Bivariate Patterns

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Answer Key & Explanations

Bivariate Patterns · Grade 8 · Worksheet 3

1. Mason recorded the number of hours he spent studying for 10 different science quizzes and the grade he received on each quiz. The data is shown in the scatter plot: (1.5, 65), (2, 70), (2.5, 72), (3, 78), (3.5, 82), (4, 85), (4.5, 88), (5, 92), (5.5, 95), (7, 70). Describe the pattern of association between study hours and quiz grades. Identify any outlier(s) and explain why they are outliers. Answer: There is a positive association between study hours and quiz grades for most of the data, but the point (7, 70) is an outlier because it has a high study time but a low grade, not following the overall trend. Solution: List all data points: (1.5, 65), (2, 70), (2.5, 72), (3, 78), (3.5, 82), (4, 85), (4.5, 88), (5, 92), (5.5, 95), (7, 70). Look at the overall trend.
   Full step-by-step solution

   Step 1: List all data points: (1.5, 65), (2, 70), (2.5, 72), (3, 78), (3.5, 82), (4, 85), (4.5, 88), (5, 92), (5.5, 95), (7, 70). Step 2: Look at the overall trend. As study hours increase from 1.5 to 5.5, quiz grades generally increase from 65 to 95. This shows a positive association. Step 3: Identify any points that do not fit the pattern. The point (7, 70) has 7 study hours, which is much higher than the others, but the grade is only 70, which is lower than the grade at 2.5 study hours. This point is far from the upward trend of the rest of the data. Step 4: Conclusion: The association is positive for most points. The outlier is (7, 70) because it has an unusually low grade for its high study time, deviating from the pattern.

2. (3x + 5)² = 121 Answer: 2 Solution: Start with the equation (3x + 5)² = 121 Take the square root of both sides: 3x + 5 = ±√121 Simplify: 3x + 5 = ±11 Case 1: 3x + 5 = 11 → 3x = 6 → x = 2 Case 2: 3x + 5 = -11 → 3x = -16 → x = -16/3 Since the problem asks for a single numerical answer and both are valid solutions, we take the…
   Full step-by-step solution

   Step 1: Start with the equation (3x + 5)² = 121 Step 2: Take the square root of both sides: 3x + 5 = ±√121 Step 3: Simplify: 3x + 5 = ±11 Step 4: Solve for both cases: Case 1: 3x + 5 = 11 → 3x = 6 → x = 2 Case 2: 3x + 5 = -11 → 3x = -16 → x = -16/3 Step 5: Since the problem asks for a single numerical answer and both are valid solutions, we take the positive integer solution x = 2.

3. A marine biologist is studying the relationship between water temperature and the number of fish observed in a coral reef. She collected data over several days and found the linear relationship can be modeled by the equation y = -12x + 240, where x represents water temperature in degrees Celsius and y represents the number of fish observed. If the water temperature reaches 18°C, how many fish would this model predict to be observed? Answer: 24 Solution: The linear equation is y = -12x + 240, where x is temperature and y is number of fish.
   Full step-by-step solution

   Step 1: The linear equation is y = -12x + 240, where x is temperature and y is number of fish. Step 2: Substitute x = 18 into the equation: y = -12(18) + 240 Step 3: Calculate -12 × 18 = -216 Step 4: Add 240 to -216: -216 + 240 = 24 Step 5: The model predicts 24 fish would be observed at 18°C water temperature.

4. Noah is a meteorologist studying the relationship between the daily high temperature (in degrees Fahrenheit) and the number of ice cream cones sold at a local beach stand. He collected data for 11 days and recorded the following pairs: (76, 41), (81, 56), (86, 71), (91, 86), (96, 101), (101, 116), (106, 131), (111, 146), (116, 161), (121, 176), and (126, 191). When he makes a scatter plot of the data, what type of association does he observe, and does the data contain any outliers? Answer: strong positive linear association with no outliers Solution: Examine the ordered pairs. As temperature (x) increases from 76 to 126, the number of cones sold (y) increases from 41 to 191.
   Full step-by-step solution

   Step 1: Examine the ordered pairs. As temperature (x) increases from 76 to 126, the number of cones sold (y) increases from 41 to 191. Step 2: Check the pattern: For every 5-degree increase in temperature, the cones sold increase by exactly 15. (From 76 to 81: +5 degrees, cones go from 41 to 56, a +15 increase. This pattern holds for every consecutive pair.) Step 3: Since the y-values increase consistently as x increases, and the points would lie on a straight line, the association is strong positive linear. Step 4: Check for outliers: All points follow the exact same pattern with no point deviating from the line. Therefore, there are no outliers. The answer is: strong positive linear association with no outliers.

5. Isabella is a city planner studying the relationship between the number of trees planted on a street (x) and the average summer temperature (in degrees Fahrenheit) on that street (y). She collected data from 8 different streets in her city. The data points are: (0, 92), (5, 88), (10, 85), (15, 81), (20, 78), (25, 74), (30, 71), and (45, 67). When Isabella creates a scatter plot of this data, she notices that most points show a clear pattern, but one point seems to not fit the pattern. Describe the type of association shown by the main cluster of points (positive, negative, or no association) and identify the point that is an outlier. Explain your reasoning for the outlier. Answer: Negative association; the outlier is (45, 67) Solution: Look at the main cluster of points: (0, 92), (5, 88), (10, 85), (15, 81), (20, 78), (25, 74), (30, 71). As the x-values (trees) increase from 0 to 30, the y-values (temperature) consistently decrease from 92 to 71.
   Full step-by-step solution

   Step 1: Look at the main cluster of points: (0, 92), (5, 88), (10, 85), (15, 81), (20, 78), (25, 74), (30, 71). As the x-values (trees) increase from 0 to 30, the y-values (temperature) consistently decrease from 92 to 71. This shows a clear downward trend. Step 2: Identify the association. Because the y-values decrease as the x-values increase, the main cluster shows a negative association. Step 3: Check the remaining point, (45, 67). Compare it to the pattern. The main points decrease by roughly 3-4 degrees for every 5 trees. If the pattern continued, at 45 trees the temperature would be around 71 - (3*3) = 62 degrees, or using a more precise slope: from 30 trees to 45 trees is an increase of 15 trees. The rate of decrease is about 3.5 degrees per 5 trees, so 15 trees would be a decrease of about 10.5 degrees, predicting 71 - 10.5 = 60.5 degrees. The actual temperature at 45 trees is 67 degrees. This is about 6.5 degrees higher than what the pattern predicts. Step 4: Conclude about the outlier. The point (45, 67) does not follow the strong negative trend of the other points. It is separated from the main cluster and does not fit the pattern, making it an outlier. The answer is: The main cluster shows a negative association. The outlier is (45, 67).

6. Emma recorded the number of pages read (x) and the time in minutes (y) for 7 students: (15, 21), (19, 27), (23, 33), (27, 39), (31, 45), (35, 51), (39, 57). Identify the type of association and any outlier if present. Answer: positive association, no outliers Solution: List the ordered pairs: (15,21), (19,27), (23,33), (27,39), (31,45), (35,51), (39,57). As x increases from 15 to 39, y increases from 21 to 57.
   Full step-by-step solution

   Step 1: List the ordered pairs: (15,21), (19,27), (23,33), (27,39), (31,45), (35,51), (39,57). Step 2: As x increases from 15 to 39, y increases from 21 to 57. The y-values increase by 6 for every increase of 4 in x (since 27-21=6, 33-27=6, etc.). This consistent increase shows a strong positive linear association. Step 3: Check for outliers: All points follow the same pattern (y = 1.5x - 1.5 approximately). No point deviates significantly from this pattern. Step 4: Conclusion: The association is positive, and there are no outliers. Answer: positive association, no outliers