Data Associations

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Answer Key & Explanations

Data Associations · Grade 11 · Worksheet 3

1. Mere is analyzing data from a local river's ecosystem. She measures the water temperature (in degrees Celsius) at various depths (in meters) and records the following paired data: Depth (m): 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. Temperature (deg C): 22, 19, 17, 14, 12, 9, 7, 4, 2, 1, 0. Describe the type of association and trend observed between depth and temperature. Answer: Negative association; as depth increases, temperature decreases. Solution: Examine the bivariate data pairs: (0,22), (5,19), (10,17), (15,14), (20,12), (25,9), (30,7), (35,4), (40,2), (45,1), (50,0).
   Full step-by-step solution

   Step 1: Examine the bivariate data pairs: (0,22), (5,19), (10,17), (15,14), (20,12), (25,9), (30,7), (35,4), (40,2), (45,1), (50,0). Step 2: Observe the trend in the independent variable (depth) as it increases from 0 to 50 meters. Step 3: Observe the corresponding values of the dependent variable (temperature): as depth increases, the temperature consistently decreases from 22 degrees Celsius down to 0 degrees Celsius. Step 4: Since one variable increases while the other decreases, this is a negative association. The trend is that temperature decreases as depth increases. Final answer: Negative association; as depth increases, temperature decreases.

2. log₇(49) + 2cos(π/3) - 7² = ? Answer: -46 Solution: Evaluate log₇(49). Since 7² = 49, log₇(49) = 2. Evaluate 2cos(π/3).
   Full step-by-step solution

   Step 1: Evaluate log₇(49). Since 7² = 49, log₇(49) = 2. Step 2: Evaluate 2cos(π/3). Since cos(π/3) = 1/2, then 2 × (1/2) = 1. Step 3: Evaluate 7². 7 × 7 = 49. Step 4: Substitute the values back into the expression: 2 + 1 - 49. Step 5: Perform the addition first: 2 + 1 = 3. Step 6: Then perform the subtraction: 3 - 49 = -46. The final answer is -46.

3. Liam is a meteorologist studying the relationship between altitude and air pressure. He collects data at various altitudes above sea level and records the corresponding air pressure in kilopascals (kPa). His data is shown in the table below: Altitude (km): 1, 3, 5, 7, 9 Air Pressure (kPa): 89, 69, 53, 41, 31 Describe the type of association (positive, negative, or none) and the trend in the relationship between altitude and air pressure based on this bivariate data. Explain what this trend means in the context of the problem. Answer: Negative association; as altitude increases, air pressure decreases. Solution: Examine the bivariate data. The two variables are altitude (independent variable, x) and air pressure (dependent variable, y). The ordered pairs are: (1, 89), (3, 69), (5, 53), (7, 41), (9, 31).
   Full step-by-step solution

   Step 1: Examine the bivariate data. The two variables are altitude (independent variable, x) and air pressure (dependent variable, y). The ordered pairs are: (1, 89), (3, 69), (5, 53), (7, 41), (9, 31). Step 2: Identify the trend. As the altitude (x) increases from 1 km to 9 km, the air pressure (y) consistently decreases from 89 kPa to 31 kPa. The values are moving in opposite directions. Step 3: Determine the type of association. Since one variable increases while the other decreases, this is a negative association (or inverse relationship). The relationship is also linear in appearance, as the pressure decreases by roughly 20 kPa for every 2 km increase in altitude. Step 4: Interpret in context. This trend means that the higher you go above sea level, the lower the air pressure becomes. This is because there is less air above you pushing down, so the atmospheric pressure drops with increasing altitude. The answer is: Negative association; as altitude increases, air pressure decreases.

4. Emma is analyzing the relationship between the number of hours spent studying (x) and the test scores (y) for 11th-grade students. She collects the following data: (3, 55), (5, 65), (7, 75), (9, 85), (11, 95). Describe the association and trend between the number of hours studied and the test scores. Answer: Positive linear association; as hours increase, test scores increase. Solution: Examine the ordered pairs: (3, 55), (5, 65), (7, 75), (9, 85), (11, 95). As x (hours studied) increases from 3 to 11, y (test score) increases from 55 to 95.
   Full step-by-step solution

   Step 1: Examine the ordered pairs: (3, 55), (5, 65), (7, 75), (9, 85), (11, 95). Step 2: As x (hours studied) increases from 3 to 11, 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 10. Step 4: This consistent pattern indicates a linear relationship. Since both variables increase together, the association is positive. Step 5: Therefore, the data shows a positive linear association: as the number of hours studied increases, the test scores tend to increase.

5. Matiu is an environmental scientist studying the relationship between the number of days since a controlled burn (x) and the number of native plant species observed in a regenerating forest plot (y). He collects the following bivariate data: (10, 14), (20, 23), (30, 31), (40, 38), (50, 42), (60, 45). Describe the association and trend shown by this data. Answer: The data shows a strong positive association; as the number of days since the controlled burn increases, the number of native plant species tends to increase, but the rate of increase appears to slow down over time, suggesting a concave down (logarithmic or quadratic) trend. Solution: Examine the ordered pairs: (10, 14), (20, 23), (30, 31), (40, 38), (50, 42), (60, 45). As x increases from 10 to 60, y increases from 14 to 45. This shows a positive association.
   Full step-by-step solution

   Step 1: Examine the ordered pairs: (10, 14), (20, 23), (30, 31), (40, 38), (50, 42), (60, 45). As x increases from 10 to 60, y increases from 14 to 45. This shows a positive association. Step 2: Check the changes in y for each increase of 10 in x: from 10 to 20, y increases by 9; from 20 to 30, y increases by 8; from 30 to 40, y increases by 7; from 40 to 50, y increases by 4; from 50 to 60, y increases by 3. The increments are decreasing. Step 3: Because the increases in y are getting smaller as x increases, the trend is not linear; it shows a curve that is increasing but at a decreasing rate. This is a concave down trend, which might be modeled by a logarithmic or quadratic function with a negative quadratic coefficient. Step 4: Conclusion: The data exhibits a strong positive association with a decreasing rate of growth, indicating a concave down trend.

6. Emma collects data on the number of hours spent studying (x) and the corresponding test scores (y) for 10 students. The scatter plot of her data shows points that roughly follow a linear pattern from (5, 55) to (25, 95). Describe the association between hours studied and test scores, including the direction, strength, and any trend you observe. Answer: Strong positive linear association; as hours studied increase, test scores tend to increase. Solution: Identify the two variables: hours spent studying (independent variable, x) and test scores (dependent variable, y). Look at the pattern of points from the lowest x (5 hours) to the highest x (25 hours).
   Full step-by-step solution

   Step 1: Identify the two variables: hours spent studying (independent variable, x) and test scores (dependent variable, y). Step 2: Look at the pattern of points from the lowest x (5 hours) to the highest x (25 hours). The y-values increase from about 55 to 95. Step 3: Since y increases as x increases, the association is positive. Step 4: The points roughly follow a straight line, indicating a linear trend. Step 5: The points are close to the line, suggesting a strong association. Conclusion: There is a strong positive linear association between hours studied and test scores. As hours studied increase, test scores tend to increase.

7. log₂(32) + cos(π/3) = ? Answer: 5.5 Solution: Evaluate log₂(32). Since 2^5 = 32, log₂(32) = 5. Evaluate cos(π/3).
   Full step-by-step solution

   Step 1: Evaluate log₂(32). Since 2^5 = 32, log₂(32) = 5. Step 2: Evaluate cos(π/3). The cosine of π/3 radians (60 degrees) is 1/2. Step 3: Add the two results: 5 + 1/2 = 5.5. The answer is 5.5.