and $\mu + 2\sigma$ is 95%. - Between $\mu - \sigma$ and $\mu + \sigma$ is 68%. We want between $\mu - 2\sigma$ and $\mu + \sigma$. This is the sum of: - From $\mu - 2\sigma$ to $\mu - \sigma$: about 13.5% (half of the 27% outside 68% range) - From $\mu - \sigma$ to $\mu + \sigma$: 68% So total: $$ 13.5\% + 68\% = 81.5\% $$ 6. **Step c) Percentage below 68:** 68 is $\mu + 2\sigma$. From the Empirical Rule, 95% of data lies between $\mu - 2\sigma$ and $\mu + 2\sigma$. Since the distribution is symmetric, the percentage below $\mu + 2\sigma$ is: $$ 50\% + 47.5\% = 97.5\% $$ (50% below mean + 47.5% between mean and $\mu + 2\sigma$) **Final answers:** a) 95% of widget weights lie between **48 and 68** ounces. b) Percentage between 48 and 63 ounces is **81.5%**. c) Percentage below 68 ounces is **97.5%**.