Data Regression

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Answer Key & Explanations

Data Regression · Grade 11 · Worksheet 3

1. Mere is analyzing the relationship between study time and test scores for her math class. She collected data from 8 students and found the linear regression equation to be y = 2.4x + 68, where x represents study time in hours and y represents the test score. If a student studies for 6 hours, what test score does the regression model predict? Answer: 82.4 Solution: The regression equation is y = 2.4x + 68 Substitute x = 6 into the equation: y = 2.4(6) + 68 Multiply: 2.4 × 6 = 14.4 Add: 14.4 + 68 = 82.4 The regression model predicts a test score of 82.4 The answer is 82.4.
   Full step-by-step solution

   Step 1: The regression equation is y = 2.4x + 68 Step 2: Substitute x = 6 into the equation: y = 2.4(6) + 68 Step 3: Multiply: 2.4 × 6 = 14.4 Step 4: Add: 14.4 + 68 = 82.4 Step 5: The regression model predicts a test score of 82.4 The answer is 82.4.

2. Sophia is studying the relationship between the number of hours students spend practicing guitar each week and their performance scores on a music assessment. She collected data from 6 students: (1, 71), (2, 76), (3, 81), (4, 86), (5, 91), (6, 96). Using linear regression, what performance score would the model predict for a student who practices 7 hours per week? Answer: 101 Solution: Calculate the mean of x-values: (1+2+3+4+5+6)/6 = 21/6 = 3.5 Calculate the mean of y-values: (71+76+81+86+91+96)/6 = 501/6 = 83.5 Calculate the slope (m) using the formula: m = Σ[(x_i - x̄)(y_i - ȳ)] / Σ[(x_i - x̄)^2] Differences from mean: x: -2.5, -1.5, -0.5, 0.5, 1.5, 2.5; y: -12.5, -7.5,…
   Full step-by-step solution

   Step 1: Calculate the mean of x-values: (1+2+3+4+5+6)/6 = 21/6 = 3.5 Step 2: Calculate the mean of y-values: (71+76+81+86+91+96)/6 = 501/6 = 83.5 Step 3: Calculate the slope (m) using the formula: m = Σ[(x_i - x̄)(y_i - ȳ)] / Σ[(x_i - x̄)^2] Differences from mean: x: -2.5, -1.5, -0.5, 0.5, 1.5, 2.5; y: -12.5, -7.5, -2.5, 2.5, 7.5, 12.5 Products: 31.25, 11.25, 1.25, 1.25, 11.25, 31.25; Sum = 87.5 Squared x-differences: 6.25, 2.25, 0.25, 0.25, 2.25, 6.25; Sum = 17.5 m = 87.5 / 17.5 = 5 Step 4: Calculate the y-intercept (b) using: b = ȳ - m*x̄ = 83.5 - 5*3.5 = 83.5 - 17.5 = 66 Step 5: The regression equation is: y = 5x + 66 Step 6: For x = 7 hours: y = 5*7 + 66 = 35 + 66 = 101 Step 7: The predicted performance score is 101

3. Emma is analyzing the relationship between study time and test scores. She collects data from 5 students: Student A studied 15 minutes and scored 55, Student B studied 30 minutes and scored 70, Student C studied 45 minutes and scored 85, Student D studied 60 minutes and scored 100, and Student E studied 75 minutes and scored 115. Using linear regression, what test score would Emma predict for a student who studies 90 minutes? Answer: 130 Solution: Examine the data points: (15,55), (30,70), (45,85), (60,100), (75,115) Calculate the slope: From (15,55) to (30,70), the score increases by 15 when study time increases by 15 minutes.
   Full step-by-step solution

   Step 1: Examine the data points: (15,55), (30,70), (45,85), (60,100), (75,115) Step 2: Calculate the slope: From (15,55) to (30,70), the score increases by 15 when study time increases by 15 minutes. Slope = 15/15 = 1 Step 3: Find the y-intercept: Using point (15,55), y = mx + b → 55 = 1(15) + b → 55 = 15 + b → b = 40 Step 4: Write the regression equation: y = 1x + 40 Step 5: Predict for 90 minutes: y = 1(90) + 40 = 90 + 40 = 130 The answer is 130.

4. Charlotte is a meteorologist studying the relationship between temperature (in °C) and the number of ice cream cones sold at a beach. Charlotte collects data for 26 days and finds that the regression line is y = 4x + 81, where x is temperature and y is number of cones sold. What does the slope 4 mean in the context of this problem? Answer: 4 Solution: The slope 4 represents the change in y (cones sold) for each one-unit increase in x (temperature).
   Full step-by-step solution

   Step 1: Understand the regression equation: y = 4x + 81. Step 2: The slope 4 represents the change in y (cones sold) for each one-unit increase in x (temperature). Step 3: In this context, a slope of 4 means that for every 1°C increase in temperature, the number of ice cream cones sold increases by 4. The slope 4 means 4.

5. Hana is analyzing the relationship between study time and test scores for her math class. She collected data from 12 students and found the linear regression equation to be y = 2.4x + 71, where x represents study time in hours and y represents the test score. If a student studied for 15 hours, what test score does the regression model predict? Answer: 107 Solution: The regression equation is y = 2.4x + 71 Substitute x = 15 into the equation: y = 2.4(15) + 71 Calculate 2.4 × 15 = 36 Add 36 + 71 = 107 The predicted test score is 107
   Full step-by-step solution

   Step 1: The regression equation is y = 2.4x + 71 Step 2: Substitute x = 15 into the equation: y = 2.4(15) + 71 Step 3: Calculate 2.4 × 15 = 36 Step 4: Add 36 + 71 = 107 Step 5: The predicted test score is 107

6. Sophia is analyzing the relationship between study time and test scores. She collected data from 8 students and found the linear regression equation to be y = 7.2x + 58.3, where x represents study time in hours and y represents the test score. If a student studies for 9 hours, what test score does the regression model predict? Answer: 123.1 Solution: The regression equation is y = 7.2x + 58.3 Substitute x = 9 (study time in hours) into the equation y = 7.2(9) + 58.3 Calculate 7.2 × 9 = 64.8 Add 64.8 + 58.3 = 123.1 The predicted test score is 123.1
   Full step-by-step solution

   Step 1: The regression equation is y = 7.2x + 58.3 Step 2: Substitute x = 9 (study time in hours) into the equation Step 3: y = 7.2(9) + 58.3 Step 4: Calculate 7.2 × 9 = 64.8 Step 5: Add 64.8 + 58.3 = 123.1 Step 6: The predicted test score is 123.1