Data Associations

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Answer Key & Explanations

Data Associations · Grade 11 · Worksheet 1

1. Emma records the number of hours studied (x) and the test score (y) for 6 students: (2, 55), (4, 60), (6, 65), (8, 70), (10, 75), (12, 80). Describe the association and trend in this bivariate data. Answer: Positive linear association; as hours studied increase, test score increases by approximately 2.5 points per hour. Solution: List the ordered pairs: (2,55), (4,60), (6,65), (8,70), (10,75), (12,80). Observe that as x (hours studied) increases from 2 to 12, y (test score) increases from 55 to 80.
   Full step-by-step solution

   Step 1: List the ordered pairs: (2,55), (4,60), (6,65), (8,70), (10,75), (12,80). Step 2: Observe that as x (hours studied) increases from 2 to 12, y (test score) increases from 55 to 80. Step 3: Calculate the change in y for each increase of 2 in x: 60-55=5, 65-60=5, 70-65=5, 75-70=5, 80-75=5. The change is constant (+5 for every +2 in x). Step 4: The slope is 5/2 = 2.5, so y increases by 2.5 for each 1-unit increase in x. Step 5: The points lie exactly on a straight line, so the association is positive and linear. Conclusion: There is a strong positive linear association; as hours studied increase, test score increases at a constant rate of 2.5 points per hour.

2. Mere is studying the relationship between the number of hours a student spends practicing a musical instrument each week and their performance score (out of 100) on a final recital. She collects data from 15 students and plots the points (hours, score): (10, 55), (12, 60), (14, 62), (16, 65), (18, 70), (20, 72), (22, 75), (24, 78), (26, 80), (28, 82), (30, 85), (32, 88), (34, 90), (36, 92), (38, 95). Describe the association (positive, negative, or none) and the trend in the data. Identify any apparent outliers or deviations from the overall pattern. Answer: Positive association; as hours increase, score increases in a roughly linear trend with no apparent outliers. Solution: List the ordered pairs: (10, 55), (12, 60), (14, 62), (16, 65), (18, 70), (20, 72), (22, 75), (24, 78), (26, 80), (28, 82), (30, 85), (32, 88), (34, 90), (36, 92), (38, 95).
   Full step-by-step solution

   Step 1: List the ordered pairs: (10, 55), (12, 60), (14, 62), (16, 65), (18, 70), (20, 72), (22, 75), (24, 78), (26, 80), (28, 82), (30, 85), (32, 88), (34, 90), (36, 92), (38, 95). Step 2: Observe the direction: As the hours (x) increase from 10 to 38, the scores (y) increase from 55 to 95. This means the points trend upward from left to right. Step 3: Determine the type of association: Since both variables increase together, the association is positive. Step 4: Examine the pattern: The increases are fairly consistent. For example, from 10 to 12 hours, score increases by 5; from 12 to 14, increase by 2; from 14 to 16, increase by 3; and so on. The overall pattern is roughly linear with a steady upward trend. Step 5: Check for outliers: All points lie close to a straight line from (10, 55) to (38, 95). No point deviates dramatically from this pattern. There are no outliers. Final answer: Positive association; as hours increase, score increases in a roughly linear trend with no apparent outliers.

3. Emma is a data analyst studying the relationship between the number of hours a plant is exposed to sunlight each day and the number of flowers it produces. She records the following data for eight plants: (2 hours, 5 flowers), (4 hours, 10 flowers), (6 hours, 15 flowers), (8 hours, 20 flowers), (10 hours, 25 flowers), (12 hours, 30 flowers), (14 hours, 35 flowers), (16 hours, 40 flowers). Describe the association and trend between sunlight exposure and flower production. Answer: Positive linear association; as sunlight increases, flower production increases at a constant rate. Solution: Identify the two variables: sunlight exposure (independent variable, x) and flower production (dependent variable, y). Examine the ordered pairs: (2,5), (4,10), (6,15), (8,20), (10,25), (12,30), (14,35), (16,40).
   Full step-by-step solution

   Step 1: Identify the two variables: sunlight exposure (independent variable, x) and flower production (dependent variable, y). Step 2: Examine the ordered pairs: (2,5), (4,10), (6,15), (8,20), (10,25), (12,30), (14,35), (16,40). Step 3: Observe that as x increases from 2 to 16, y increases from 5 to 40. The direction of the relationship is upward, so the association is positive. Step 4: Calculate the differences: for each increase of 2 hours in sunlight, flower production increases by 5 flowers. This constant rate of change indicates a linear trend. Step 5: The data points lie exactly on a straight line (since the ratio is constant), confirming a positive linear association. Final answer: Positive linear association; as sunlight increases, flower production increases at a constant rate.

4. Aroha collects data on the number of hours studied (x) and the test score (y) for 7 students: (1, 55), (3, 65), (5, 71), (7, 78), (9, 83), (11, 89), (13, 95). Describe the association and trend in this bivariate data. Answer: Strong positive linear association; as hours studied increase, test score tends to increase. Solution: List the ordered pairs: (1,55), (3,65), (5,71), (7,78), (9,83), (11,89), (13,95). Observe that as x (hours studied) increases from 1 to 13, y (test score) increases from 55 to 95.
   Full step-by-step solution

   Step 1: List the ordered pairs: (1,55), (3,65), (5,71), (7,78), (9,83), (11,89), (13,95). Step 2: Observe that as x (hours studied) increases from 1 to 13, y (test score) increases from 55 to 95. Step 3: The increase in y is consistent — each time x increases by 2, y increases by roughly 10, 6, 7, 5, 6, 6. This shows a clear upward trend. Step 4: The points lie close to a straight line, indicating a strong linear relationship. Step 5: Since y increases when x increases, the association is positive. Conclusion: There is a strong positive linear association between hours studied and test score; as study time increases, test score tends to increase.

5. Charlotte collects data on the number of hours spent studying per week (x) and the corresponding exam scores (y) for 10 students in her Grade 11 class. The scatter plot shows the following ordered pairs: (8, 65), (10, 70), (12, 72), (15, 80), (18, 85), (20, 88), (22, 90), (25, 95), (28, 98), (30, 100). Describe the type of association between hours spent studying and exam scores, and explain the trend in the data. Answer: The association is positive, strong, and approximately linear. As the number of hours spent studying increases, the exam scores tend to increase. Solution: Plot the points mentally on a coordinate plane with x (study hours) ranging from 8 to 30 and y (exam scores) ranging from 65 to 100. Observe the pattern from left to right.
   Full step-by-step solution

   Step 1: Plot the points mentally on a coordinate plane with x (study hours) ranging from 8 to 30 and y (exam scores) ranging from 65 to 100. Step 2: Observe the pattern from left to right. For each increase in x, the y values are also increasing: (8,65) to (10,70) to (12,72), etc. This indicates a positive association. Step 3: Check how closely the points follow a straight line. The points show a consistent upward pattern without major deviations. For example, from x=8 to x=30, the scores rise from 65 to 100, which is nearly a straight line with a slope of about (100-65)/(30-8) = 35/22 ≈ 1.59. This suggests a strong, approximately linear trend. Step 4: Conclude the association: The relationship is positive, strong, and approximately linear. The trend is that as study hours increase, exam scores tend to increase. Final answer: The association is positive, strong, and approximately linear. As the number of hours spent studying increases, the exam scores tend to increase.

6. Charlotte is a meteorologist studying the relationship between the average daily temperature (in degrees Celsius) and the number of ice cream cones sold at a beachside shop over 12 days. She collects the following bivariate data: (12°C, 47 cones), (17°C, 52 cones), (22°C, 57 cones), (27°C, 62 cones), (32°C, 67 cones), (37°C, 72 cones), (42°C, 77 cones), (47°C, 82 cones), (52°C, 87 cones), (57°C, 92 cones), (62°C, 97 cones), (67°C, 102 cones). Describe the association and trend between temperature and ice cream sales. Is it positive, negative, or no association? What trend do you observe as temperature increases? Answer: Positive association; as temperature increases, ice cream cone sales tend to increase. Solution: List the ordered pairs: (12,47), (17,52), (22,57), (27,62), (32,67), (37,72), (42,77), (47,82), (52,87), (57,92), (62,97), (67,102).
   Full step-by-step solution

   Step 1: List the ordered pairs: (12,47), (17,52), (22,57), (27,62), (32,67), (37,72), (42,77), (47,82), (52,87), (57,92), (62,97), (67,102). Step 2: Observe that as the temperature (x) increases from 12 to 67, the number of cones sold (y) increases from 47 to 102. Step 3: The y-values increase by a consistent amount (5 cones) for every 5°C increase in temperature, showing a clear linear pattern. Step 4: Since both variables increase together, this indicates a positive association. Step 5: The trend is that as temperature rises, ice cream sales tend to rise in a predictable, increasing manner. Final answer: Positive association; as temperature increases, ice cream cone sales tend to increase.