Data Regression

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Answer Key & Explanations

Data Regression · Grade 11 · Worksheet 2

1. Olivia is studying the relationship between study time and test scores for her math class. She collected data from 9 students and calculated the linear regression equation as y = 1.7x + 71.3, where x represents study time in hours and y represents the test score. If a student studied for 5 hours, what would be the predicted test score according to this model? Answer: C. 79.8 Solution: The regression equation is y = 1.7x + 71.3 Substitute x = 5 (hours of study time) into the equation y = 1.7(5) + 71.3 y = 8.5 + 71.3 y = 79.8 The predicted test score for 5 hours of study is 79.8 The correct answer is 79.8.
   Full step-by-step solution

   Step 1: The regression equation is y = 1.7x + 71.3 Step 2: Substitute x = 5 (hours of study time) into the equation Step 3: y = 1.7(5) + 71.3 Step 4: y = 8.5 + 71.3 Step 5: y = 79.8 Step 6: The predicted test score for 5 hours of study is 79.8 The correct answer is 79.8.

2. Tane is analyzing the relationship between study time and test scores for his math class. He collected data from 7 students and found the linear regression equation to be y = 3.5x + 65, where x represents study time in hours and y represents the test score. If a student studies for 5 hours, what test score does the regression model predict? Answer: 82.5 Solution: The regression equation is y = 3.5x + 65 Substitute x = 5 into the equation: y = 3.5(5) + 65 Calculate 3.5 × 5 = 17.5 Add the y-intercept: 17.5 + 65 = 82.5 The predicted test score is 82.5
   Full step-by-step solution

   Step 1: The regression equation is y = 3.5x + 65 Step 2: Substitute x = 5 into the equation: y = 3.5(5) + 65 Step 3: Calculate 3.5 × 5 = 17.5 Step 4: Add the y-intercept: 17.5 + 65 = 82.5 Step 5: The predicted test score is 82.5

3. Olivia is studying the relationship between the number of hours students spend practicing piano each week and their scores on a music theory test. She collected data from 7 students and calculated the linear regression equation as y = 3.5x + 47, where x represents practice hours and y represents test scores. If a student practices for 9 hours per week, what test score does the regression model predict? Answer: 78.5 Solution: The regression equation is y = 3.5x + 47, where x is practice hours and y is test score.
   Full step-by-step solution

   Step 1: The regression equation is y = 3.5x + 47, where x is practice hours and y is test score. Step 2: Substitute x = 9 into the equation: y = 3.5(9) + 47 Step 3: Calculate 3.5 × 9 = 31.5 Step 4: Add 47 to 31.5: 31.5 + 47 = 78.5 Step 5: The predicted test score is 78.5

4. Sophia 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 = 2.8x + 71, where x represents study time in hours and y represents the test score. If a student studied for 7 hours, what test score does the regression model predict? Answer: 90.6 Solution: The regression equation is y = 2.8x + 71 Substitute x = 7 into the equation: y = 2.8(7) + 71 Calculate 2.8 × 7 = 19.6 Add 19.6 + 71 = 90.6 The predicted test score is 90.6
   Full step-by-step solution

   Step 1: The regression equation is y = 2.8x + 71 Step 2: Substitute x = 7 into the equation: y = 2.8(7) + 71 Step 3: Calculate 2.8 × 7 = 19.6 Step 4: Add 19.6 + 71 = 90.6 Step 5: The predicted test score is 90.6

5. Sophia is studying the relationship between the amount of time students spend practicing guitar (in hours per week) and their performance scores on a music assessment. She collected data from 6 students: (1, 66), (6, 71), (11, 76), (16, 81), (21, 86), (26, 91). Using linear regression, what performance score would the model predict for a student who practices 31 hours per week? Answer: 96 Solution: Calculate the slope (m) using the formula m = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²) Where n = 6, Σx = 1+6+11+16+21+26 = 81, Σy = 66+71+76+81+86+91 = 471 Σxy = (1×66)+(6×71)+(11×76)+(16×81)+(21×86)+(26×91) = 66+426+836+1296+1806+2366 = 6796 Σx² = 1²+6²+11²+16²+21²+26² = 1+36+121+256+441+676 = 1531 m =…
   Full step-by-step solution

   Step 1: Calculate the slope (m) using the formula m = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²) Where n = 6, Σx = 1+6+11+16+21+26 = 81, Σy = 66+71+76+81+86+91 = 471 Σxy = (1×66)+(6×71)+(11×76)+(16×81)+(21×86)+(26×91) = 66+426+836+1296+1806+2366 = 6796 Σx² = 1²+6²+11²+16²+21²+26² = 1+36+121+256+441+676 = 1531 m = (6×6796 - 81×471) / (6×1531 - 81²) = (40776 - 38151) / (9186 - 6561) = 2625 / 2625 = 1 Step 2: Calculate the y-intercept (b) using the formula b = (Σy - mΣx) / n b = (471 - 1×81) / 6 = (471 - 81) / 6 = 390 / 6 = 65 Step 3: Write the regression equation: y = mx + b = 1x + 65 Step 4: Substitute x = 31 into the equation: y = 1×31 + 65 = 31 + 65 = 96 The predicted performance score for 31 hours of practice is 96.

6. 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.