Scatter Plots

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Scatter Plots · Grade 8 · Worksheet 3

1. A biologist is studying the relationship between the wingspan of monarch butterflies (in millimeters) and their flight speed (in meters per second). The data shows a strong positive linear correlation. If a butterfly with a wingspan of 8.5 × 10^1 mm flies at 2.2 m/s, and a butterfly with a wingspan of 9.5 × 10^1 mm flies at 2.8 m/s, what would be the expected flight speed for a butterfly with a wingspan of 8.8 × 10^1 mm? Answer: 2.38 Solution: Identify the two data points: (85, 2.2) and (95, 2.8) where x is wingspan in mm and y is speed in m/s Calculate the slope (rate of change): m = (2.8 - 2.2) / (95 - 85) = 0.6 / 10 = 0.06 m/s per mm Use point-slope form with the first point: y - 2.2 = 0.06(x - 85) Substitute x = 88: y - 2.2 =…
   Full step-by-step solution

   Step 1: Identify the two data points: (85, 2.2) and (95, 2.8) where x is wingspan in mm and y is speed in m/s Step 2: Calculate the slope (rate of change): m = (2.8 - 2.2) / (95 - 85) = 0.6 / 10 = 0.06 m/s per mm Step 3: Use point-slope form with the first point: y - 2.2 = 0.06(x - 85) Step 4: Substitute x = 88: y - 2.2 = 0.06(88 - 85) = 0.06 × 3 = 0.18 Step 5: Solve for y: y = 2.2 + 0.18 = 2.38 Step 6: The expected flight speed is 2.38 m/s The answer is 2.38.

2. Emma records the number of pages read (x) and the time in minutes (y) for 7 books: (15, 21), (25, 35), (35, 49), (45, 63), (55, 77), (65, 91), (75, 105). Plot these points on a scatter plot and describe the association between pages read and time. Answer: Strong positive linear association Solution: Plot each point on a coordinate grid with x-axis labeled 'Pages Read' and y-axis labeled 'Time (minutes)'. Points: (15,21), (25,35), (35,49), (45,63), (55,77), (65,91), (75,105).
   Full step-by-step solution

   Step 1: Plot each point on a coordinate grid with x-axis labeled 'Pages Read' and y-axis labeled 'Time (minutes)'. Step 2: Points: (15,21), (25,35), (35,49), (45,63), (55,77), (65,91), (75,105). Step 3: As x increases from 15 to 75, y increases from 21 to 105. Both variables increase together. Step 4: The points all lie exactly on the line y = 1.4x (since 21/15 = 1.4, 35/25 = 1.4, etc.). Step 5: Because the points form a perfect straight line going upward, the association is strong, positive, and linear. Answer: Strong positive linear association.

3. Emma records the number of hours studied (x) and the test score (y) for 7 students: (1, 55), (3, 65), (5, 75), (7, 85), (9, 95), (11, 105), (13, 115). Describe the association between hours studied and test score, including direction, form, and strength. Answer: Strong positive linear association Solution: List the ordered pairs: (1,55), (3,65), (5,75), (7,85), (9,95), (11,105), (13,115). As x (hours studied) increases from 1 to 13, y (test score) increases from 55 to 115.
   Full step-by-step solution

   Step 1: List the ordered pairs: (1,55), (3,65), (5,75), (7,85), (9,95), (11,105), (13,115). Step 2: As x (hours studied) increases from 1 to 13, y (test score) increases from 55 to 115. This shows a positive direction (as one variable increases, the other also increases). Step 3: Check the pattern: For each increase of 2 hours in x, y increases by exactly 10 points. This constant rate of change means the points lie perfectly on a straight line, so the form is linear. Step 4: Since all points lie exactly on the line with no scatter, the strength is strong. Step 5: Combine: The association is strong, positive, and linear. The answer is: Strong positive linear association.

4. Emma collected data on the number of hours she practiced the piano each week and the number of new pieces she learned that week for 10 weeks. She recorded the data as ordered pairs (hours practiced, new pieces learned): (2, 1), (4, 2), (5, 2), (6, 3), (7, 4), (8, 4), (9, 5), (10, 5), (11, 6), (12, 7). She then created a scatter plot of this bivariate data. Describe the association shown in the scatter plot, including its direction, form, and strength. Then, draw a line of best fit through the points (4, 2) and (11, 6) and calculate its slope. Answer: Positive, linear, strong association with a slope of 4/7. Solution: Describe the association. As hours practiced increase, new pieces learned also increase, so the direction is positive. To find the slope of the line of best fit through (4, 2) and (11, 6), use the slope formula: slope = (y2 - y1) / (x2 - x1).
   Full step-by-step solution

   Step 1: Describe the association. As hours practiced increase, new pieces learned also increase, so the direction is positive. The points appear to follow a straight-line pattern (form is linear) and are closely clustered around an imaginary line (strength is strong). Step 2: To find the slope of the line of best fit through (4, 2) and (11, 6), use the slope formula: slope = (y2 - y1) / (x2 - x1). Step 3: Substitute the coordinates: slope = (6 - 2) / (11 - 4). Step 4: Calculate the numerator: 6 - 2 = 4. Step 5: Calculate the denominator: 11 - 4 = 7. Step 6: Slope = 4/7. The answer is a positive, linear, strong association with a slope of 4/7.

5. Olivia is studying the relationship between the number of hours students spend reading per week and their scores on a reading comprehension test. She collects data from 10 classmates and creates a scatter plot with hours reading on the x-axis and test scores (out of 100) on the y-axis. The scatter plot shows a positive linear association. Olivia draws a line of best fit that passes through the points (5, 70) and (15, 90). Based on this trend, what test score would you predict for a student who reads for 10 hours per week? Answer: 80 Solution: Find the slope of the line of best fit using the two given points (5, 70) and (15, 90).
   Full step-by-step solution

   Step 1: Find the slope of the line of best fit using the two given points (5, 70) and (15, 90). Slope = (y2 - y1) / (x2 - x1) = (90 - 70) / (15 - 5) = 20 / 10 = 2 This means for each additional hour of reading, the test score increases by 2 points. Step 2: Use the point-slope form to find the equation of the line. Using point (5, 70): y - 70 = 2(x - 5) Step 3: Simplify to slope-intercept form: y - 70 = 2x - 10 y = 2x + 60 Step 4: Substitute x = 10 (hours of reading) into the equation: y = 2(10) + 60 y = 20 + 60 y = 80 The predicted test score for 10 hours of reading per week is 80.

6. A scatter plot shows the relationship between study time (hours per week) and test scores (percentage) for 20 students. The data points form a pattern where as study time increases, test scores generally increase. The line of best fit passes through points (2, 60) and (8, 90). What is the slope of this line of best fit? Answer: 5 Solution: The slope of a line through two points (x1, y1) and (x2, y2) is: slope = (y2 - y1) / (x2 - x1) Identify the two given points.
   Full step-by-step solution

   Step 1: Understand the slope formula. The slope of a line through two points (x1, y1) and (x2, y2) is: slope = (y2 - y1) / (x2 - x1) Step 2: Identify the two given points. Point 1: (2, 60) → x1 = 2, y1 = 60 Point 2: (8, 90) → x2 = 8, y2 = 90 Step 3: Calculate the change in y (vertical change). y2 - y1 = 90 - 60 = 30 Step 4: Calculate the change in x (horizontal change). x2 - x1 = 8 - 2 = 6 Step 5: Divide the change in y by the change in x. slope = 30 / 6 Step 6: Simplify the fraction. 30 / 6 = 5 Step 7: Interpret the result. The slope of the line of best fit is 5. This means that for each additional hour of study time per week, the test score increases by 5 percentage points on average. Final answer: 5