1. **State the problem:** We need to determine which battery brand is the most consistent in battery life based on the given test results. 2. **Understanding consistency:** Consistency means the values vary the least. We measure this using the variance or standard deviation of the battery life data for each brand. 3. **Calculate the mean for each brand:** - Brand A mean: $$\frac{8.3 + 7.6 + 7.6 + 7.6 + 6.9}{5} = \frac{37.9}{5} = 7.58$$ - Brand B mean: $$\frac{8.4 + 8.4 + 7.2 + 7 + 7}{5} = \frac{38}{5} = 7.6$$ - Brand C mean: $$\frac{8.5 + 8.4 + 7.1 + 7.1 + 7.1}{5} = \frac{38.2}{5} = 7.64$$ - Brand D mean: $$\frac{8.6 + 8.6 + 8.6 + 8.5 + 7.2}{5} = \frac{41.5}{5} = 8.3$$ 4. **Calculate variance for each brand:** Variance formula: $$\text{variance} = \frac{\sum (x_i - \bar{x})^2}{n}$$ where $x_i$ are data points and $\bar{x}$ is the mean. - Brand A variance: $$\frac{(8.3-7.58)^2 + (7.6-7.58)^2 + (7.6-7.58)^2 + (7.6-7.58)^2 + (6.9-7.58)^2}{5}$$ $$= \frac{0.5184 + 0.0004 + 0.0004 + 0.0004 + 0.4624}{5} = \frac{0.982}{5} = 0.1964$$ - Brand B variance: $$\frac{(8.4-7.6)^2 + (8.4-7.6)^2 + (7.2-7.6)^2 + (7-7.6)^2 + (7-7.6)^2}{5}$$ $$= \frac{0.64 + 0.64 + 0.16 + 0.36 + 0.36}{5} = \frac{2.12}{5} = 0.424$$ - Brand C variance: $$\frac{(8.5-7.64)^2 + (8.4-7.64)^2 + (7.1-7.64)^2 + (7.1-7.64)^2 + (7.1-7.64)^2}{5}$$ $$= \frac{0.7396 + 0.5776 + 0.2916 + 0.2916 + 0.2916}{5} = \frac{2.192}{5} = 0.4384$$ - Brand D variance: $$\frac{(8.6-8.3)^2 + (8.6-8.3)^2 + (8.6-8.3)^2 + (8.5-8.3)^2 + (7.2-8.3)^2}{5}$$ $$= \frac{0.09 + 0.09 + 0.09 + 0.04 + 1.21}{5} = \frac{1.52}{5} = 0.304$$ 5. **Compare variances:** Lower variance means more consistency. - Brand A: 0.1964 - Brand B: 0.424 - Brand C: 0.4384 - Brand D: 0.304 6. **Conclusion:** Brand A has the lowest variance and is the most consistent battery brand. **Final answer:** Brand A is the most consistent.