Bivariate Data

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Answer Key & Explanations

Bivariate Data · Grade 11 · Worksheet 1

1. A researcher is studying the relationship between study time (x, in hours) and test scores (y, out of 100). After collecting data from 50 students, the regression equation is calculated as ŷ = 65.2 + 2.8x. The correlation coefficient is r = 0.76. What percentage of the variation in test scores can be explained by the linear relationship with study time? Answer: 57.76 Solution: Understand what the question is asking. We are given a regression problem with correlation coefficient r = 0.76.
   Full step-by-step solution

   Step 1: Understand what the question is asking. We are given a regression problem with correlation coefficient r = 0.76. We want the percentage of variation in test scores (y) explained by the linear relationship with study time (x). Step 2: Recall the statistical measure for this. The proportion of variation in y explained by x in a linear regression is given by the **coefficient of determination**, which is r^2 (r squared). Step 3: Calculate r^2. r = 0.76 r^2 = (0.76)^2 = 0.5776 Step 4: Convert to a percentage. 0.5776 × 100 = 57.76 Step 5: Interpret the result. This means 57.76% of the variation in test scores can be explained by the linear relationship with study time. Final Answer: 57.76

2. Construct a two-way table for Sophia's survey of 200 students on preferred study method (visual, auditory, kinesthetic) and performance level (high, average, low). 48 students prefer visual and have high performance, 32 prefer visual and have average performance, 20 prefer visual and have low performance. 24 prefer auditory and have high performance, 36 prefer auditory and have average performance, 20 prefer auditory and have low performance. 8 prefer kinesthetic and have high performance, 12 prefer kinesthetic and have average performance, 0 prefer kinesthetic and have low performance. Calculate the marginal relative frequency of students who prefer visual study methods. Answer: 0.50 Solution: Find the total number of students who prefer visual methods: 48 + 32 + 20 = 100. Step 2: The overall total is 200. Step 3: Marginal relative frequency = 100 / 200 = 0.50.
   Full step-by-step solution

   Step 1: Find the total number of students who prefer visual methods: 48 + 32 + 20 = 100. Step 2: The overall total is 200. Step 3: Marginal relative frequency = 100 / 200 = 0.50. The answer is 0.50.

3. Create a scatter plot for Liam's data: (1,5), (5,13), (9,21), (13,29), (17,37), (21,45), (25,53), (29,61), (33,69), (37,77). Calculate the correlation coefficient r. Answer: 1.0 Solution: Calculate the mean of x-values: (1+5+9+13+17+21+25+29+33+37)/10 = 190/10 = 19 Calculate the mean of y-values: (5+13+21+29+37+45+53+61+69+77)/10 = 410/10 = 41 Calculate the sum of (x-mean_x)(y-mean_y): (1-19)(5-41) + (5-19)(13-41) + (9-19)(21-41) + (13-19)(29-41) + (17-19)(37-41) + (21-19)(45-41)…
   Full step-by-step solution

   Step 1: Calculate the mean of x-values: (1+5+9+13+17+21+25+29+33+37)/10 = 190/10 = 19 Step 2: Calculate the mean of y-values: (5+13+21+29+37+45+53+61+69+77)/10 = 410/10 = 41 Step 3: Calculate the sum of (x-mean_x)(y-mean_y): (1-19)(5-41) + (5-19)(13-41) + (9-19)(21-41) + (13-19)(29-41) + (17-19)(37-41) + (21-19)(45-41) + (25-19)(53-41) + (29-19)(61-41) + (33-19)(69-41) + (37-19)(77-41) = (-18)(-36) + (-14)(-28) + (-10)(-20) + (-6)(-12) + (-2)(-4) + (2)(4) + (6)(12) + (10)(20) + (14)(28) + (18)(36) = 648 + 392 + 200 + 72 + 8 + 8 + 72 + 200 + 392 + 648 = 2640 Step 4: Calculate the sum of (x-mean_x)^2: (-18)^2 + (-14)^2 + (-10)^2 + (-6)^2 + (-2)^2 + (2)^2 + (6)^2 + (10)^2 + (14)^2 + (18)^2 = 324 + 196 + 100 + 36 + 4 + 4 + 36 + 100 + 196 + 324 = 1320 Step 5: Calculate the sum of (y-mean_y)^2: (-36)^2 + (-28)^2 + (-20)^2 + (-12)^2 + (-4)^2 + (4)^2 + (12)^2 + (20)^2 + (28)^2 + (36)^2 = 1296 + 784 + 400 + 144 + 16 + 16 + 144 + 400 + 784 + 1296 = 5280 Step 6: Calculate r = sum[(x-mean_x)(y-mean_y)] / sqrt[sum(x-mean_x)^2 * sum(y-mean_y)^2] = 2640 / sqrt(1320 * 5280) = 2640 / sqrt(6969600) = 2640 / 2640 = 1.0 The correlation coefficient is 1.0, indicating a perfect positive linear relationship.

4. An environmental scientist is studying the relationship between air pollution levels (measured in PM2.5 micrograms per cubic meter) and respiratory illness rates (per 1000 people) across 15 different cities. After collecting the data, she calculates a linear regression equation of ŷ = 25 + 0.8x, where x represents PM2.5 levels and ŷ represents predicted illness rates. The correlation coefficient is r = 0.75. If a city has a PM2.5 level of 30 micrograms per cubic meter, what would be the predicted respiratory illness rate according to this model? Answer: 49 Solution: Identify the regression equation: ŷ = 25 + 0.8x Identify the given PM2.5 level: x = 30 Substitute x = 30 into the equation: ŷ = 25 + 0.8 * 30 Perform the multiplication: 0.8 * 30 = 24 Perform the addition: 25 + 24 = 49 The predicted respiratory illness rate is 49 per 1000 people.
   Full step-by-step solution

   Step 1: Identify the regression equation: ŷ = 25 + 0.8x Step 2: Identify the given PM2.5 level: x = 30 Step 3: Substitute x = 30 into the equation: ŷ = 25 + 0.8 * 30 Step 4: Perform the multiplication: 0.8 * 30 = 24 Step 5: Perform the addition: 25 + 24 = 49 Step 6: The predicted respiratory illness rate is 49 per 1000 people. The answer is 49.

5. A scatter plot displays the relationship between the number of hours students spend on extracurricular activities per week (x) and their GPA (y) for 60 high school students. The data points form an approximately linear pattern with a correlation coefficient of r = -0.72. The least squares regression line equation is ŷ = -0.15x + 3.85. According to this model, what GPA would be predicted for a student who spends 6 hours per week on extracurricular activities? Answer: 2.95 Solution: The regression equation is given as ŷ = -0.15x + 3.85, where x represents hours spent on extracurricular activities and ŷ represents the predicted GPA.
   Full step-by-step solution

   Step 1: The regression equation is given as ŷ = -0.15x + 3.85, where x represents hours spent on extracurricular activities and ŷ represents the predicted GPA. Step 2: We are asked to predict the GPA for a student who spends 6 hours per week on extracurricular activities, so we substitute x = 6 into the equation. Step 3: Calculate ŷ = -0.15(6) + 3.85 Step 4: Compute the multiplication: -0.15 × 6 = -0.90 Step 5: Add the result to the y-intercept: -0.90 + 3.85 = 2.95 Step 6: The predicted GPA is 2.95. The answer is 2.95.

6. Create a scatter plot for Aroha's data: (9,14), (11,18), (13,22), (15,26), (17,30), (19,34), (21,38), (23,42), (25,46), (27,50). Describe the correlation and estimate the line of best fit. Answer: Strong positive correlation, y = 2x - 4 Solution: Plot the points (9,14), (11,18), (13,22), (15,26), (17,30), (19,34), (21,38), (23,42), (25,46), (27,50) Observe that as x increases by 2, y increases by 4 consistently Calculate slope: (18-14)/(11-9) = 4/2 = 2 Use point-slope form with (9,14): y - 14 = 2(x - 9) Simplify: y - 14 = 2x - 18 y = 2x…
   Full step-by-step solution

   Step 1: Plot the points (9,14), (11,18), (13,22), (15,26), (17,30), (19,34), (21,38), (23,42), (25,46), (27,50) Step 2: Observe that as x increases by 2, y increases by 4 consistently Step 3: Calculate slope: (18-14)/(11-9) = 4/2 = 2 Step 4: Use point-slope form with (9,14): y - 14 = 2(x - 9) Step 5: Simplify: y - 14 = 2x - 18 Step 6: y = 2x - 4 Step 7: The correlation is strong positive because all points lie perfectly on a straight line with positive slope The line of best fit is y = 2x - 4