1. **State the problem:** We have calorie counts for different types of donuts: one type with 560 calories, four types with 380 calories, two types with 430 calories, four types with 460 calories, and three types with 490 calories. We need to find the range and the sample standard deviation of these calorie counts. 2. **Find the range:** The range is the difference between the maximum and minimum values. Maximum calories = 560 Minimum calories = 380 Range = $560 - 380 = 180$ 3. **Find the sample standard deviation:** The formula for sample standard deviation $s$ is: $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}$$ where $n$ is the total number of data points and $\bar{x}$ is the sample mean. 4. **Calculate total number of donuts $n$:** $n = 1 + 4 + 2 + 4 + 3 = 14$ 5. **Calculate the sample mean $\bar{x}$:** $$\bar{x} = \frac{1\times560 + 4\times380 + 2\times430 + 4\times460 + 3\times490}{14}$$ $$= \frac{560 + 1520 + 860 + 1840 + 1470}{14} = \frac{6250}{14} \approx 446.43$$ 6. **Calculate each squared deviation multiplied by frequency:** - $(560 - 446.43)^2 \times 1 = (113.57)^2 \times 1 = 12896.75$ - $(380 - 446.43)^2 \times 4 = (-66.43)^2 \times 4 = 4412.15 \times 4 = 17648.6$ - $(430 - 446.43)^2 \times 2 = (-16.43)^2 \times 2 = 269.92 \times 2 = 539.84$ - $(460 - 446.43)^2 \times 4 = (13.57)^2 \times 4 = 184.12 \times 4 = 736.48$ - $(490 - 446.43)^2 \times 3 = (43.57)^2 \times 3 = 1898.25 \times 3 = 5694.75$ 7. **Sum of squared deviations:** $$12896.75 + 17648.6 + 539.84 + 736.48 + 5694.75 = 37516.42$$ 8. **Calculate sample variance:** $$s^2 = \frac{37516.42}{14 - 1} = \frac{37516.42}{13} \approx 2885.11$$ 9. **Calculate sample standard deviation:** $$s = \sqrt{2885.11} \approx 53.7$$ **Final answers:** - Range = 180 calories - Sample standard deviation $\approx 53.7$ calories