Bivariate Patterns

lessonbunny.com

Answer Key & Explanations

Bivariate Patterns · Grade 8 · Worksheet 1

1. Liam is analyzing the relationship between study time and test scores. He collected data from 8 classmates and found the linear regression 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 score would this model predict? Answer: 80 Solution: y = 2.5x + 65 where x = study time in hours, y = test score. Identify the given value of x. The problem says the student studies for 6 hours, so x = 6.
   Full step-by-step solution

   We are given the linear regression equation: y = 2.5x + 65 where x = study time in hours, y = test score. Step 1: Identify the given value of x. The problem says the 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 y = 80 Step 5: Interpret the result. The model predicts a test score of 80 when a student studies for 6 hours. Final answer: 80

2. 3² × (4 + 5) - √81 = ? Answer: 72 Solution: Calculate inside the parentheses: (4 + 5) = 9 Calculate the exponent: 3² = 9 Multiply: 9 × 9 = 81 Calculate the square root: √81 = 9 Subtract: 81 - 9 = 72 The answer is 72.
   Full step-by-step solution

   Step 1: Calculate inside the parentheses: (4 + 5) = 9 Step 2: Calculate the exponent: 3² = 9 Step 3: Multiply: 9 × 9 = 81 Step 4: Calculate the square root: √81 = 9 Step 5: Subtract: 81 - 9 = 72 The answer is 72.

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 that for every additional hour studied, the test score increases by 4 points. If a student who studied for 2 hours scored 72 points, write a linear equation in slope-intercept form that represents this relationship, where x represents hours studied and y represents test score. Answer: y = 4x + 64 Solution: - For every additional hour studied, the score increases by 4 points → This is the slope \( m \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We are told: - For every additional hour studied, the score increases by 4 points → This is the **slope** \( m \). - \( x \) = hours studied - \( y \) = test score - When \( x = 2 \), \( y = 72 \) - We need a linear equation in slope-intercept form: \( y = mx + b \) --- **Step 2: Identify the slope** "For every additional hour studied, the test score increases by 4 points" means: \[ m = 4 \] So the equation so far is: \[ y = 4x + b \] --- **Step 3: Use the given point to find \( b \)** We know: \( x = 2 \), \( y = 72 \) Substitute into \( y = 4x + b \): \[ 72 = 4(2) + b \] \[ 72 = 8 + b \] \[ b = 72 - 8 \] \[ b = 64 \] --- **Step 4: Write the final equation** Substitute \( m = 4 \) and \( b = 64 \) into \( y = mx + b \): \[ y = 4x + 64 \] --- **Final answer:** y = 4x + 64

4. A marine biologist is studying the relationship between water temperature and the number of fish observed in a coral reef. She recorded data over several days and found the linear regression equation to be y = -12x + 240, where x represents water temperature in degrees Celsius and y represents the number of fish observed. According to this model, how many fish would be predicted when the water temperature reaches 18°C? Answer: 24 Solution: Identify the given equation: y = -12x + 240 Identify what x represents: water temperature in °C Substitute x = 18 into the equation: y = -12(18) + 240 Calculate -12 × 18 = -216 Add 240 to -216: 240 + (-216) = 24 The predicted number of fish is 24 The answer is 24.
   Full step-by-step solution

   Step 1: Identify the given equation: y = -12x + 240 Step 2: Identify what x represents: water temperature in °C Step 3: Substitute x = 18 into the equation: y = -12(18) + 240 Step 4: Calculate -12 × 18 = -216 Step 5: Add 240 to -216: 240 + (-216) = 24 Step 6: The predicted number of fish is 24 The answer is 24.

5. A scientist studies the relationship between the number of hours students study for a math test and their test scores. The data shows a strong positive linear association with a correlation coefficient of 0.92. If a student studies for 3.5 hours, the line of best fit predicts a score of 84 points. What is the residual for a student who actually scored 88 points after studying for 3.5 hours? Answer: 4 Solution: Understand what a residual is. A residual is the difference between the actual observed value and the predicted value from the line of best fit. Formula: residual = actual value − predicted value.
   Full step-by-step solution

   Step 1: Understand what a residual is. A residual is the difference between the actual observed value and the predicted value from the line of best fit. Formula: residual = actual value − predicted value. Step 2: Identify the given values. Predicted score (from line of best fit) for 3.5 hours of study = 84 points. Actual score = 88 points. Step 3: Apply the residual formula. residual = 88 − 84 = 4. Step 4: Interpret the result. Since the residual is positive (4), it means the actual score is 4 points higher than the predicted score for that student. Final answer: 4

6. Liam is tracking the relationship between study time and test scores. He recorded data for 6 students: (1, 65), (2, 70), (3, 75), (4, 80), (5, 85), (6, 90). When he plots these points, he notices they form a perfect linear pattern. What equation in slope-intercept form represents this relationship between hours studied (x) and test score (y)? Answer: y = 5x + 60 Solution: Pick two points, for example (1, 65) and (2, 70). Slope formula: m = (y2 - y1) / (x2 - x1) m = (70 - 65) / (2 - 1) m = 5 / 1 m = 5.
   Full step-by-step solution

   Let's find the equation in slope-intercept form (y = mx + b) using the given points: (1, 65), (2, 70), (3, 75), (4, 80), (5, 85), (6, 90). --- **Step 1: Find the slope (m)** Pick two points, for example (1, 65) and (2, 70). Slope formula: m = (y2 - y1) / (x2 - x1) m = (70 - 65) / (2 - 1) m = 5 / 1 m = 5. You can check with another pair, e.g., (3, 75) and (4, 80): m = (80 - 75) / (4 - 3) = 5 / 1 = 5. So the slope is constant: m = 5. --- **Step 2: Find the y-intercept (b)** Use y = mx + b with m = 5 and any point, say (1, 65): 65 = 5 * 1 + b 65 = 5 + b b = 65 - 5 b = 60. Check with another point, say (6, 90): 90 = 5 * 6 + b 90 = 30 + b b = 60. It matches. --- **Step 3: Write the equation** y = 5x + 60. --- **Final Answer:** y = 5x + 60

7. √(3x - 5) = 7 Answer: 18 Solution: The equation is √(3x - 5) = 7 Square both sides to eliminate the square root: (√(3x - 5))² = 7² This simplifies to 3x - 5 = 49 Add 5 to both sides: 3x = 54 Divide both sides by 3: x = 18 Check the solution: √(3×18 - 5) = √(54 - 5) = √49 = 7 The answer is 18.
   Full step-by-step solution

   Step 1: The equation is √(3x - 5) = 7 Step 2: Square both sides to eliminate the square root: (√(3x - 5))² = 7² Step 3: This simplifies to 3x - 5 = 49 Step 4: Add 5 to both sides: 3x = 54 Step 5: Divide both sides by 3: x = 18 Step 6: Check the solution: √(3×18 - 5) = √(54 - 5) = √49 = 7 The answer is 18.