Data Regression

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Answer Key & Explanations

Data Regression · Grade 11 · Worksheet 1

1. Kaia is studying the relationship between daily temperature and ice cream sales at her shop. She collected data for 10 days and calculated the linear regression equation as y = 9.2x + 15.8, where x is temperature in degrees Celsius and y is ice cream sales in dollars. If the temperature is forecast to be 28°C tomorrow, how many dollars in ice cream sales does the regression model predict? Answer: 273.4 Solution: The regression equation is y = 9.2x + 15.8 Substitute x = 28 into the equation: y = 9.2(28) + 15.8 Calculate 9.2 × 28 = 257.6 Add 15.8: 257.6 + 15.8 = 273.4 The predicted ice cream sales are $273.40 The answer is 273.4.
   Full step-by-step solution

   Step 1: The regression equation is y = 9.2x + 15.8 Step 2: Substitute x = 28 into the equation: y = 9.2(28) + 15.8 Step 3: Calculate 9.2 × 28 = 257.6 Step 4: Add 15.8: 257.6 + 15.8 = 273.4 Step 5: The predicted ice cream sales are $273.40 The answer is 273.4.

2. Olivia is analyzing the relationship between the number of hours students study for their physics test and their test scores. She collected data from 7 students: (1, 65), (3, 71), (5, 77), (7, 83), (9, 89), (11, 95), (13, 101). Using linear regression, what test score would the model predict for a student who studies for 15 hours? Answer: 107 Solution: x̄ = (1 + 3 + 5 + 7 + 9 + 11 + 13)/7 = 49/7 = 7 ȳ = (65 + 71 + 77 + 83 + 89 + 95 + 101)/7 = 581/7 = 83 Calculate the slope (m) using the formula m = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)^2] Σ[(x - x̄)(y - ȳ)] = (1-7)(65-83) + (3-7)(71-83) + (5-7)(77-83) + (7-7)(83-83) + (9-7)(89-83) + (11-7)(95-83) +…
   Full step-by-step solution

   Step 1: Calculate the means of x and y values x̄ = (1 + 3 + 5 + 7 + 9 + 11 + 13)/7 = 49/7 = 7 ȳ = (65 + 71 + 77 + 83 + 89 + 95 + 101)/7 = 581/7 = 83 Step 2: Calculate the slope (m) using the formula m = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)^2] Σ[(x - x̄)(y - ȳ)] = (1-7)(65-83) + (3-7)(71-83) + (5-7)(77-83) + (7-7)(83-83) + (9-7)(89-83) + (11-7)(95-83) + (13-7)(101-83) = (-6)(-18) + (-4)(-12) + (-2)(-6) + (0)(0) + (2)(6) + (4)(12) + (6)(18) = 108 + 48 + 12 + 0 + 12 + 48 + 108 = 336 Σ[(x - x̄)^2] = (1-7)^2 + (3-7)^2 + (5-7)^2 + (7-7)^2 + (9-7)^2 + (11-7)^2 + (13-7)^2 = 36 + 16 + 4 + 0 + 4 + 16 + 36 = 112 m = 336/112 = 3 Step 3: Calculate the y-intercept (b) using the formula b = ȳ - m*x̄ b = 83 - 3*7 = 83 - 21 = 62 Step 4: Write the regression equation y = 3x + 62 Step 5: Substitute x = 15 into the equation y = 3*15 + 62 = 45 + 62 = 107 The predicted test score for 15 hours of study is 107.

3. Emma is analyzing the relationship between study time and test scores. She collected data from 11 students: (1, 65), (3, 71), (5, 77), (7, 83), (9, 89), (11, 95), (13, 101), (15, 107), (17, 113), (19, 119), (21, 125). The linear regression equation for this data is y = 3x + b. What is the value of the y-intercept b? Answer: 62 Solution: The regression equation is given as y = 3x + b with slope 3. To find b, we can use the fact that the regression line passes through the mean point (x̄, ȳ).
   Full step-by-step solution

   Step 1: The regression equation is given as y = 3x + b with slope 3. Step 2: To find b, we can use the fact that the regression line passes through the mean point (x̄, ȳ). Step 3: Calculate the mean of x-values: (1+3+5+7+9+11+13+15+17+19+21)/11 = 121/11 = 11 Step 4: Calculate the mean of y-values: (65+71+77+83+89+95+101+107+113+119+125)/11 = 1045/11 = 95 Step 5: The mean point is (11, 95). Substitute into the regression equation: 95 = 3(11) + b Step 6: Calculate: 95 = 33 + b Step 7: Solve for b: b = 95 - 33 = 62 The answer is 62.

4. 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.

5. Charlotte is studying the relationship between the number of hours students spend studying for a math test and their test scores. Charlotte collected data from 15 students and found the equation of the least-squares regression line to be: predicted score = 76 + 8 * (hours studied). What is the meaning of the slope of this regression line in the context of this study? Answer: 8 Solution: Recall that in a linear regression equation of the form y = b + mx, the slope m represents the change in the predicted value of y for each one-unit increase in x.
   Full step-by-step solution

   Step 1: Recall that in a linear regression equation of the form y = b + mx, the slope m represents the change in the predicted value of y for each one-unit increase in x. Step 2: In this equation, predicted score = 76 + 8 * (hours studied), the slope is 8. Step 3: The slope 8 means that for each additional hour a student studies, the model predicts their test score will increase by 8 points. The answer is 8.

6. Kaia is analyzing the relationship between study time and test scores for her math class. She collected data from 11 students and found the linear regression equation to be y = 1.7x + 63, where x is study time in hours and y is the test score. If a student studied for 5 hours, what would be the predicted test score? Answer: C. 71.5 Solution: The regression equation is y = 1.7x + 63 Substitute x = 5 into the equation: y = 1.7(5) + 63 Calculate 1.7 × 5 = 8.5 Add 8.5 + 63 = 71.5 The predicted test score is 71.5 The correct answer is 71.5.
   Full step-by-step solution

   Step 1: The regression equation is y = 1.7x + 63 Step 2: Substitute x = 5 into the equation: y = 1.7(5) + 63 Step 3: Calculate 1.7 × 5 = 8.5 Step 4: Add 8.5 + 63 = 71.5 Step 5: The predicted test score is 71.5 The correct answer is 71.5.