Data Regression

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Answer Key & Explanations

Data Regression · Grade 11 · Worksheet 3

1. 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 7 students: (8, 72), (10, 78), (12, 84), (14, 90), (16, 96), (18, 102), (20, 108). Using linear regression, what performance score would the model predict for a student who practices 15 hours per week? Answer: 93 Solution: Calculate the mean of x-values (practice hours): (8+10+12+14+16+18+20)/7 = 98/7 = 14 Calculate the mean of y-values (scores): (72+78+84+90+96+102+108)/7 = 630/7 = 90 Calculate the slope (m) using the formula m = Σ[(x_i - x̄)(y_i - ȳ)] / Σ[(x_i - x̄)^2] (8-14)(72-90) = (-6)(-18) = 108…
   Full step-by-step solution

   Step 1: Calculate the mean of x-values (practice hours): (8+10+12+14+16+18+20)/7 = 98/7 = 14 Step 2: Calculate the mean of y-values (scores): (72+78+84+90+96+102+108)/7 = 630/7 = 90 Step 3: Calculate the slope (m) using the formula m = Σ[(x_i - x̄)(y_i - ȳ)] / Σ[(x_i - x̄)^2] Step 4: For each point, calculate (x_i - x̄)(y_i - ȳ): (8-14)(72-90) = (-6)(-18) = 108 (10-14)(78-90) = (-4)(-12) = 48 (12-14)(84-90) = (-2)(-6) = 12 (14-14)(90-90) = (0)(0) = 0 (16-14)(96-90) = (2)(6) = 12 (18-14)(102-90) = (4)(12) = 48 (20-14)(108-90) = (6)(18) = 108 Sum = 108+48+12+0+12+48+108 = 336 Step 5: Calculate Σ[(x_i - x̄)^2]: (8-14)^2 = 36 (10-14)^2 = 16 (12-14)^2 = 4 (14-14)^2 = 0 (16-14)^2 = 4 (18-14)^2 = 16 (20-14)^2 = 36 Sum = 36+16+4+0+4+16+36 = 112 Step 6: Calculate slope m = 336/112 = 3 Step 7: Calculate y-intercept b = ȳ - m*x̄ = 90 - 3*14 = 90 - 42 = 48 Step 8: The regression equation is y = 3x + 48 Step 9: For x = 15 hours, y = 3*15 + 48 = 45 + 48 = 93 The predicted score is 93.

2. Isabella is analyzing the relationship between study time and test scores. She collected data from 12 students and found the linear regression equation: y = 8.5x + 42, where x represents study time in hours and y represents test score. If a student studies for 9 hours, what test score does the regression model predict? Answer: 118.5 Solution: The regression equation is y = 8.5x + 42 Substitute x = 9 into the equation: y = 8.5(9) + 42 Calculate 8.5 × 9 = 76.5 Add 42 to 76.5: 76.5 + 42 = 118.5 The predicted test score is 118.5
   Full step-by-step solution

   Step 1: The regression equation is y = 8.5x + 42 Step 2: Substitute x = 9 into the equation: y = 8.5(9) + 42 Step 3: Calculate 8.5 × 9 = 76.5 Step 4: Add 42 to 76.5: 76.5 + 42 = 118.5 Step 5: The predicted test score is 118.5

3. Aroha is studying the relationship between study time and test scores in her physics class. She collected data from 9 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 12 hours, what test score does the regression model predict? Answer: 99.8 Solution: The regression equation is y = 2.4x + 71 Substitute x = 12 into the equation: y = 2.4(12) + 71 Calculate 2.4 × 12 = 28.8 Add the y-intercept: 28.8 + 71 = 99.8 The predicted test score is 99.8
   Full step-by-step solution

   Step 1: The regression equation is y = 2.4x + 71 Step 2: Substitute x = 12 into the equation: y = 2.4(12) + 71 Step 3: Calculate 2.4 × 12 = 28.8 Step 4: Add the y-intercept: 28.8 + 71 = 99.8 Step 5: The predicted test score is 99.8

4. Sophia is studying the relationship between study time and test scores. She collected data from 6 students: (1, 61), (2, 66), (3, 71), (4, 76), (5, 81), (6, 86). These points form a perfect linear pattern when plotted on a scatter plot. Using linear regression, what would be Sophia's predicted test score for a student who studies for 7 hours? Answer: 91 Solution: Examine the data points: (1, 61), (2, 66), (3, 71), (4, 76), (5, 81), (6, 86) Calculate the slope by finding how much y increases when x increases by 1: 66-61=5, 71-66=5, 76-71=5, 81-76=5, 86-81=5 The slope is consistently 5, meaning for each additional hour of study, the test score increases by…
   Full step-by-step solution

   Step 1: Examine the data points: (1, 61), (2, 66), (3, 71), (4, 76), (5, 81), (6, 86) Step 2: Calculate the slope by finding how much y increases when x increases by 1: 66-61=5, 71-66=5, 76-71=5, 81-76=5, 86-81=5 Step 3: The slope is consistently 5, meaning for each additional hour of study, the test score increases by 5 points Step 4: Find the y-intercept using the first point (1, 61): 61 = 5(1) + b → 61 = 5 + b → b = 56 Step 5: The regression equation is y = 5x + 56 Step 6: For x = 7 hours: y = 5(7) + 56 = 35 + 56 = 91 Step 7: The predicted test score for 7 hours of study is 91

5. Sophia is studying the relationship between study time and test scores in her math class. She collected data from 12 students and performed linear regression analysis. The regression equation she obtained was y = 2.8x + 67, where x represents study time in hours and y represents test score. If Noah studied for 8 hours, what score would this regression model predict for him? Answer: A. 89.4 Solution: Linear regression equations are used to predict outcomes based on input variables. The equation takes the form y = mx + b, where m is the slope (rate of change) and b is the y-intercept (starting value when x=0).
   Full step-by-step solution

   Linear regression equations are used to predict outcomes based on input variables. The equation takes the form y = mx + b, where m is the slope (rate of change) and b is the y-intercept (starting value when x=0). To make a prediction, you substitute the known x-value into the equation and calculate the corresponding y-value. This process is similar to evaluating any linear function, but it's important to remember that regression predictions are estimates based on patterns in data, not guaranteed outcomes.

6. Sophia is studying the relationship between the number of hours students spend on homework and their test scores. She collected data from 8 students and calculated the linear regression equation as y = 8.2x + 72, where x is the number of homework hours and y is the test score. If a student spends 11 hours on homework, what test score does the regression model predict? Answer: 162.2 Solution: The regression equation is y = 8.2x + 72 Substitute x = 11 into the equation: y = 8.2(11) + 72 Calculate 8.2 × 11 = 90.2 Add 72 to 90.2: 90.2 + 72 = 162.2 The predicted test score is 162.2
   Full step-by-step solution

   Step 1: The regression equation is y = 8.2x + 72 Step 2: Substitute x = 11 into the equation: y = 8.2(11) + 72 Step 3: Calculate 8.2 × 11 = 90.2 Step 4: Add 72 to 90.2: 90.2 + 72 = 162.2 Step 5: The predicted test score is 162.2