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Bivariate Patterns

Grade 8 · Statistics · Worksheet 2

  1. Emma is analyzing the relationship between the number of practice problems completed and math test scores in her class. She collected data from 7 students and found the line of best fit to be y = 1.8x + 68, where x represents the number of practice problems completed and y represents the test score. According to this model, what test score would be predicted for a student who completes 25 practice problems? Answer: ______________
  2. 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). When he plots the points, they appear to form a straight line. What linear equation in slope-intercept form (y = mx + b) best models the relationship between hours studied (x) and test score (y)? Answer: ______________
  3. A scientist is studying the relationship between the number of hours students study for a math test and their test scores. The data shows a linear pattern. When a student studies for 2 hours, they score 70%. When a student studies for 4 hours, they score 85%. If this linear relationship continues, what score would a student be predicted to get if they studied for 6 hours? Answer: ______________
  4. Liam is analyzing the relationship between study time and test scores for 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 the test score. If a student studies for 4 hours, what test score does the model predict? Answer: ______________
  5. Emma is analyzing the relationship between the number of practice problems completed and math test scores for her classmates. She collected data and found the line of best fit to be y = 1.8x + 68, where x represents the number of practice problems completed and y represents the test score percentage. According to this model, what test score would be predicted for a student who completes 25 practice problems? Answer: ______________
  6. A marine biologist is studying the relationship between water temperature and coral bleaching. She records data showing that for every 1°C increase in water temperature above 29°C, the percentage of bleached coral increases by 15%. If the water temperature is 31.5°C, what percentage of coral is bleached according to this linear relationship? Assume 0% bleaching occurs at 29°C. Answer: ______________
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Answer Key & Explanations

Bivariate Patterns · Grade 8 · Worksheet 2

  1. Emma is analyzing the relationship between the number of practice problems completed and math test scores in her class. She collected data from 7 students and found the line of best fit to be y = 1.8x + 68, where x represents the number of practice problems completed and y represents the test score. According to this model, what test score would be predicted for a student who completes 25 practice problems? Answer: 113 Solution: Identify the given linear equation: y = 1.8x + 68 Substitute x = 25 (the number of practice problems) into the equation: y = 1.8 * 25 + 68 Calculate 1.8 * 25 = 45 Add 68 to the result: 45 + 68 = 113 The predicted test score is 113.
    Full step-by-step solution

    Step 1: Identify the given linear equation: y = 1.8x + 68 Step 2: Substitute x = 25 (the number of practice problems) into the equation: y = 1.8 * 25 + 68 Step 3: Calculate 1.8 * 25 = 45 Step 4: Add 68 to the result: 45 + 68 = 113 Step 5: The predicted test score is 113. The answer is 113.

  2. 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). When he plots the points, they appear to form a straight line. What linear equation in slope-intercept form (y = mx + b) best models the relationship between hours studied (x) and test score (y)? Answer: y = 5x + 60 Solution: m = (change in y) / (change in x) Pick two points: (1, 65) and (2, 70) Change in y = 70 - 65 = 5 Change in x = 2 - 1 = 1 m = 5 / 1 = 5 Check with another pair: (3, 75) and (4, 80) Change in y = 80 - 75 = 5 Change in x = 4 - 3 = 1 m = 5 / 1 = 5 So the slope m = 5.
    Full step-by-step solution

    Let's find the linear equation that models the data: (1, 65), (2, 70), (3, 75), (4, 80), (5, 85). --- **Step 1: Identify the slope (m)** The slope formula is: m = (change in y) / (change in x) Pick two points: (1, 65) and (2, 70) Change in y = 70 - 65 = 5 Change in x = 2 - 1 = 1 m = 5 / 1 = 5 Check with another pair: (3, 75) and (4, 80) Change in y = 80 - 75 = 5 Change in x = 4 - 3 = 1 m = 5 / 1 = 5 So the slope m = 5. --- **Step 2: Use slope-intercept form y = mx + b** We know m = 5, so: y = 5x + b --- **Step 3: Find the y-intercept (b)** Pick any point, say (1, 65): 65 = 5(1) + b 65 = 5 + b b = 65 - 5 b = 60 Check with another point, say (5, 85): 85 = 5(5) + b 85 = 25 + b b = 85 - 25 b = 60 Same result. --- **Step 4: Write the equation** y = 5x + 60 --- **Step 5: Verify with all points** (1, 65): 5(1) + 60 = 65 ✓ (2, 70): 5(2) + 60 = 70 ✓ (3, 75): 5(3) + 60 = 75 ✓ (4, 80): 5(4) + 60 = 80 ✓ (5, 85): 5(5) + 60 = 85 ✓ All points satisfy the equation. --- **Final answer:** y = 5x + 60

  3. A scientist is studying the relationship between the number of hours students study for a math test and their test scores. The data shows a linear pattern. When a student studies for 2 hours, they score 70%. When a student studies for 4 hours, they score 85%. If this linear relationship continues, what score would a student be predicted to get if they studied for 6 hours? Answer: 100% Solution: - 2 hours → 70% score - 4 hours → 85% score So the points are (2, 70) and (4, 85).
    Full step-by-step solution

    Let's solve this step by step. --- **Step 1: Identify the two data points given** We are told: - 2 hours → 70% score - 4 hours → 85% score So the points are (2, 70) and (4, 85). --- **Step 2: Find the slope of the linear relationship** Slope m = (change in score) / (change in hours) m = (85 - 70) / (4 - 2) m = 15 / 2 m = 7.5 This means for each additional hour of study, the score increases by 7.5%. --- **Step 3: Write the equation of the line** Use point-slope form: y - y1 = m(x - x1) Using (2, 70): y - 70 = 7.5(x - 2) Simplify: y - 70 = 7.5x - 15 y = 7.5x - 15 + 70 y = 7.5x + 55 --- **Step 4: Predict the score for 6 hours** Plug x = 6 into y = 7.5x + 55: y = 7.5 * 6 + 55 y = 45 + 55 y = 100 --- **Step 5: Conclusion** If a student studies for 6 hours, the predicted score is 100%.

  4. Liam is analyzing the relationship between study time and test scores for 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 the test score. If a student studies for 4 hours, what test score does the model predict? Answer: 75 Solution: We are given the equation of the line of best fit: y = 2.5x + 65 Here, x = study time in hours, y = predicted test score. We are told x = 4 hours, and we need to find y. Substitute the given study time into the equation.
    Full step-by-step solution

    Step 1: Understand the problem. We are given the equation of the line of best fit: y = 2.5x + 65 Here, x = study time in hours, y = predicted test score. We are told x = 4 hours, and we need to find y. Step 2: Substitute the given study time into the equation. Replace x with 4: y = 2.5 * 4 + 65 Step 3: Perform the multiplication first (order of operations). 2.5 * 4 = 10 So now: y = 10 + 65 Step 4: Perform the addition. 10 + 65 = 75 So y = 75. Step 5: Interpret the result. The model predicts that a student who studies for 4 hours will score 75 on the test. Final answer: 75

  5. Emma is analyzing the relationship between the number of practice problems completed and math test scores for her classmates. She collected data and found the line of best fit to be y = 1.8x + 68, where x represents the number of practice problems completed and y represents the test score percentage. According to this model, what test score would be predicted for a student who completes 25 practice problems? Answer: 113 Solution: Identify the linear equation: y = 1.8x + 68 Substitute x = 25 into the equation: y = 1.8(25) + 68 Multiply: 1.8 × 25 = 45 Add: 45 + 68 = 113 The predicted test score is 113% The answer is 113.
    Full step-by-step solution

    Step 1: Identify the linear equation: y = 1.8x + 68 Step 2: Substitute x = 25 into the equation: y = 1.8(25) + 68 Step 3: Multiply: 1.8 × 25 = 45 Step 4: Add: 45 + 68 = 113 Step 5: The predicted test score is 113% The answer is 113.

  6. A marine biologist is studying the relationship between water temperature and coral bleaching. She records data showing that for every 1°C increase in water temperature above 29°C, the percentage of bleached coral increases by 15%. If the water temperature is 31.5°C, what percentage of coral is bleached according to this linear relationship? Assume 0% bleaching occurs at 29°C. Answer: 37.5% Solution: - 0% bleaching occurs at 29°C. - For every 1°C increase above 29°C, bleaching increases by 15%.
    Full step-by-step solution

    Let's go step-by-step. --- **Step 1: Understand the problem** We are told: - 0% bleaching occurs at 29°C. - For every 1°C increase above 29°C, bleaching increases by 15%. - This is a linear relationship: Bleaching % = 15 × (Temperature above 29°C) --- **Step 2: Find the temperature above 29°C** Given temperature = 31.5°C Temperature above 29°C = 31.5 − 29 = 2.5°C --- **Step 3: Apply the linear relationship** Bleaching % = 15% per °C × 2.5°C --- **Step 4: Calculate** 15 × 2.5 = 37.5 --- **Step 5: Interpret the result** At 31.5°C, the coral bleaching percentage is 37.5%. --- **Final Answer:** 37.5%