Data Associations

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Answer Key & Explanations

Data Associations · Grade 11 · Worksheet 2

1. Emma records the average monthly temperature (in degrees Celsius) and the number of ice cream cones sold at her shop over 8 months. The data points are: (5, 30), (10, 50), (15, 70), (20, 100), (25, 130), (30, 160), (35, 190), (40, 220). Describe the association and trend between temperature and ice cream sales. Answer: positive association, linear trend Solution: List the ordered pairs: (5,30), (10,50), (15,70), (20,100), (25,130), (30,160), (35,190), (40,220). Step 2: As x (temperature) increases from 5 to 40, y (ice cream sales) increases from 30 to 220.
   Full step-by-step solution

   Step 1: List the ordered pairs: (5,30), (10,50), (15,70), (20,100), (25,130), (30,160), (35,190), (40,220). Step 2: As x (temperature) increases from 5 to 40, y (ice cream sales) increases from 30 to 220. Step 3: The increase is consistent: each time x increases by 5, y increases by 20 (except from 15 to 20 where it increases by 30, but overall the pattern is roughly linear). Step 4: Therefore, the association is positive (both variables increase together) and the trend is linear (points fall close to a straight line). The answer is positive association, linear trend.

2. Hana is investigating the relationship between the number of hours a student spends on social media per day (x) and their average exam score (y) for a class of 22 Grade 11 students. She plots the data on a scatter plot. The data points show a clear curved pattern: as x increases from 0 to 2 hours, y decreases slowly from 88 to 82. As x increases from 2 to 6 hours, y decreases more rapidly from 82 to 48. As x increases beyond 6 hours, y continues to decrease but at a slowing rate, approaching a minimum of 30. Describe the overall association and trend between hours on social media and exam scores, including any specific features of the trend visible in the plot. Answer: The scatter plot shows a negative, non-linear association with a decreasing trend that is not constant. The trend is curved, showing a rapid decrease in exam scores as social media hours increase from 2 to 6, and a leveling off (asymptotic behavior) as hours exceed 6, approaching a minimum score near 30. Solution: Identify the type of data. We have two variables: hours on social media (x, independent) and exam score (y, dependent). The rate of decrease changes: slow from 0 to 2 hours, then faster from 2 to 6 hours, then slower again beyond 6 hours.
   Full step-by-step solution

   Step 1: Identify the type of data. We have two variables: hours on social media (x, independent) and exam score (y, dependent). This is bivariate data. Step 2: Describe the direction of the association. As x increases, y decreases overall. This indicates a negative association. Step 3: Determine if the association is linear or non-linear. The description says the points show a 'clear curved pattern', not a straight line. The rate of decrease changes: slow from 0 to 2 hours, then faster from 2 to 6 hours, then slower again beyond 6 hours. This means the association is non-linear (curvilinear). Step 4: Describe the specific trend features. From x=0 to x=2, the decline is gentle (y drops 6 points over 2 hours). From x=2 to x=6, the decline is steep (y drops 34 points over 4 hours, which is 8.5 points per hour on average). Beyond x=6, the decline flattens out, and y approaches 30 but does not go below it, suggesting a horizontal asymptote or a floor effect. Step 5: Summarize. The scatter plot reveals a negative, non-linear association where the trend is a decreasing curve that steepens in the middle range of x and then levels off at higher x values. The final answer is: The scatter plot shows a negative, non-linear association with a decreasing trend that is not constant. The trend is curved, showing a rapid decrease in exam scores as social media hours increase from 2 to 6, and a leveling off (asymptotic behavior) as hours exceed 6, approaching a minimum score near 30.

3. log₃(81) + 2sin(π/6) - 4² = ? Answer: -11 Solution: Evaluate log₃(81). Since 3⁴ = 81, log₃(81) = 4. Evaluate 2sin(π/6).
   Full step-by-step solution

   Step 1: Evaluate log₃(81). Since 3⁴ = 81, log₃(81) = 4. Step 2: Evaluate 2sin(π/6). Since sin(π/6) = 1/2, then 2 × (1/2) = 1. Step 3: Evaluate 4². 4 × 4 = 16. Step 4: Combine all terms: 4 + 1 - 16 = 5 - 16 = -11. The final answer is -11.

4. Matiu is an ecologist studying the relationship between the average annual temperature (in degrees Celsius) and the number of bird species observed in different forest reserves across New Zealand. He collects the following bivariate data from eight reserves: (10, 24), (12, 28), (14, 32), (16, 34), (18, 38), (20, 40), (22, 42), (24, 44). Describe the association and trend between average annual temperature and the number of bird species observed. Answer: positive association, linear trend Solution: Examine the data pairs: (10,24), (12,28), (14,32), (16,34), (18,38), (20,40), (22,42), (24,44). Step 2: Observe the x-values (temperature) increasing from 10 to 24.
   Full step-by-step solution

   Step 1: Examine the data pairs: (10,24), (12,28), (14,32), (16,34), (18,38), (20,40), (22,42), (24,44). Step 2: Observe the x-values (temperature) increasing from 10 to 24. Step 3: Observe the y-values (bird species) also increasing from 24 to 44. Step 4: Since both variables increase together, the association is positive. Step 5: Check for a linear trend: as temperature increases by 2 degrees, the number of bird species increases by 4, 4, 2, 4, 2, 2, 2 — approximately constant increments, suggesting a roughly linear relationship. Step 6: The scatter plot would show points rising from left to right in a roughly straight line pattern. Final answer: The association is positive and the trend is linear.

5. Isabella is studying the relationship between the number of hours spent on practice and the score achieved on a music performance test. She collects data from 12 students: (2, 47), (3, 52), (5, 57), (7, 62), (8, 67), (10, 72), (12, 77), (13, 82), (15, 87), (17, 92), (18, 97), (20, 102). Describe the type of association (positive, negative, or no association) and the trend in the data as hours of practice increase. Answer: Positive association; as hours of practice increase, the score tends to increase. Solution: Examine the ordered pairs. The x-values (hours of practice) are: 2, 3, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20. The corresponding y-values (scores) are: 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97, 102.
   Full step-by-step solution

   Step 1: Examine the ordered pairs. The x-values (hours of practice) are: 2, 3, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20. The corresponding y-values (scores) are: 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97, 102. Step 2: Observe the relationship. As each x-value increases, the y-value also increases. For example, when x goes from 2 to 3, y goes from 47 to 52. When x goes from 10 to 12, y goes from 72 to 77. This pattern continues for all data points. Step 3: Determine the type of association. Since both variables increase together, the association is positive. The trend is that as the number of hours spent on practice increases, the music performance test score also increases. Final answer: The data shows a positive association; as hours of practice increase, the score tends to increase.

6. Liam records the number of hours spent studying (x) and the corresponding test scores (y) for 7 students. The data points are: (1, 55), (3, 65), (5, 75), (7, 85), (9, 95), (11, 105), (13, 115). Describe the association between hours studied and test score, and identify the trend as linear or nonlinear. Answer: Positive linear association; as x increases, y increases at a constant rate of 5 points per hour. Solution: List the data points: (1,55), (3,65), (5,75), (7,85), (9,95), (11,105), (13,115). As x increases from 1 to 3 (increase of 2), y increases from 55 to 65 (increase of 10).
   Full step-by-step solution

   Step 1: List the data points: (1,55), (3,65), (5,75), (7,85), (9,95), (11,105), (13,115). Step 2: As x increases from 1 to 3 (increase of 2), y increases from 55 to 65 (increase of 10). As x increases from 3 to 5 (increase of 2), y increases from 65 to 75 (increase of 10). This pattern continues: each increase of 2 hours corresponds to an increase of 10 points. Step 3: The rate of change is constant: 10 points per 2 hours = 5 points per hour. Step 4: Since y increases at a constant rate as x increases, the association is positive and linear. Step 5: Conclusion: There is a strong positive linear association. As the number of hours studied increases, the test score increases by 5 points for every additional hour.