1. **Problem 19:** Given the data set 2, 4, 5, 8, 11, determine the mean, median, mode, range, variance, and standard deviation. 2. **Mean:** The mean is the average of the data points. $$\text{Mean} = \frac{2 + 4 + 5 + 8 + 11}{5} = \frac{30}{5} = 6$$ 3. **Median:** The median is the middle value when data is ordered. Ordered data: 2, 4, 5, 8, 11 Middle value (3rd) is 5. $$\text{Median} = 5$$ 4. **Mode:** The mode is the most frequent value. All values appear once, so no mode. $$\text{Mode} = \text{None}$$ 5. **Range:** Difference between max and min values. $$\text{Range} = 11 - 2 = 9$$ 6. **Variance:** Use formula for sample variance: $$s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$$ Calculate squared deviations: $(2-6)^2=16$, $(4-6)^2=4$, $(5-6)^2=1$, $(8-6)^2=4$, $(11-6)^2=25$ Sum: $16+4+1+4+25=50$ $$s^2 = \frac{50}{4} = 12.5$$ 7. **Standard deviation:** Square root of variance. $$s = \sqrt{12.5} \approx 3.54$$ --- 8. **Problem 20:** Given matrices $$A = \begin{bmatrix}1 & 0 & -1 \\ 2 & 1 & 3 \\ 0 & 1 & 2\end{bmatrix}, \quad B = \begin{bmatrix}2 & -1 & 0 \\ 0 & 1 & 3 \\ -1 & 0 & 1\end{bmatrix}$$ 9. a) Compute $A - B$: $$A - B = \begin{bmatrix}1-2 & 0-(-1) & -1-0 \\ 2-0 & 1-1 & 3-3 \\ 0-(-1) & 1-0 & 2-1\end{bmatrix} = \begin{bmatrix}-1 & 1 & -1 \\ 2 & 0 & 0 \\ 1 & 1 & 1\end{bmatrix}$$ 10. b) Compute $2A + B$: $$2A = \begin{bmatrix}2 & 0 & -2 \\ 4 & 2 & 6 \\ 0 & 2 & 4\end{bmatrix}$$ $$2A + B = \begin{bmatrix}2+2 & 0+(-1) & -2+0 \\ 4+0 & 2+1 & 6+3 \\ 0+(-1) & 2+0 & 4+1\end{bmatrix} = \begin{bmatrix}4 & -1 & -2 \\ 4 & 3 & 9 \\ -1 & 2 & 5\end{bmatrix}$$ 11. c) Compute $A^T - 4B$: Transpose of $A$: $$A^T = \begin{bmatrix}1 & 2 & 0 \\ 0 & 1 & 1 \\ -1 & 3 & 2\end{bmatrix}$$ $$4B = \begin{bmatrix}8 & -4 & 0 \\ 0 & 4 & 12 \\ -4 & 0 & 4\end{bmatrix}$$ $$A^T - 4B = \begin{bmatrix}1-8 & 2-(-4) & 0-0 \\ 0-0 & 1-4 & 1-12 \\ -1-(-4) & 3-0 & 2-4\end{bmatrix} = \begin{bmatrix}-7 & 6 & 0 \\ 0 & -3 & -11 \\ 3 & 3 & -2\end{bmatrix}$$ --- 12. **Problem 21:** 13. a) Evaluate: $$\frac{5}{6} + \frac{1}{2} \times \frac{3}{7} \div \frac{1}{3}$$ Order of operations: $$\frac{1}{2} \times \frac{3}{7} = \frac{3}{14}$$ Divide by $\frac{1}{3}$ is multiply by 3: $$\frac{3}{14} \times 3 = \frac{9}{14}$$ Add $\frac{5}{6}$: Find common denominator 42: $$\frac{5}{6} = \frac{35}{42}, \quad \frac{9}{14} = \frac{27}{42}$$ Sum: $$\frac{35}{42} + \frac{27}{42} = \frac{62}{42} = \frac{31}{21}$$ 14. b) Express 0.35 as a percentage: $$0.35 \times 100 = 35\%$$