Data Regression

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Answer Key & Explanations

Data Regression · Grade 11 · Worksheet 2

1. Moana is a meteorologist studying the relationship between temperature (in °C) and the number of ice cream cones sold at a beach. Moana collects data for 7 days and finds that the regression line is y = 20x + 57, where x is temperature and y is number of cones sold. What does the slope 20 mean in the context of this problem? Answer: 20 Solution: The slope 20 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 = 20x + 57. Step 2: The slope 20 represents the change in y (cones sold) for each one-unit increase in x (temperature). Step 3: In this context, a slope of 20 means that for every 1°C increase in temperature, the number of ice cream cones sold increases by 20. The slope 20 means 20.

2. Sophia is analyzing the relationship between study time and test scores. She collects data from 6 students: (1, 66), (2, 71), (3, 76), (4, 81), (5, 86), (6, 91) where x represents hours studied and y represents test score. Using linear regression, what test score would the model predict for a student who studies 7 hours? Answer: 96 Solution: Calculate the slope (m) of the regression line using the formula m = [n(∑xy) - (∑x)(∑y)] / [n(∑x²) - (∑x)²] ∑x = 1 + 2 + 3 + 4 + 5 + 6 = 21 ∑y = 66 + 71 + 76 + 81 + 86 + 91 = 471 ∑xy = (1×66) + (2×71) + (3×76) + (4×81) + (5×86) + (6×91) = 66 + 142 + 228 + 324 + 430 + 546 = 1736 ∑x² = 1² + 2² +…
   Full step-by-step solution

   Step 1: Calculate the slope (m) of the regression line using the formula m = [n(∑xy) - (∑x)(∑y)] / [n(∑x²) - (∑x)²] Step 2: Calculate the necessary sums: ∑x = 1 + 2 + 3 + 4 + 5 + 6 = 21 ∑y = 66 + 71 + 76 + 81 + 86 + 91 = 471 ∑xy = (1×66) + (2×71) + (3×76) + (4×81) + (5×86) + (6×91) = 66 + 142 + 228 + 324 + 430 + 546 = 1736 ∑x² = 1² + 2² + 3² + 4² + 5² + 6² = 1 + 4 + 9 + 16 + 25 + 36 = 91 n = 6 Step 3: Calculate the slope: m = [6(1736) - (21)(471)] / [6(91) - (21)²] = [10416 - 9891] / [546 - 441] = 525 / 105 = 5 Step 4: Calculate the y-intercept (b) using the formula b = (∑y - m∑x) / n b = (471 - 5×21) / 6 = (471 - 105) / 6 = 366 / 6 = 61 Step 5: Write the regression equation: y = 5x + 61 Step 6: Substitute x = 7 into the equation: y = 5(7) + 61 = 35 + 61 = 96 The predicted test score for 7 hours of study is 96.

3. Mason is analyzing the relationship between study time and test scores. He collected data from 10 students and found the linear regression equation: y = 2.8x + 72.5, where x represents study time in hours and y represents test score. If Isabella studied for 9 hours, what test score does the regression model predict for her? Answer: 97.7 Solution: The regression equation is y = 2.8x + 72.5 Isabella studied for 9 hours, so x = 9 Substitute x = 9 into the equation: y = 2.8(9) + 72.5 Calculate 2.8 × 9 = 25.2 Add 72.5: 25.2 + 72.5 = 97.7 The predicted test score is 97.7
   Full step-by-step solution

   Step 1: The regression equation is y = 2.8x + 72.5 Step 2: Isabella studied for 9 hours, so x = 9 Step 3: Substitute x = 9 into the equation: y = 2.8(9) + 72.5 Step 4: Calculate 2.8 × 9 = 25.2 Step 5: Add 72.5: 25.2 + 72.5 = 97.7 Step 6: The predicted test score is 97.7

4. Emma is analyzing the relationship between study time and test scores. She collected data from 10 students and found the linear regression equation to be y = 2.5x + 65, where x is study time in hours and y is the test score. If a student studies for 8 hours, what test score does the regression model predict? Answer: 85 Solution: The regression equation is y = 2.5x + 65 Substitute x = 8 (study time in hours) into the equation y = 2.5(8) + 65 Calculate 2.5 × 8 = 20 Add 20 + 65 = 85 The predicted test score is 85
   Full step-by-step solution

   Step 1: The regression equation is y = 2.5x + 65 Step 2: Substitute x = 8 (study time in hours) into the equation Step 3: y = 2.5(8) + 65 Step 4: Calculate 2.5 × 8 = 20 Step 5: Add 20 + 65 = 85 Step 6: The predicted test score is 85

5. Aroha is designing a triangular garden plot with sides measuring 12 m, 16 m, and 20 m. She needs to calculate the area of the triangular plot to determine how much fertilizer to purchase. What is the area of Aroha's triangular garden in square meters? Answer: 96 Solution: s = (12 + 16 + 20) / 2 s = 48 / 2 s = 24 m Area = sqrt[s(s - a)(s - b)(s - c)] Area = sqrt[24(24 - 12)(24 - 16)(24 - 20)] Area = sqrt[24 × 12 × 8 × 4] 24 × 12 = 288 288 × 8 = 2304 2304 × 4 = 9216 Area = sqrt(9216) = 96 The area of Aroha's triangular garden is 96 square meters.
   Full step-by-step solution

   Step 1: Calculate the semi-perimeter (s) of the triangle s = (12 + 16 + 20) / 2 s = 48 / 2 s = 24 m Step 2: Apply Heron's formula Area = sqrt[s(s - a)(s - b)(s - c)] Area = sqrt[24(24 - 12)(24 - 16)(24 - 20)] Area = sqrt[24 × 12 × 8 × 4] Step 3: Calculate each multiplication 24 × 12 = 288 288 × 8 = 2304 2304 × 4 = 9216 Step 4: Take the square root Area = sqrt(9216) = 96 The area of Aroha's triangular garden is 96 square meters.

6. 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