Variance Standard Deviation 9836C4

1. **State the problem:** We need to find the sample variance and sample standard deviation for the battery usage hours of three companies: EverSoBright, Dependable, and Beacon. 2. **Recall formulas:** - Sample mean: $$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$$ - Sample variance: $$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$ - Sample standard deviation: $$s = \sqrt{s^2}$$ 3. **Calculate for EverSoBright:** - Data: $6.2, 6.4, 7.1, 5.9, 8.3, 5.3, 7.5, 9.3$ - Number of data points: $n=8$ - Mean: $$\bar{x} = \frac{6.2 + 6.4 + 7.1 + 5.9 + 8.3 + 5.3 + 7.5 + 9.3}{8} = \frac{55}{8} = 6.875$$ - Calculate squared deviations: $$ (6.2 - 6.875)^2 = 0.4556 $$ $$ (6.4 - 6.875)^2 = 0.2256 $$ $$ (7.1 - 6.875)^2 = 0.0506 $$ $$ (5.9 - 6.875)^2 = 0.9506 $$ $$ (8.3 - 6.875)^2 = 2.0306 $$ $$ (5.3 - 6.875)^2 = 2.4806 $$ $$ (7.5 - 6.875)^2 = 0.3906 $$ $$ (9.3 - 6.875)^2 = 5.8906 $$ - Sum of squared deviations: $$0.4556 + 0.2256 + 0.0506 + 0.9506 + 2.0306 + 2.4806 + 0.3906 + 5.8906 = 12.4748$$ - Sample variance: $$s^2 = \frac{12.4748}{8-1} = \frac{12.4748}{7} = 1.7821$$ - Sample standard deviation: $$s = \sqrt{1.7821} = 1.334$$ 4. **Calculate for Dependable:** - Data: $6.8, 6.2, 7.2, 5.9, 7.0, 7.4, 7.3, 8.2$ - $n=8$ - Mean: $$\bar{x} = \frac{6.8 + 6.2 + 7.2 + 5.9 + 7.0 + 7.4 + 7.3 + 8.2}{8} = \frac{55}{8} = 6.875$$ - Squared deviations: $$ (6.8 - 6.875)^2 = 0.0056 $$ $$ (6.2 - 6.875)^2 = 0.4556 $$ $$ (7.2 - 6.875)^2 = 0.1056 $$ $$ (5.9 - 6.875)^2 = 0.9506 $$ $$ (7.0 - 6.875)^2 = 0.0156 $$ $$ (7.4 - 6.875)^2 = 0.2756 $$ $$ (7.3 - 6.875)^2 = 0.1806 $$ $$ (8.2 - 6.875)^2 = 1.7556 $$ - Sum: $$0.0056 + 0.4556 + 0.1056 + 0.9506 + 0.0156 + 0.2756 + 0.1806 + 1.7556 = 3.7448$$ - Variance: $$s^2 = \frac{3.7448}{7} = 0.535$$ - Standard deviation: $$s = \sqrt{0.535} = 0.731$$ 5. **Calculate for Beacon:** - Data: $6.1, 6.6, 7.3, 5.7, 7.1, 7.6, 7.1, 8.5$ - $n=8$ - Mean: $$\bar{x} = \frac{6.1 + 6.6 + 7.3 + 5.7 + 7.1 + 7.6 + 7.1 + 8.5}{8} = \frac{56}{8} = 7.0$$ - Squared deviations: $$ (6.1 - 7.0)^2 = 0.81 $$ $$ (6.6 - 7.0)^2 = 0.16 $$ $$ (7.3 - 7.0)^2 = 0.09 $$ $$ (5.7 - 7.0)^2 = 1.69 $$ $$ (7.1 - 7.0)^2 = 0.01 $$ $$ (7.6 - 7.0)^2 = 0.36 $$ $$ (7.1 - 7.0)^2 = 0.01 $$ $$ (8.5 - 7.0)^2 = 2.25 $$ - Sum: $$0.81 + 0.16 + 0.09 + 1.69 + 0.01 + 0.36 + 0.01 + 2.25 = 5.38$$ - Variance: $$s^2 = \frac{5.38}{7} = 0.7686$$ - Standard deviation: $$s = \sqrt{0.7686} = 0.877$$ **Final answers:** - EverSoBright variance: $1.7821$, standard deviation: $1.334$ - Dependable variance: $0.535$, standard deviation: $0.731$ - Beacon variance: $0.7686$, standard deviation: $0.877$