Bivariate Patterns

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Answer Key & Explanations

Bivariate Patterns · Grade 8 · Worksheet 3

1. 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**

2. A scatter plot shows the relationship between hours spent practicing basketball per week and free throw percentage for 15 middle school players. The data points form a linear pattern with a positive association. The line of best fit passes through points (3, 45) and (7, 65). What is the slope of this line of best fit? Answer: 5 Solution: Identify the coordinates of the two points: (3, 45) and (7, 65) Recall the slope formula: slope = (y2 - y1) / (x2 - x1) Substitute the values: slope = (65 - 45) / (7 - 3) Calculate the numerator: 65 - 45 = 20 Calculate the denominator: 7 - 3 = 4 Divide: 20 / 4 = 5 The slope of the line of best…
   Full step-by-step solution

   Step 1: Identify the coordinates of the two points: (3, 45) and (7, 65) Step 2: Recall the slope formula: slope = (y2 - y1) / (x2 - x1) Step 3: Substitute the values: slope = (65 - 45) / (7 - 3) Step 4: Calculate the numerator: 65 - 45 = 20 Step 5: Calculate the denominator: 7 - 3 = 4 Step 6: Divide: 20 / 4 = 5 The slope of the line of best fit is 5.

3. Emma is analyzing the relationship between daily screen time and hours of sleep for teenagers. She collected data from 7 students and created a scatter plot showing a negative linear association. The line of best fit has the equation y = -0.8x + 9.2, where x represents screen time in hours and y represents sleep in hours. According to this model, how many hours of sleep would a teenager get if they had 4 hours of screen time? Answer: 6 Solution: The equation given is y = -0.8x + 9.2, where x is screen time and y is sleep time.
   Full step-by-step solution

   Step 1: The equation given is y = -0.8x + 9.2, where x is screen time and y is sleep time. Step 2: Substitute x = 4 into the equation: y = -0.8(4) + 9.2 Step 3: Calculate -0.8 × 4 = -3.2 Step 4: Add 9.2 to -3.2: -3.2 + 9.2 = 6 Step 5: The predicted sleep time is 6 hours. The answer is 6.

4. Liam is analyzing the relationship between study time and test scores in his science class. He collected data from 8 students and created a scatter plot. The line of best fit has the equation y = 2.5x + 65, where x represents study time in hours and y represents test scores. If a student studies for 6 hours, what test score does the model predict? Answer: 80 Solution: y = 2.5x + 65 - x = study time in hours - y = predicted test score Identify the given value of x. The problem says a student studies for 6 hours, so x = 6. Substitute x = 6 into the equation.
   Full step-by-step solution

   We are given the line of best fit equation: y = 2.5x + 65 Here: - x = study time in hours - y = predicted test score Step 1: Identify the given value of x. The problem says a student studies for 6 hours, so x = 6. Step 2: Substitute x = 6 into the equation. y = 2.5 * 6 + 65 Step 3: Perform the multiplication. 2.5 * 6 = 15 Step 4: Add the result to 65. y = 15 + 65 Step 5: Calculate the final value. y = 80 Step 6: Interpret the result. The model predicts that a student who studies for 6 hours will score 80 on the test. Final answer: 80

5. (2x + 5)² = 81 Answer: 2 Solution: Start with the equation (2x + 5)² = 81 Take the square root of both sides: 2x + 5 = ±9 Solve for the positive case: 2x + 5 = 9 → 2x = 4 → x = 2 Solve for the negative case: 2x + 5 = -9 → 2x = -14 → x = -7 The equation has two solutions: x = 2 and x = -7 Since this is asking for the positive…
   Full step-by-step solution

   Step 1: Start with the equation (2x + 5)² = 81 Step 2: Take the square root of both sides: 2x + 5 = ±9 Step 3: Solve for the positive case: 2x + 5 = 9 → 2x = 4 → x = 2 Step 4: Solve for the negative case: 2x + 5 = -9 → 2x = -14 → x = -7 Step 5: The equation has two solutions: x = 2 and x = -7 Since this is asking for the positive solution, the answer is 2.

6. A marine biologist is tracking the growth of a coral colony over time. She records the coral's diameter in centimeters each month for 6 months. The data shows a strong positive linear association with a correlation coefficient of 0.92. If the line of best fit is represented by the equation y = 2.5x + 15, where x represents months and y represents diameter in centimeters, what was the coral's predicted diameter when she first started measuring it? Answer: 15 centimeters Solution: y = 2.5x + 15 Here, x = number of months since the biologist started measuring, and y = predicted diameter in centimeters. Identify what is being asked.
   Full step-by-step solution

   Step 1: Understand the problem. We are given the line of best fit: y = 2.5x + 15 Here, x = number of months since the biologist started measuring, and y = predicted diameter in centimeters. Step 2: Identify what is being asked. The question is: What was the coral's predicted diameter when she first started measuring it? That means we need the diameter at the very beginning, when x = 0 (month zero). Step 3: Substitute x = 0 into the equation. y = 2.5 * 0 + 15 Step 4: Perform the calculation. 2.5 * 0 = 0 0 + 15 = 15 So y = 15. Step 5: Interpret the result. When x = 0 months, the predicted diameter is 15 centimeters. This is the y-intercept of the line of best fit, which represents the starting value when time is zero. Final answer: 15 centimeters

7. A scatter plot shows the relationship between the number of hours students spend on homework per week and their math test scores. The data points form a linear pattern with a positive association. The line of best fit passes through the points (3, 72) and (7, 88). What is the slope of this line of best fit? Answer: 4 Solution: Identify the coordinates of the two points: (3, 72) and (7, 88) Recall the slope formula: slope = (y2 - y1) / (x2 - x1) Substitute the values: slope = (88 - 72) / (7 - 3) Calculate the numerator: 88 - 72 = 16 Calculate the denominator: 7 - 3 = 4 Divide: 16 / 4 = 4 The slope of the line of best…
   Full step-by-step solution

   Step 1: Identify the coordinates of the two points: (3, 72) and (7, 88) Step 2: Recall the slope formula: slope = (y2 - y1) / (x2 - x1) Step 3: Substitute the values: slope = (88 - 72) / (7 - 3) Step 4: Calculate the numerator: 88 - 72 = 16 Step 5: Calculate the denominator: 7 - 3 = 4 Step 6: Divide: 16 / 4 = 4 The slope of the line of best fit is 4.