1. Angela's Weighted GPA Calculation: Step 1: State the problem: Calculate Angela's weighted GPA using her grades and unit weights. Step 2: Multiply each grade by its unit weight: $$91 \times 4 = 364$$ $$87 \times 3 = 261$$ $$85 \times 2 = 170$$ $$90 \times 3 = 270$$ $$93 \times 1 = 93$$ Step 3: Sum the weighted grades: $$364 + 261 + 170 + 270 + 93 = 1158$$ Step 4: Sum the total units: $$4 + 3 + 2 + 3 + 1 = 13$$ Step 5: Calculate weighted GPA: $$\frac{1158}{13} = 89.08$$ Angela's weighted GPA is 89.08. 2. Samantha's Weighted GPA Calculation: Step 1: State the problem: Calculate Samantha's weighted GPA using her grades and unit weights. Step 2: Multiply each grade by its unit weight: $$94 \times 3 = 282$$ $$88 \times 4 = 352$$ $$90 \times 3 = 270$$ $$85 \times 2 = 170$$ $$92 \times 3 = 276$$ Step 3: Sum the weighted grades: $$282 + 352 + 270 + 170 + 276 = 1350$$ Step 4: Sum the total units: $$3 + 4 + 3 + 2 + 3 = 15$$ Step 5: Calculate weighted GPA: $$\frac{1350}{15} = 90$$ Samantha's weighted GPA is 90. 3. Variance and Standard Deviation of Daily Wages: Step 1: State the problem: Find variance and standard deviation of wages: 600, 450, 580, 520, 750, 700, 900, 500. Step 2: Calculate the mean: $$\bar{x} = \frac{600 + 450 + 580 + 520 + 750 + 700 + 900 + 500}{8} = \frac{5000}{8} = 625$$ Step 3: Calculate squared differences from the mean: $$(600-625)^2 = 625$$ $$(450-625)^2 = 30625$$ $$(580-625)^2 = 2025$$ $$(520-625)^2 = 11025$$ $$(750-625)^2 = 15625$$ $$(700-625)^2 = 5625$$ $$(900-625)^2 = 75625$$ $$(500-625)^2 = 15625$$ Step 4: Sum squared differences: $$625 + 30625 + 2025 + 11025 + 15625 + 5625 + 75625 + 15625 = 157775$$ Step 5: Calculate variance (population): $$\sigma^2 = \frac{157775}{8} = 19721.875$$ Step 6: Calculate standard deviation: $$\sigma = \sqrt{19721.875} \approx 140.44$$ Variance is 19721.88 and standard deviation is approximately 140.44. 4. Variance and Standard Deviation of Quiz Scores (Population and Sample): Scores: 18, 22, 25, 30, 15, 27, 35, 20 Step 1: Calculate mean: $$\bar{x} = \frac{18 + 22 + 25 + 30 + 15 + 27 + 35 + 20}{8} = \frac{192}{8} = 24$$ Step 2: Calculate squared differences: $$(18-24)^2=36$$ $$(22-24)^2=4$$ $$(25-24)^2=1$$ $$(30-24)^2=36$$ $$(15-24)^2=81$$ $$(27-24)^2=9$$ $$(35-24)^2=121$$ $$(20-24)^2=16$$ Step 3: Sum squared differences: $$36 + 4 + 1 + 36 + 81 + 9 + 121 + 16 = 304$$ Step 4: Population variance: $$\sigma^2 = \frac{304}{8} = 38$$ Step 5: Population standard deviation: $$\sigma = \sqrt{38} \approx 6.16$$ Step 6: Sample variance: $$s^2 = \frac{304}{7} \approx 43.43$$ Step 7: Sample standard deviation: $$s = \sqrt{43.43} \approx 6.59$$ Population variance is 38, population standard deviation approx 6.16. Sample variance approx 43.43, sample standard deviation approx 6.59. 5. Spelling Quiz Scores Variance and Standard Deviation: Scores: 17, 20, 22, 18, 25, 19, 21, 23, 16 Step 1: Calculate mean: $$\bar{x} = \frac{17 + 20 + 22 + 18 + 25 + 19 + 21 + 23 + 16}{9} = \frac{181}{9} \approx 20.11$$ Step 2: Calculate squared differences: $$(17-20.11)^2 \approx 9.67$$ $$(20-20.11)^2 \approx 0.01$$ $$(22-20.11)^2 \approx 3.57$$ $$(18-20.11)^2 \approx 4.45$$ $$(25-20.11)^2 \approx 23.89$$ $$(19-20.11)^2 \approx 1.23$$ $$(21-20.11)^2 \approx 0.79$$ $$(23-20.11)^2 \approx 8.35$$ $$(16-20.11)^2 \approx 16.89$$ Step 3: Sum squared differences: $$9.67 + 0.01 + 3.57 + 4.45 + 23.89 + 1.23 + 0.79 + 8.35 + 16.89 = 68.85$$ Step 4: Population variance: $$\sigma^2 = \frac{68.85}{9} \approx 7.65$$ Step 5: Population standard deviation: $$\sigma = \sqrt{7.65} \approx 2.77$$ Step 6: Sample variance: $$s^2 = \frac{68.85}{8} \approx 8.61$$ Step 7: Sample standard deviation: $$s = \sqrt{8.61} \approx 2.93$$ Population variance is approximately 7.65, population standard deviation approx 2.77. Sample variance approx 8.61, sample standard deviation approx 2.93.