Population Comparison

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Answer Key & Explanations

Population Comparison · Grade 7 · Worksheet 1

1. Liam is comparing the test scores of two different classes. Class A has 28 students with a mean score of 82, and Class B has 32 students with a mean score of 78. What is the combined mean score for both classes? Answer: 79.87 Solution: Understand what "mean score" means. The mean score is the total sum of all scores divided by the number of students. Class A: 28 students, mean = 82 Total sum for Class A = 28 × 82 28 × 82 = 28 × 80 + 28 × 2 = 2240 + 56 = 2296 Find the total sum of scores for Class B.
   Full step-by-step solution

   Step 1: Understand what "mean score" means. The mean score is the total sum of all scores divided by the number of students. Step 2: Find the total sum of scores for Class A. Class A: 28 students, mean = 82 Total sum for Class A = 28 × 82 28 × 82 = 28 × 80 + 28 × 2 = 2240 + 56 = 2296 Step 3: Find the total sum of scores for Class B. Class B: 32 students, mean = 78 Total sum for Class B = 32 × 78 32 × 78 = 32 × 80 − 32 × 2 = 2560 − 64 = 2496 Step 4: Find the total sum of scores for both classes. Total sum = 2296 + 2496 = 4792 Step 5: Find the total number of students. Total students = 28 + 32 = 60 Step 6: Find the combined mean. Combined mean = Total sum / Total students = 4792 / 60 Step 7: Perform the division. 4792 ÷ 60: 60 × 79 = 4740 Subtract: 4792 − 4740 = 52 So 4792 / 60 = 79 + 52/60 Step 8: Simplify 52/60. Divide numerator and denominator by 4: 52/60 = 13/15 Step 9: Convert 13/15 to decimal. 13 ÷ 15 = 0.8666... ≈ 0.87 (rounded to two decimal places) Step 10: Add to integer part. 79 + 0.87 = 79.87 Final Answer: The combined mean score is 79.87.

2. Population A: mean = 67, median = 64, IQR = 18. Population B: mean = 74, median = 76, IQR = 12. Compare the centers and variabilities of the two populations. Answer: Population B has a higher center (mean 74 vs 67, median 76 vs 64) and less variability (IQR 12 vs 18) than Population A. Solution: Compare centers. Population A mean = 67, median = 64. Population B mean = 74, median = 76.
   Full step-by-step solution

   Step 1: Compare centers. Population A mean = 67, median = 64. Population B mean = 74, median = 76. Both mean and median are higher for Population B, so Population B has a greater center. Step 2: Compare variability. Population A IQR = 18. Population B IQR = 12. Since 12 < 18, Population B has less variability (more consistent data) than Population A. Step 3: Conclusion. Population B tends to have higher values and is more consistent, while Population A has lower values and is more spread out. The answer is: Population B has a higher center (mean 74 vs 67, median 76 vs 64) and less variability (IQR 12 vs 18) than Population A.

3. A rectangular city park is drawn on a coordinate plane with corners at (0, 0), (120, 0), (120, 80), and (0, 80). A diagonal walking path is drawn from point (0, 0) to point (120, 80), dividing the park into two triangular sections. What is the area of the larger triangular section in square meters? Answer: 4800 Solution: Find the area of the entire rectangular park. The rectangle has length 120 meters and width 80 meters. Area of rectangle = length × width = 120 × 80 = 9600 square meters.
   Full step-by-step solution

   Step 1: Find the area of the entire rectangular park. The rectangle has length 120 meters and width 80 meters. Area of rectangle = length × width = 120 × 80 = 9600 square meters. Step 2: Understand how the diagonal divides the rectangle. A diagonal drawn from one corner of a rectangle to the opposite corner divides the rectangle into two congruent triangles of equal area. Step 3: Calculate the area of one triangular section. Since both triangles have equal area, each triangle's area is half of the rectangle's area. Area of one triangle = 9600 ÷ 2 = 4800 square meters. Step 4: Identify the larger triangular section. Since both triangles are congruent (identical in size and shape), they have the same area. Therefore, the larger triangular section has area 4800 square meters. The answer is 4800.

4. A rectangular swimming pool is drawn on a coordinate plane with corners at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular safety zone is marked off inside the pool with vertices at (0, 0), (20, 0), and (10, 8). What is the area of the pool that is available for swimming (outside the safety zone)? Answer: 160 Solution: Calculate the area of the rectangular pool. The rectangle has length from x=0 to x=20, so length = 20 units. The rectangle has width from y=0 to y=12, so width = 12 units.
   Full step-by-step solution

   Step 1: Calculate the area of the rectangular pool. The rectangle has length from x=0 to x=20, so length = 20 units. The rectangle has width from y=0 to y=12, so width = 12 units. Area of rectangle = length × width = 20 × 12 = 240 square units. Step 2: Calculate the area of the triangular safety zone. The triangle has vertices at (0, 0), (20, 0), and (10, 8). Using the formula for area of a triangle with coordinates: Area = 1/2 × |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| Area = 1/2 × |0(0-8) + 20(8-0) + 10(0-0)| Area = 1/2 × |0 + 20×8 + 0| Area = 1/2 × |160| Area = 1/2 × 160 = 80 square units. Step 3: Calculate the swimming area (outside the safety zone). Swimming area = Total pool area - Safety zone area Swimming area = 240 - 80 = 160 square units. The answer is 160.

5. Population A: mean = 73, median = 71, range = 27. Population B: mean = 79, median = 81, range = 19. Compare. Answer: Population B has a higher mean and median, and a smaller range, indicating it is generally higher and more consistent. Solution: Compare the means. Population A has a mean of 73, and Population B has a mean of 79. Since 79 > 73, Population B has a higher average value.
   Full step-by-step solution

   Step 1: Compare the means. Population A has a mean of 73, and Population B has a mean of 79. Since 79 > 73, Population B has a higher average value. Step 2: Compare the medians. Population A has a median of 71, and Population B has a median of 81. Since 81 > 71, the middle value of Population B is higher. Step 3: Compare the ranges. Population A has a range of 27, and Population B has a range of 19. Since 19 < 27, Population B has less variability or spread in its data. Step 4: Conclusion. Population B has a higher mean and median, and a smaller range, meaning its values are generally higher and more consistent (less spread out) than Population A.

6. (-3)² × 4 + 18 ÷ (-2) - 5 = ? Answer: 22 Solution: Calculate the exponent first: (-3)² = 9 Perform multiplication: 9 × 4 = 36 Perform division: 18 ÷ (-2) = -9 Now the expression is: 36 + (-9) - 5 Add and subtract from left to right: 36 + (-9) = 27 27 - 5 = 22 The answer is 22.
   Full step-by-step solution

   Step 1: Calculate the exponent first: (-3)² = 9 Step 2: Perform multiplication: 9 × 4 = 36 Step 3: Perform division: 18 ÷ (-2) = -9 Step 4: Now the expression is: 36 + (-9) - 5 Step 5: Add and subtract from left to right: 36 + (-9) = 27 Step 6: 27 - 5 = 22 The answer is 22.

7. The double bar graph below shows the number of books read by two Grade 7 classes, Ms. Emma's class and Mr. Liam's class, over five months. For Ms. Emma's class, the numbers of books read each month are: 15, 20, 25, 30, 35. For Mr. Liam's class, the numbers are: 25, 30, 35, 40, 45. Compare the two populations by calculating the mean and range for each class. Which class has a higher mean, and which class has a greater variability (range)? Answer: Ms. Emma's class has a mean of 25 and a range of 20. Mr. Liam's class has a mean of 35 and a range of 20. Mr. Liam's class has a higher mean, and both classes have the same variability (range). Solution: Calculate the mean for Ms. Emma's class. Sum of books = 15 + 20 + 25 + 30 + 35 = 125.
   Full step-by-step solution

   Step 1: Calculate the mean for Ms. Emma's class. Sum of books = 15 + 20 + 25 + 30 + 35 = 125. Number of months = 5. Mean = 125 / 5 = 25. Step 2: Calculate the range for Ms. Emma's class. Maximum = 35, Minimum = 15. Range = 35 - 15 = 20. Step 3: Calculate the mean for Mr. Liam's class. Sum of books = 25 + 30 + 35 + 40 + 45 = 175. Number of months = 5. Mean = 175 / 5 = 35. Step 4: Calculate the range for Mr. Liam's class. Maximum = 45, Minimum = 25. Range = 45 - 25 = 20. Step 5: Compare. Mr. Liam's class has a higher mean (35 > 25), so on average they read more books per month. Both classes have the same range (20), so the variability in the number of books read is equal. The answer is: Ms. Emma's class has a mean of 25 and a range of 20. Mr. Liam's class has a mean of 35 and a range of 20. Mr. Liam's class has a higher mean, and both classes have the same variability (range).

8. Population A: mean = 72, median = 70, range = 27. Population B: mean = 77, median = 78, range = 22. Compare the two populations using measures of center and variability. Answer: Population B has a higher mean (77 vs 72) and median (78 vs 70), indicating generally higher scores. Population A has greater variability (range 27 vs 22), meaning its scores are more spread out. Solution: Compare measures of center. Population A has a mean of 72 and a median of 70. Population B has a mean of 77 and a median of 78.
   Full step-by-step solution

   Step 1: Compare measures of center. Population A has a mean of 72 and a median of 70. Population B has a mean of 77 and a median of 78. Both the mean and median are higher for Population B, indicating that, on average, Population B has higher values. Step 2: Compare measures of variability. Population A has a range of 27 (from 72 - 27/2 to 72 + 27/2, or roughly 58.5 to 85.5). Population B has a range of 22 (from 77 - 11 to 77 + 11, or 66 to 88). The range for Population A is larger, meaning its data values are more spread out and less consistent. Step 3: Conclusion. Population B has higher central tendency (mean and median) and less variability (smaller range), so it is both higher-performing and more consistent. Population A has lower central tendency and greater variability, so it is lower-performing and less consistent. The answer is: Population B has a higher mean (77 vs 72) and median (78 vs 70), indicating generally higher scores. Population A has greater variability (range 27 vs 22), meaning its scores are more spread out.