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Bivariate Data

Grade 11 · Statistics · Worksheet 2

  1. Mason surveyed 32 students at his school about their preferred study method (visual or textual) and their performance on a recent math test. He recorded the following data: 12 students preferred visual learning and scored above 82%, while 7 students preferred visual learning and scored 82% or below. Among those who preferred textual learning, 8 scored above 82% and 5 scored 82% or below. Construct a two-way frequency table for this bivariate categorical data. Then, calculate the conditional relative frequency of scoring above 82% among students who prefer visual learning, and interpret its meaning in context. Answer: ______________
  2. Create a scatter plot for Hana's data: (2,8), (4,12), (6,16), (8,20), (10,24), (12,28), (14,32), (16,36), (18,40), (20,44). Identify the correlation type and calculate the slope of the line of best fit. Answer: ______________
  3. Create a scatter plot for Tane's data: (5,13), (9,21), (13,29), (17,37), (21,45), (25,53), (29,61), (33,69), (37,77), (41,85). Calculate the correlation coefficient r. Answer: ______________
  4. A scatter plot displays the relationship between daily screen time (x, in hours) and average sleep duration (y, in hours) for 60 high school students. The data points form an approximately linear pattern with a correlation coefficient of r = -0.78. The regression line equation is ŷ = -0.35x + 8.2. If a student reports 6 hours of daily screen time, what sleep duration does the regression model predict? Answer: ______________
  5. Sophia, a high school science student, is investigating whether there is a relationship between the number of hours students spend on social media per week and their average exam score (out of 100). She surveys 30 students and records the following bivariate data. The data is summarized in the two-way table below, where hours on social media per week is categorized as Low (0–7 hours), Medium (8–14 hours), or High (15+ hours), and exam score is categorized as Low (below 70), Medium (70–84), or High (85–100). | Exam Score → | Low | Medium | High | Total | |--------------|-----|--------|------|-------| | Low Social Media | 2 | 7 | 5 | 14 | | Medium Social Media | 4 | 6 | 2 | 12 | | High Social Media | 3 | 1 | 0 | 4 | | Total | 9 | 14 | 7 | 30 | Based on this table, what percentage of students who spend High hours on social media scored in the High exam score category? Express your answer as a percentage (rounded to one decimal place, if necessary). Answer: ______________
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Answer Key & Explanations

Bivariate Data · Grade 11 · Worksheet 2

  1. Mason surveyed 32 students at his school about their preferred study method (visual or textual) and their performance on a recent math test. He recorded the following data: 12 students preferred visual learning and scored above 82%, while 7 students preferred visual learning and scored 82% or below. Among those who preferred textual learning, 8 scored above 82% and 5 scored 82% or below. Construct a two-way frequency table for this bivariate categorical data. Then, calculate the conditional relative frequency of scoring above 82% among students who prefer visual learning, and interpret its meaning in context. Answer: 0.6316 or 63.16% Solution: Create a two-way frequency table. Rows: Study Method (Visual, Textual). Columns: Test Performance (Above 82%, 82% or Below).
    Full step-by-step solution

    Step 1: Create a two-way frequency table. Rows: Study Method (Visual, Textual). Columns: Test Performance (Above 82%, 82% or Below). Fill in the given values: - Visual and Above 82%: 12 - Visual and 82% or Below: 7 - Textual and Above 82%: 8 - Textual and 82% or Below: 5 Step 2: Compute row totals: Visual total = 12 + 7 = 19 Textual total = 8 + 5 = 13 Step 3: Compute column totals: Above 82% total = 12 + 8 = 20 82% or Below total = 7 + 5 = 12 Step 4: Grand total = 19 + 13 = 32 (or 20 + 12 = 32). Step 5: Conditional relative frequency of scoring above 82% among visual learners: Number of visual learners who scored above 82% = 12 Total number of visual learners = 19 Conditional relative frequency = 12 / 19 = 0.631578... Rounded to four decimal places: 0.6316, or 63.16%. Interpretation: Among students who prefer visual learning, approximately 63.16% scored above 82% on the math test. The answer is 0.6316 or 63.16%.

  2. Create a scatter plot for Hana's data: (2,8), (4,12), (6,16), (8,20), (10,24), (12,28), (14,32), (16,36), (18,40), (20,44). Identify the correlation type and calculate the slope of the line of best fit. Answer: 2 Solution: Examine the data points: (2,8), (4,12), (6,16), (8,20), (10,24), (12,28), (14,32), (16,36), (18,40), (20,44) As x increases by 2, y increases by 4 consistently, indicating a perfect positive linear correlation To calculate slope, use two points: (2,8) and (4,12) Slope = (y2 - y1)/(x2 - x1) = (12…
    Full step-by-step solution

    Step 1: Examine the data points: (2,8), (4,12), (6,16), (8,20), (10,24), (12,28), (14,32), (16,36), (18,40), (20,44) Step 2: As x increases by 2, y increases by 4 consistently, indicating a perfect positive linear correlation Step 3: To calculate slope, use two points: (2,8) and (4,12) Step 4: Slope = (y2 - y1)/(x2 - x1) = (12 - 8)/(4 - 2) = 4/2 = 2 Step 5: The slope of the line of best fit is 2

  3. Create a scatter plot for Tane's data: (5,13), (9,21), (13,29), (17,37), (21,45), (25,53), (29,61), (33,69), (37,77), (41,85). Calculate the correlation coefficient r. Answer: 1.0 Solution: Calculate the mean of x-values: (5+9+13+17+21+25+29+33+37+41)/10 = 230/10 = 23. Step 2: Calculate the mean of y-values: (13+21+29+37+45+53+61+69+77+85)/10 = 490/10 = 49.
    Full step-by-step solution

    Step 1: Calculate the mean of x-values: (5+9+13+17+21+25+29+33+37+41)/10 = 230/10 = 23. Step 2: Calculate the mean of y-values: (13+21+29+37+45+53+61+69+77+85)/10 = 490/10 = 49. Step 3: Calculate the sum of (x-mean_x)(y-mean_y): (5-23)(13-49) + (9-23)(21-49) + (13-23)(29-49) + (17-23)(37-49) + (21-23)(45-49) + (25-23)(53-49) + (29-23)(61-49) + (33-23)(69-49) + (37-23)(77-49) + (41-23)(85-49) = (-18)(-36) + (-14)(-28) + (-10)(-20) + (-6)(-12) + (-2)(-4) + (2)(4) + (6)(12) + (10)(20) + (14)(28) + (18)(36) = 648 + 392 + 200 + 72 + 8 + 8 + 72 + 200 + 392 + 648 = 2640. Step 4: Calculate the sum of (x-mean_x)^2: (-18)^2 + (-14)^2 + (-10)^2 + (-6)^2 + (-2)^2 + (2)^2 + (6)^2 + (10)^2 + (14)^2 + (18)^2 = 324 + 196 + 100 + 36 + 4 + 4 + 36 + 100 + 196 + 324 = 1320. Step 5: Calculate the sum of (y-mean_y)^2: (-36)^2 + (-28)^2 + (-20)^2 + (-12)^2 + (-4)^2 + (4)^2 + (12)^2 + (20)^2 + (28)^2 + (36)^2 = 1296 + 784 + 400 + 144 + 16 + 16 + 144 + 400 + 784 + 1296 = 5280. Step 6: Calculate r = sum[(x-mean_x)(y-mean_y)] / sqrt[sum(x-mean_x)^2 * sum(y-mean_y)^2] = 2640 / sqrt(1320 * 5280) = 2640 / sqrt(6969600) = 2640 / 2640 = 1.0. The correlation coefficient is 1.0, indicating a perfect positive linear relationship.

  4. A scatter plot displays the relationship between daily screen time (x, in hours) and average sleep duration (y, in hours) for 60 high school students. The data points form an approximately linear pattern with a correlation coefficient of r = -0.78. The regression line equation is ŷ = -0.35x + 8.2. If a student reports 6 hours of daily screen time, what sleep duration does the regression model predict? Answer: 6.1 Solution: Identify the regression equation: ŷ = -0.35x + 8.2 Substitute x = 6 (hours of screen time) into the equation: ŷ = -0.35(6) + 8.2 Calculate -0.35 × 6 = -2.1 Add the result to the y-intercept: -2.1 + 8.2 = 6.1 The regression model predicts 6.1 hours of sleep for a student with 6 hours of daily…
    Full step-by-step solution

    Step 1: Identify the regression equation: ŷ = -0.35x + 8.2 Step 2: Substitute x = 6 (hours of screen time) into the equation: ŷ = -0.35(6) + 8.2 Step 3: Calculate -0.35 × 6 = -2.1 Step 4: Add the result to the y-intercept: -2.1 + 8.2 = 6.1 Step 5: The regression model predicts 6.1 hours of sleep for a student with 6 hours of daily screen time. The answer is 6.1.

  5. Sophia, a high school science student, is investigating whether there is a relationship between the number of hours students spend on social media per week and their average exam score (out of 100). She surveys 30 students and records the following bivariate data. The data is summarized in the two-way table below, where hours on social media per week is categorized as Low (0–7 hours), Medium (8–14 hours), or High (15+ hours), and exam score is categorized as Low (below 70), Medium (70–84), or High (85–100). | Exam Score → | Low | Medium | High | Total | |--------------|-----|--------|------|-------| | Low Social Media | 2 | 7 | 5 | 14 | | Medium Social Media | 4 | 6 | 2 | 12 | | High Social Media | 3 | 1 | 0 | 4 | | Total | 9 | 14 | 7 | 30 | Based on this table, what percentage of students who spend High hours on social media scored in the High exam score category? Express your answer as a percentage (rounded to one decimal place, if necessary). Answer: 0.0% Solution: Identify the row for High Social Media. The row shows: Low exam = 3, Medium exam = 1, High exam = 0, Total = 4. The number of High Social Media students who scored High is 0.
    Full step-by-step solution

    Step 1: Identify the row for High Social Media. The row shows: Low exam = 3, Medium exam = 1, High exam = 0, Total = 4. Step 2: The number of High Social Media students who scored High is 0. Step 3: The total number of High Social Media students is 4. Step 4: Calculate the percentage: (0 / 4) * 100% = 0%. Step 5: Rounded to one decimal place, the answer is 0.0%. The answer is 0.0%.