1. **Problem Statement:** Find the range, variance, and standard deviation of the net worth data (in billions): 59, 52, 28, 26, 19, 19, 18, 17, 17, 17. 2. **Range:** The range is the difference between the maximum and minimum values. $$\text{Range} = 59 - 17 = 42$$ 3. **Variance and Standard Deviation:** Use the formulas: $$\text{Mean} = \frac{\sum x_i}{n}$$ $$\text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n}$$ $$\text{Standard Deviation} = \sqrt{\text{Variance}}$$ 4. Calculate the mean: $$\bar{x} = \frac{59 + 52 + 28 + 26 + 19 + 19 + 18 + 17 + 17 + 17}{10} = \frac{272}{10} = 27.2$$ 5. Calculate each squared deviation: $$(59 - 27.2)^2 = 31.8^2 = 1011.24$$ $$(52 - 27.2)^2 = 24.8^2 = 615.04$$ $$(28 - 27.2)^2 = 0.8^2 = 0.64$$ $$(26 - 27.2)^2 = (-1.2)^2 = 1.44$$ $$(19 - 27.2)^2 = (-8.2)^2 = 67.24$$ $$(19 - 27.2)^2 = 67.24$$ $$(18 - 27.2)^2 = (-9.2)^2 = 84.64$$ $$(17 - 27.2)^2 = (-10.2)^2 = 104.04$$ $$(17 - 27.2)^2 = 104.04$$ $$(17 - 27.2)^2 = 104.04$$ 6. Sum the squared deviations: $$1011.24 + 615.04 + 0.64 + 1.44 + 67.24 + 67.24 + 84.64 + 104.04 + 104.04 + 104.04 = 2159.6$$ 7. Calculate variance: $$\text{Variance} = \frac{2159.6}{10} = 215.96$$ 8. Calculate standard deviation: $$\text{Standard Deviation} = \sqrt{215.96} \approx 14.7$$ --- 1. **Problem Statement:** Find the variance and standard deviation for the inches of rain data with given frequency distribution and sums: Given: $$n = 25, \sum f x_m = 12225, \sum f x_m^2 = 329431$$ 2. Calculate the mean: $$\bar{x} = \frac{\sum f x_m}{n} = \frac{12225}{25} = 489$$ 3. Calculate variance using formula: $$\text{Variance} = \frac{\sum f x_m^2}{n} - \bar{x}^2 = \frac{329431}{25} - 489^2$$ 4. Calculate each term: $$\frac{329431}{25} = 13177.24$$ $$489^2 = 239121$$ 5. Calculate variance: $$13177.24 - 239121 = -225943.76$$ This negative variance is impossible, indicating a misinterpretation of the data. 6. Correct interpretation: The sum $\sum f x_m = 12225$ likely represents $\sum f x_m^2$ and $\sum f x_m^2 = 329431$ is $\sum f x_m^3$ or a misprint. 7. Using the given sums as: $$\sum f x_m = 329431, \sum f x_m^2 = 12225$$ Calculate mean: $$\bar{x} = \frac{329431}{25} = 13177.24$$ Calculate variance: $$\text{Variance} = \frac{12225}{25} - (13177.24)^2 = 489 - 173654000 \approx -173653511$$ Still negative, so data likely misaligned. 8. Since the problem states $n=25$, $\sum f x_m = 12225$, and $\sum f x_m^2 = 329431$, use these: $$\bar{x} = \frac{12225}{25} = 489$$ $$\text{Variance} = \frac{329431}{25} - 489^2 = 13177.24 - 239121 = -225943.76$$ Negative variance again. 9. Conclusion: The data for problem #5 is inconsistent or incorrectly transcribed, so variance and standard deviation cannot be reliably calculated. **Final answers:** - Problem #4 Range: 42 - Problem #4 Variance: 215.96 - Problem #4 Standard Deviation: 14.7 - Problem #5 Variance and Standard Deviation: Cannot be determined due to inconsistent data.