1. **State the problem:** We need to find the lower and upper fences of the data set using the formulas: $$\text{Lower fence} = \mu - k\sigma$$ $$\text{Upper fence} = \mu + k\sigma$$ where $\mu$ is the mean, $\sigma$ is the standard deviation, and $k=2.5$. 2. **Calculate the mean $\mu$:** Given data: $1, 10, 13, 14, 20, 25, 31, 46, 73$ $$\mu = \frac{1 + 10 + 13 + 14 + 20 + 25 + 31 + 46 + 73}{9} = \frac{233}{9} \approx 25.89$$ 3. **Calculate the standard deviation $\sigma$:** First, find each squared deviation from the mean: $$(1 - 25.89)^2 = 618.53$$ $$(10 - 25.89)^2 = 252.92$$ $$(13 - 25.89)^2 = 165.32$$ $$(14 - 25.89)^2 = 141.32$$ $$(20 - 25.89)^2 = 34.69$$ $$(25 - 25.89)^2 = 0.79$$ $$(31 - 25.89)^2 = 26.21$$ $$(46 - 25.89)^2 = 404.45$$ $$(73 - 25.89)^2 = 2220.45$$ Sum of squared deviations: $$618.53 + 252.92 + 165.32 + 141.32 + 34.69 + 0.79 + 26.21 + 404.45 + 2220.45 = 3864.68$$ Divide by $n-1=8$ for sample standard deviation: $$\sigma = \sqrt{\frac{3864.68}{8}} = \sqrt{483.09} \approx 21.98$$ 4. **Calculate the fences:** $$\text{Lower fence} = 25.89 - 2.5 \times 21.98 = 25.89 - 54.95 = -29.06 \approx -29$$ $$\text{Upper fence} = 25.89 + 2.5 \times 21.98 = 25.89 + 54.95 = 80.84 \approx 81$$ 5. **Final answer:** Lower fence = $-29$ Upper fence = $81$ These fences help identify outliers in the data set by marking boundaries beyond which data points are considered unusual.