1. **State the problem:** We have a group of butterfly incubation times in days: 6.7, 5.7, 4.6, 9.4, 9.4, 9.5, 6.8, 8.3, 3.3, 7.8, 4, 8.8, 9.6. We need to find: - The range of incubation times (difference between maximum and minimum). - The standard deviation of these times. 2. **Find the range:** - The minimum value is $3.3$ days. - The maximum value is $9.6$ days. - Range formula: $$\text{Range} = \text{max} - \text{min}$$ - Calculate: $$9.6 - 3.3 = 6.3$$ 3. **Calculate the standard deviation:** - Step 1: Find the mean (average) $$\bar{x} = \frac{\sum x_i}{n}$$ where $n=13$. - Sum of values: $$6.7 + 5.7 + 4.6 + 9.4 + 9.4 + 9.5 + 6.8 + 8.3 + 3.3 + 7.8 + 4 + 8.8 + 9.6 = 93.9$$ - Mean: $$\bar{x} = \frac{93.9}{13} \approx 7.223$$ - Step 2: Calculate each squared deviation $(x_i - \bar{x})^2$ and sum them: - $(6.7 - 7.223)^2 = 0.274$ - $(5.7 - 7.223)^2 = 2.324$ - $(4.6 - 7.223)^2 = 6.876$ - $(9.4 - 7.223)^2 = 4.722$ - $(9.4 - 7.223)^2 = 4.722$ - $(9.5 - 7.223)^2 = 5.182$ - $(6.8 - 7.223)^2 = 0.180$ - $(8.3 - 7.223)^2 = 1.158$ - $(3.3 - 7.223)^2 = 15.384$ - $(7.8 - 7.223)^2 = 0.333$ - $(4 - 7.223)^2 = 10.404$ - $(8.8 - 7.223)^2 = 2.488$ - $(9.6 - 7.223)^2 = 5.664$ - Sum of squared deviations: $$0.274 + 2.324 + 6.876 + 4.722 + 4.722 + 5.182 + 0.180 + 1.158 + 15.384 + 0.333 + 10.404 + 2.488 + 5.664 = 59.511$$ - Step 3: Calculate variance (using $n-1=12$ for sample standard deviation): $$s^2 = \frac{59.511}{12} = 4.959$$ - Step 4: Calculate standard deviation: $$s = \sqrt{4.959} \approx 2.227$$ **Final answers:** - Range: $6.3$ days - Standard deviation: $2.227$ days