Bivariate Data

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Answer Key & Explanations

Bivariate Data · Grade 11 · Worksheet 1

1. A researcher is studying the relationship between study hours and test scores. The regression equation is ŷ = 2.5x + 65, where x represents study hours and ŷ represents predicted test score. The correlation coefficient is r = 0.85. If a student studies for 6 hours, what is their predicted test score according to this model? Answer: 80 Solution: Identify the regression equation. ŷ = 2.5x + 65 Here, ŷ is the predicted test score, and x is the number of study hours. Identify the given value of x.
   Full step-by-step solution

   Step 1: Identify the regression equation. The problem gives the regression equation as: ŷ = 2.5x + 65 Here, ŷ is the predicted test score, and x is the number of study hours. Step 2: Identify the given value of x. The student studies for 6 hours, so x = 6. Step 3: Substitute x = 6 into the regression equation. ŷ = 2.5 * 6 + 65 Step 4: Perform the multiplication first. 2.5 * 6 = 15 Step 5: Add the result to 65. 15 + 65 = 80 Step 6: Interpret the result. The predicted test score for a student who studies 6 hours is 80. Note: The correlation coefficient r = 0.85 is not needed for this calculation; it only tells us the strength and direction of the relationship, but the regression equation already gives the prediction formula. Final Answer: 80

2. A marine biologist is studying the relationship between water temperature (°C) and coral growth rate (mm/year) across 25 reef sites. She calculates a linear regression equation of ŷ = 12.5 - 0.4x, where x represents water temperature and ŷ represents predicted coral growth rate. The standard error of the slope is 0.08. Test the hypothesis that water temperature affects coral growth at the α = 0.01 significance level by calculating the t-statistic for the slope coefficient. Answer: -5.0 Solution: Identify the slope coefficient from the regression equation ŷ = 12.5 - 0.4x The slope coefficient (b) is -0.4 The standard error (SE) is given as 0.08 Calculate the t-statistic using the formula: t = (b - 0) / SE Since we're testing if the slope is different from zero, we use: t = (-0.4 - 0) /…
   Full step-by-step solution

   Step 1: Identify the slope coefficient from the regression equation ŷ = 12.5 - 0.4x The slope coefficient (b) is -0.4 Step 2: Identify the standard error of the slope The standard error (SE) is given as 0.08 Step 3: Calculate the t-statistic using the formula: t = (b - 0) / SE Since we're testing if the slope is different from zero, we use: t = (-0.4 - 0) / 0.08 Step 4: Perform the calculation: t = -0.4 / 0.08 = -5.0 The t-statistic is -5.0, which we would compare to critical values from the t-distribution to determine statistical significance.

3. Isabella, a high school science teacher, surveyed 112 Grade 11 students to investigate the relationship between whether a student plays a musical instrument and whether they participate in a school sports team. She recorded the data in a two-way table. Of the 48 students who play a musical instrument, 28 also play a sport. Of the 64 students who do not play a musical instrument, 48 play a sport. Construct a two-way table to represent this bivariate categorical data distribution, then calculate the percentage of students who play a musical instrument but do not play a sport, rounded to the nearest whole percent. Answer: 18% Solution: Define the categories. Rows: Plays instrument (Yes, No). Total students = 112.
   Full step-by-step solution

   Step 1: Define the categories. Rows: Plays instrument (Yes, No). Columns: Plays sport (Yes, No). Step 2: Total students = 112. Instrument Yes total = 48. Instrument No total = 64. Step 3: Instrument Yes and Sport Yes = 28. Instrument Yes and Sport No = 48 - 28 = 20. Step 4: Instrument No and Sport Yes = 48. Instrument No and Sport No = 64 - 48 = 16. Step 5: Two-way table: Sport Yes Sport No Total Instrument Yes 28 20 48 Instrument No 48 16 64 Total 76 36 112 Step 6: Number who play instrument but no sport = 20. Percentage = (20 / 112) * 100 = 17.857... Rounded to nearest whole percent = 18%.

4. 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

5. Construct a two-way table for Hana's survey of 200 students on preferred study method (Visual or Auditory) and performance level (High, Medium, Low). 48 students prefer Visual and have High performance, 32 prefer Visual and have Medium performance, 20 prefer Visual and have Low performance, 24 prefer Auditory and have High performance, 40 prefer Auditory and have Medium performance, 36 prefer Auditory and have Low performance. Calculate the marginal relative frequency of students who prefer Visual study method. Answer: 0.5 Solution: Create the two-way table with row totals and column totals. Visual 48 32 20 100 Auditory 24 40 36 100 Total 72 72 56 200 The marginal relative frequency for Visual is the total number of Visual students divided by the grand total.
   Full step-by-step solution

   Step 1: Create the two-way table with row totals and column totals. High Medium Low Total Visual 48 32 20 100 Auditory 24 40 36 100 Total 72 72 56 200 Step 2: The marginal relative frequency for Visual is the total number of Visual students divided by the grand total. Marginal relative frequency = 100 / 200 = 0.5 The answer is 0.5.

6. Create a scatter plot for the data: (7, 18), (9, 22), (11, 26), (13, 30), (15, 34), (17, 38). Describe the correlation and estimate the line of best fit. Answer: Strong positive correlation; y = 2x + 4 Solution: Plot the points (7,18), (9,22), (11,26), (13,30), (15,34), (17,38) on a coordinate plane. Observe the pattern: As x increases by 2, y increases by 4. This indicates a strong positive linear correlation.
   Full step-by-step solution

   Step 1: Plot the points (7,18), (9,22), (11,26), (13,30), (15,34), (17,38) on a coordinate plane. Step 2: Observe the pattern: As x increases by 2, y increases by 4. This indicates a strong positive linear correlation. Step 3: Calculate the slope: change in y / change in x = 4/2 = 2. Step 4: Use point-slope form with (7,18): y - 18 = 2(x - 7) Step 5: Simplify: y - 18 = 2x - 14 → y = 2x + 4 Step 6: Verify with another point: For x=9, y=2(9)+4=22, which matches the data. The line of best fit is y = 2x + 4 with strong positive correlation.