Bivariate Data

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Answer Key & Explanations

Bivariate Data · Grade 11 · Worksheet 3

1. Emma is studying the relationship between the number of hours of sunlight per day (x) and the height (in cm) of sunflower plants (y) after 30 days. She collects data from 11 plants and records the following ordered pairs: (5, 13), (7, 17), (9, 21), (11, 25), (13, 29), (15, 33), (17, 37), (19, 41), (21, 45), (23, 49), (25, 53). Create a scatter plot of this bivariate data, describe the form, direction, and strength of the distribution, and determine the linear regression equation that models the relationship. Answer: y = 2x + 3 Solution: Create a scatter plot by plotting each (x, y) point on a coordinate plane with x-axis for sunlight hours and y-axis for height. Points: (5,13), (7,17), (9,21), (11,25), (13,29), (15,33), (17,37), (19,41), (21,45), (23,49), (25,53).
   Full step-by-step solution

   Step 1: Create a scatter plot by plotting each (x, y) point on a coordinate plane with x-axis for sunlight hours and y-axis for height. Points: (5,13), (7,17), (9,21), (11,25), (13,29), (15,33), (17,37), (19,41), (21,45), (23,49), (25,53). All points form a straight line. Step 2: Describe the distribution. Form: linear (points fall on a straight line). Direction: positive (as sunlight hours increase, height increases). Strength: perfect (all points lie exactly on the line). Step 3: Find the slope m. Choose two points, e.g., (5,13) and (7,17). Slope = (17-13)/(7-5) = 4/2 = 2. Step 4: Find the y-intercept b using y = mx + b. Substitute (5,13): 13 = 2(5) + b => 13 = 10 + b => b = 3. Step 5: Write the regression equation: y = 2x + 3. The answer is y = 2x + 3.

2. A medical researcher is studying the relationship between daily exercise time (in minutes) and resting heart rate (in beats per minute) for adults. After collecting data from 50 participants, the researcher calculates a linear regression equation of ŷ = 72 - 0.15x, where x represents daily exercise time and ŷ represents predicted resting heart rate. The correlation coefficient is -0.68. If a new participant exercises for 40 minutes daily, what would be their predicted resting heart rate according to this model? Answer: 66 Solution: ŷ = 72 − 0.15x - x = daily exercise time (in minutes) - ŷ = predicted resting heart rate (in beats per minute) The problem says the new participant exercises for 40 minutes daily, so: x = 40 ŷ = 72 − 0.15 × 40 0.15 × 40 = 6 ŷ = 72 − 6 72 − 6 = 66 The predicted resting heart rate for a person who…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the regression equation** The regression equation given is: ŷ = 72 − 0.15x Where: - x = daily exercise time (in minutes) - ŷ = predicted resting heart rate (in beats per minute) --- **Step 2: Identify the value of x for the new participant** The problem says the new participant exercises for 40 minutes daily, so: x = 40 --- **Step 3: Substitute x into the equation** ŷ = 72 − 0.15 × 40 --- **Step 4: Perform the multiplication** 0.15 × 40 = 6 So: ŷ = 72 − 6 --- **Step 5: Perform the subtraction** 72 − 6 = 66 --- **Step 6: Interpret the result** The predicted resting heart rate for a person who exercises 40 minutes daily is 66 beats per minute. --- **Final Answer:** 66

3. Noah, a city planner, surveyed 81 residents in two neighborhoods, Eastside and Westside, about their preferred mode of transportation for commuting to work: car, bus, or bicycle. The results showed that in Eastside, 16 residents preferred cars, 11 preferred buses, and 6 preferred bicycles. In Westside, 21 residents preferred cars, 16 preferred buses, and 11 preferred bicycles. Construct a two-way table to represent this bivariate data with neighborhood and transportation preference as the two categorical variables. Then, calculate the conditional relative frequency of residents who prefer bicycles given that they live in Westside, expressed as a percentage rounded to the nearest whole number. Answer: 23% Solution: Construct a two-way table with rows for Eastside and Westside, and columns for Car, Bus, and Bicycle, plus a Total row and column. - Eastside: Car = 16, Bus = 11, Bicycle = 6. Total Eastside = 16 + 11 + 6 = 33.
   Full step-by-step solution

   Step 1: Construct a two-way table with rows for Eastside and Westside, and columns for Car, Bus, and Bicycle, plus a Total row and column. - Eastside: Car = 16, Bus = 11, Bicycle = 6. Total Eastside = 16 + 11 + 6 = 33. - Westside: Car = 21, Bus = 16, Bicycle = 11. Total Westside = 21 + 16 + 11 = 48. - Total for Car = 16 + 21 = 37. - Total for Bus = 11 + 16 = 27. - Total for Bicycle = 6 + 11 = 17. - Grand Total = 33 + 48 = 81. Step 2: The conditional relative frequency of residents who prefer bicycles given they live in Westside is the number of Westside residents who prefer bicycles divided by the total number of Westside residents: 11 / 48. Step 3: Calculate 11 / 48 = 0.229166... Multiply by 100 to get a percentage: 0.229166 * 100 = 22.9166%. Step 4: Round to the nearest whole number: 23%. The answer is 23%.

4. Olivia, a high school environmental science student, is investigating whether there is a relationship between the number of hours spent studying per week and the final exam score (out of 100) for 15 students in her class. She collects the following data: Hours studied (x): 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80 Exam score (y): 45, 50, 60, 65, 70, 75, 80, 85, 88, 90, 92, 93, 95, 96, 98 Construct a scatter plot of the data and describe the distribution in terms of form, direction, and strength. Then, identify any potential outliers. Answer: The scatter plot shows a strong, positive, curved (non-linear) relationship with no obvious outliers. Solution: Construct the scatter plot. - Label x-axis: Hours studied (from 10 to 80) - Label y-axis: Exam score (from 40 to 100) - Plot each point: (10,45), (15,50), (20,60), (25,65), (30,70), (35,75), (40,80), (45,85), (50,88), (55,90), (60,92), (65,93), (70,95), (75,96), (80,98) Describe the form.
   Full step-by-step solution

   Step 1: Construct the scatter plot. - Label x-axis: Hours studied (from 10 to 80) - Label y-axis: Exam score (from 40 to 100) - Plot each point: (10,45), (15,50), (20,60), (25,65), (30,70), (35,75), (40,80), (45,85), (50,88), (55,90), (60,92), (65,93), (70,95), (75,96), (80,98) Step 2: Describe the form. The points do not follow a straight line; they show a curve that rises quickly at first and then levels off. The form is curved (non-linear). Step 3: Describe the direction. As hours studied increase, exam scores increase. The direction is positive. Step 4: Describe the strength. The points are very tightly clustered around a smooth curve, with little scatter. The strength is strong. Step 5: Identify outliers. All points follow the same general curved pattern; none are far from the curve. There are no obvious outliers. The answer is: The scatter plot shows a strong, positive, curved (non-linear) relationship with no obvious outliers.

5. A scatter plot shows the relationship between study hours (x) and test scores (y) for 50 students. The data points form an approximately linear pattern with a correlation coefficient of r = 0.85. The regression line equation is y = 3.2x + 65. If a student studies for 8 hours, what test score does the regression model predict? Answer: 90.6 Solution: y = 3.2x + 65 - x = study hours - y = predicted test score We want the predicted test score for a student who studies 8 hours, so we substitute x = 8 into the equation.
   Full step-by-step solution

   We are given the regression line equation: y = 3.2x + 65 Here, - x = study hours - y = predicted test score We want the predicted test score for a student who studies 8 hours, so we substitute x = 8 into the equation. Step 1: Write the equation with x = 8 y = 3.2 * 8 + 65 Step 2: Multiply 3.2 by 8 3.2 * 8 = 25.6 Step 3: Add 65 to the result 25.6 + 65 = 90.6 So the regression model predicts a test score of 90.6 for 8 hours of study. The correlation coefficient r = 0.85 is not needed for this calculation because the regression equation is already given.