1. **Problem Statement:** Calculate the sample mean operating time and sample variance for the given machine hours data: 3, 6, 2, 5, 5, 4, 5. 2. **Formula for Sample Mean:** $$\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$$ where $x_i$ are the observed values and $n$ is the number of observations. 3. **Calculate the Sample Mean:** Sum of hours: $3 + 6 + 2 + 5 + 5 + 4 + 5 = 30$ Number of machines: $n = 7$ $$\bar{x} = \frac{30}{7} \approx 4.2857$$ 4. **Formula for Sample Variance:** $$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$ 5. **Calculate Each Squared Deviation:** $(3 - 4.2857)^2 = 1.6531$ $(6 - 4.2857)^2 = 2.9388$ $(2 - 4.2857)^2 = 5.2245$ $(5 - 4.2857)^2 = 0.5102$ $(5 - 4.2857)^2 = 0.5102$ $(4 - 4.2857)^2 = 0.0816$ $(5 - 4.2857)^2 = 0.5102$ Sum of squared deviations: $$1.6531 + 2.9388 + 5.2245 + 0.5102 + 0.5102 + 0.0816 + 0.5102 = 11.4286$$ 6. **Calculate Sample Variance:** $$s^2 = \frac{11.4286}{7 - 1} = \frac{11.4286}{6} \approx 1.9048$$ **Final answers:** - Sample mean operating time: $\bar{x} \approx 4.29$ hours - Sample variance: $s^2 \approx 1.90$ hours squared