Population Comparison

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Answer Key & Explanations

Population Comparison · Grade 7 · Worksheet 2

1. A city is comparing the average monthly rainfall in two neighborhoods over a year. Neighborhood A had rainfall amounts of 85 mm, 92 mm, 78 mm, 105 mm, 110 mm, 95 mm, 88 mm, 82 mm, 76 mm, 98 mm, 102 mm, and 90 mm. Neighborhood B had rainfall amounts of 72 mm, 85 mm, 79 mm, 94 mm, 108 mm, 112 mm, 96 mm, 84 mm, 77 mm, 101 mm, 106 mm, and 89 mm. Which neighborhood had the higher mean monthly rainfall, and by how much? Answer: Neighborhood A by 1.5 mm Solution: 85 + 92 + 78 + 105 + 110 + 95 + 88 + 82 + 76 + 98 + 102 + 90 = 1101 mm 1101 ÷ 12 = 91.75 mm 72 + 85 + 79 + 94 + 108 + 112 + 96 + 84 + 77 + 101 + 106 + 89 = 1103 mm 1103 ÷ 12 = 91.9166...
   Full step-by-step solution

   Step 1: Calculate total rainfall for Neighborhood A 85 + 92 + 78 + 105 + 110 + 95 + 88 + 82 + 76 + 98 + 102 + 90 = 1101 mm Step 2: Calculate mean rainfall for Neighborhood A 1101 ÷ 12 = 91.75 mm Step 3: Calculate total rainfall for Neighborhood B 72 + 85 + 79 + 94 + 108 + 112 + 96 + 84 + 77 + 101 + 106 + 89 = 1103 mm Step 4: Calculate mean rainfall for Neighborhood B 1103 ÷ 12 = 91.9166... mm ≈ 91.92 mm Step 5: Compare the means Neighborhood A: 91.75 mm Neighborhood B: 91.92 mm Neighborhood B has a higher mean Step 6: Calculate the difference 91.92 - 91.75 = 0.17 mm The answer is Neighborhood B by 0.17 mm.

2. Population A: mean = 102, median = 98, IQR = 24. Population B: mean = 115, median = 112, IQR = 15. Compare the two populations using measures of center and variability. Answer: Population B has a higher center (mean 115 vs 102, median 112 vs 98) and less variability (IQR 15 vs 24) than Population A. Solution: Compare measures of center. Population A has a mean of 102 and median of 98. Population B has a mean of 115 and median of 112.
   Full step-by-step solution

   Step 1: Compare measures of center. Population A has a mean of 102 and median of 98. Population B has a mean of 115 and median of 112. Both the mean and median are higher for Population B, indicating that the typical value in Population B is greater than in Population A. Step 2: Compare measures of variability. Population A has an IQR of 24. Population B has an IQR of 15. A smaller IQR means the middle 50% of the data is more tightly clustered. Population B has less variability (more consistent) than Population A. Step 3: Conclusion. Population B has a higher center (mean 115 vs 102, median 112 vs 98) and less variability (IQR 15 vs 24) compared to Population A. This means Population B tends to have larger values that are more consistent, while Population A has smaller values with more spread.

3. (-3)² - 4 × (-2) + 12 ÷ (-4) = ? Answer: 14 Solution: Calculate the exponent: (-3)² = 9 Calculate the multiplication: 4 × (-2) = -8 Calculate the division: 12 ÷ (-4) = -3 Substitute back into the expression: 9 - (-8) + (-3) Simplify the subtraction of a negative: 9 + 8 + (-3) Add from left to right: 9 + 8 = 17, then 17 + (-3) = 14 The answer is 14.
   Full step-by-step solution

   Step 1: Calculate the exponent: (-3)² = 9 Step 2: Calculate the multiplication: 4 × (-2) = -8 Step 3: Calculate the division: 12 ÷ (-4) = -3 Step 4: Substitute back into the expression: 9 - (-8) + (-3) Step 5: Simplify the subtraction of a negative: 9 + 8 + (-3) Step 6: Add from left to right: 9 + 8 = 17, then 17 + (-3) = 14 The answer is 14.

4. Population A: mean = 125, median = 118, range = 42. Population B: mean = 140, median = 136, range = 28. Compare the centers and variabilities of the two populations. Answer: Population B has a higher center (mean 140 > 125, median 136 > 118) and less variability (range 28 < 42) than Population A. Solution: Compare the means. Population A mean = 125, Population B mean = 140. Since 140 > 125, Population B has a higher average value.
   Full step-by-step solution

   Step 1: Compare the means. Population A mean = 125, Population B mean = 140. Since 140 > 125, Population B has a higher average value. Step 2: Compare the medians. Population A median = 118, Population B median = 136. Since 136 > 118, the middle value of Population B is also higher. Step 3: Compare the ranges. Population A range = 42, Population B range = 28. Since 28 < 42, Population B has less spread or variability in its data. Conclusion: Population B has a higher center (both mean and median are larger) and is more consistent (smaller range) than Population A.

5. Population A: mean = 91, median = 87, range = 33. Population B: mean = 79, median = 83, range = 19. Compare the two populations using measures of center and variability. Answer: Population A has a higher mean (91 vs 79) and median (87 vs 83) but greater variability (range 33 vs 19). Population B is more consistent with lower spread. Solution: Compare measures of center. Population A has a mean of 91 and median of 87. Population B has a mean of 79 and median of 83.
   Full step-by-step solution

   Step 1: Compare measures of center. Population A has a mean of 91 and median of 87. Population B has a mean of 79 and median of 83. Population A has a higher mean (91 > 79) and a higher median (87 > 83), indicating that the typical score in Population A is higher than in Population B. Step 2: Compare measures of variability. Population A has a range of 33. Population B has a range of 19. Since 33 > 19, Population A has greater variability, meaning its scores are more spread out. Population B is more consistent with less variation. Step 3: Combine the comparisons. Overall, Population A tends to have higher scores (higher mean and median) but is less consistent (larger range). Population B has lower scores on average but is more tightly clustered around its center. The answer is: Population A has a higher mean (91 vs 79) and median (87 vs 83) but greater variability (range 33 vs 19). Population B is more consistent with lower spread.

6. (-3)² × 4 + 2³ ÷ (-2) - 5 = ? Answer: 27 Solution: Calculate exponents: (-3)² = 9 and 2³ = 8 Perform multiplication: 9 × 4 = 36 Perform division: 8 ÷ (-2) = -4 Rewrite the expression: 36 + (-4) - 5 Add and subtract from left to right: 36 + (-4) = 32, then 32 - 5 = 27 The answer is 27.
   Full step-by-step solution

   Step 1: Calculate exponents: (-3)² = 9 and 2³ = 8 Step 2: Perform multiplication: 9 × 4 = 36 Step 3: Perform division: 8 ÷ (-2) = -4 Step 4: Rewrite the expression: 36 + (-4) - 5 Step 5: Add and subtract from left to right: 36 + (-4) = 32, then 32 - 5 = 27 The answer is 27.

7. (-2)³ + 5 × (3 - 7) ÷ 2 = ? Answer: -18 Solution: Start with the expression: (-2)³ + 5 × (3 - 7) ÷ 2 Calculate the exponent: (-2)³ = -2 × -2 × -2 = 4 × -2 = -8 Calculate inside the parentheses: (3 - 7) = -4 Now we have: -8 + 5 × (-4) ÷ 2 Perform multiplication and division from left to right: 5 × (-4) = -20 Then: -20 ÷ 2 = -10 Now we have: -8 +…
   Full step-by-step solution

   Step 1: Start with the expression: (-2)³ + 5 × (3 - 7) ÷ 2 Step 2: Calculate the exponent: (-2)³ = -2 × -2 × -2 = 4 × -2 = -8 Step 3: Calculate inside the parentheses: (3 - 7) = -4 Step 4: Now we have: -8 + 5 × (-4) ÷ 2 Step 5: Perform multiplication and division from left to right: 5 × (-4) = -20 Step 6: Then: -20 ÷ 2 = -10 Step 7: Now we have: -8 + (-10) = -18 Step 8: The final answer is -18.

8. Population A: mean = 92, median = 88, range = 34. Population B: mean = 85, median = 90, range = 22. Which population has a higher typical value according to the mean, and which has less variability? Answer: Population A has a higher typical value by mean; Population B has less variability. Solution: Compare means. Population A mean = 92, Population B mean = 85. Since 92 > 85, Population A has a higher typical value by mean.
   Full step-by-step solution

   Step 1: Compare means. Population A mean = 92, Population B mean = 85. Since 92 > 85, Population A has a higher typical value by mean. Step 2: Compare ranges. Population A range = 34, Population B range = 22. Since 22 < 34, Population B has less variability (scores are more consistent). Final answer: Population A has a higher typical value by mean; Population B has less variability.