Bivariate Patterns

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Answer Key & Explanations

Bivariate Patterns · Grade 8 · Worksheet 2

1. Mere recorded the temperature (in °C) and the number of hot chocolates sold each day at her stall. The data is: (12, 48), (14, 44), (16, 40), (18, 36), (20, 32), (22, 28), (24, 24), (26, 20), (28, 16), (30, 12), (32, 8), (34, 4). Describe the pattern of association and identify any outliers. Answer: Negative association; no outliers Solution: List the ordered pairs: (12, 48), (14, 44), (16, 40), (18, 36), (20, 32), (22, 28), (24, 24), (26, 20), (28, 16), (30, 12), (32, 8), (34, 4).
   Full step-by-step solution

   Step 1: List the ordered pairs: (12, 48), (14, 44), (16, 40), (18, 36), (20, 32), (22, 28), (24, 24), (26, 20), (28, 16), (30, 12), (32, 8), (34, 4). Step 2: Observe the trend: As temperature increases by 2°C each time, the number of hot chocolates sold decreases by 4 each time. This is a consistent, linear pattern. Step 3: Since one variable increases while the other decreases in a steady way, the association is negative. Step 4: Check for outliers: All points lie exactly on a straight line (the change is perfectly consistent). There are no points that deviate from this pattern. Step 5: Conclusion: The data shows a strong negative association with no outliers.

2. Emma is tracking the relationship between the number of hours she practices piano each week and her performance score on a scale of 1-100. Her data shows that for every additional hour of practice, her score increases by 6 points. If she scored 68 when practicing for 3 hours, what would be her predicted score if she practices for 7 hours, assuming this linear relationship continues? Answer: 92 Solution: Identify the rate of change - score increases by 6 points per hour of practice, so slope m = 6 Use the given point (3, 68) to find the y-intercept: 68 = 6(3) + b Calculate: 68 = 18 + b, so b = 68 - 18 = 50 Write the equation: y = 6x + 50, where x is hours practiced and y is score Substitute x =…
   Full step-by-step solution

   Step 1: Identify the rate of change - score increases by 6 points per hour of practice, so slope m = 6 Step 2: Use the given point (3, 68) to find the y-intercept: 68 = 6(3) + b Step 3: Calculate: 68 = 18 + b, so b = 68 - 18 = 50 Step 4: Write the equation: y = 6x + 50, where x is hours practiced and y is score Step 5: Substitute x = 7: y = 6(7) + 50 = 42 + 50 = 92 The answer is 92.

3. Aroha recorded the number of hours studied and the test scores for 10 students. The data is: (2, 55), (4, 65), (6, 75), (8, 85), (10, 95), (3, 58), (5, 70), (7, 80), (9, 90), (12, 100). Describe the association pattern and identify any outlier. Answer: Positive association; no outliers Solution: List the ordered pairs: (2,55), (4,65), (6,75), (8,85), (10,95), (3,58), (5,70), (7,80), (9,90), (12,100). Observe the pattern: As hours studied increase, test scores increase consistently.
   Full step-by-step solution

   Step 1: List the ordered pairs: (2,55), (4,65), (6,75), (8,85), (10,95), (3,58), (5,70), (7,80), (9,90), (12,100). Step 2: Observe the pattern: As hours studied increase, test scores increase consistently. This is a positive association. Step 3: Check for outliers: All points follow a clear linear pattern (scores increase by about 5 points per hour studied). The point (12,100) continues this pattern, so there are no outliers. The answer is: Positive association; no outliers.

4. Mason recorded the number of pages read (x) and the time in minutes (y) for 10 students: (12, 15), (15, 18), (18, 22), (21, 25), (24, 28), (27, 31), (30, 34), (33, 37), (36, 40), (50, 30). Describe the association pattern and identify any outlier. Answer: Positive association with an outlier at (50, 30) Solution: List the ordered pairs: (12,15), (15,18), (18,22), (21,25), (24,28), (27,31), (30,34), (33,37), (36,40), (50,30). As x increases from 12 to 36, y increases from 15 to 40.
   Full step-by-step solution

   Step 1: List the ordered pairs: (12,15), (15,18), (18,22), (21,25), (24,28), (27,31), (30,34), (33,37), (36,40), (50,30). Step 2: As x increases from 12 to 36, y increases from 15 to 40. This shows a positive association — as pages read increase, time increases. Step 3: The last point (50, 30) has a much larger x-value (50) but a smaller y-value (30) than the previous point (36, 40). This does not fit the pattern. Step 4: The association is positive, and the outlier is (50, 30).

5. 2³ × 3² - 5² = ? Answer: 47 Solution: Calculate 2³ = 2 × 2 × 2 = 8 Calculate 3² = 3 × 3 = 9 Calculate 5² = 5 × 5 = 25 Multiply the results: 8 × 9 = 72 Subtract: 72 - 25 = 47 The answer is 47.
   Full step-by-step solution

   Step 1: Calculate 2³ = 2 × 2 × 2 = 8 Step 2: Calculate 3² = 3 × 3 = 9 Step 3: Calculate 5² = 5 × 5 = 25 Step 4: Multiply the results: 8 × 9 = 72 Step 5: Subtract: 72 - 25 = 47 The answer is 47.

6. The table shows the number of pages read (y) by Emma each day (x). Identify the association pattern and any outlier: (1, 55), (3, 65), (5, 75), (7, 85), (9, 95), (11, 105), (13, 45). Answer: positive association with an outlier at (13, 45) Solution: List the ordered pairs: (1,55), (3,65), (5,75), (7,85), (9,95), (11,105), (13,45). Check the pattern: as x increases from 1 to 11, y increases by 10 each time (55 to 65 to 75 to 85 to 95 to 105).
   Full step-by-step solution

   Step 1: List the ordered pairs: (1,55), (3,65), (5,75), (7,85), (9,95), (11,105), (13,45). Step 2: Check the pattern: as x increases from 1 to 11, y increases by 10 each time (55 to 65 to 75 to 85 to 95 to 105). This shows a positive association. Step 3: The last point (13,45) has y = 45, which is much lower than the expected value (115 if the pattern continued). This point does not follow the trend. Step 4: Conclusion: The data shows a positive association with an outlier at (13, 45).

7. 2³ × 3² + √144 = ? Answer: 84 Solution: Calculate the exponents first: 2³ = 8 and 3² = 9 Calculate the square root: √144 = 12 Perform multiplication: 8 × 9 = 72 Perform addition: 72 + 12 = 84 The answer is 84.
   Full step-by-step solution

   Step 1: Calculate the exponents first: 2³ = 8 and 3² = 9 Step 2: Calculate the square root: √144 = 12 Step 3: Perform multiplication: 8 × 9 = 72 Step 4: Perform addition: 72 + 12 = 84 The answer is 84.

8. Liam is tracking the relationship between the number of hours he studies and his test scores. He recorded this data: (1, 65), (2, 70), (3, 75), (4, 80), (5, 85). If this linear pattern continues, what test score would Liam likely get if he studies for 7 hours? Answer: 95 Solution: (1, 65), (2, 70), (3, 75), (4, 80), (5, 85) Look at how the score changes when hours increase by 1: From (1, 65) to (2, 70): score increases by 5 From (2, 70) to (3, 75): score increases by 5 From (3, 75) to (4, 80): score increases by 5 From (4, 80) to (5, 85): score increases by 5 Change in…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the pattern in the data** The data points are: (1, 65), (2, 70), (3, 75), (4, 80), (5, 85) Look at how the score changes when hours increase by 1: From (1, 65) to (2, 70): score increases by 5 From (2, 70) to (3, 75): score increases by 5 From (3, 75) to (4, 80): score increases by 5 From (4, 80) to (5, 85): score increases by 5 So the slope (rate of change) is constant: Change in score / Change in hours = 5 / 1 = 5 points per hour. --- **Step 2: Find the equation of the line** A linear relationship can be written as: Score = m * (hours) + b We know m = 5. Use one point to find b. Let's use (1, 65): 65 = 5 * 1 + b 65 = 5 + b b = 65 - 5 b = 60 So the equation is: Score = 5 * (hours) + 60 --- **Step 3: Check with other points** For hours = 2: Score = 5*2 + 60 = 70 ✓ For hours = 5: Score = 5*5 + 60 = 85 ✓ --- **Step 4: Predict for 7 hours** Score = 5 * 7 + 60 Score = 35 + 60 Score = 95 --- **Step 5: Conclusion** If the pattern continues, Liam would likely score 95 after studying for 7 hours. **Final answer: 95**