Bivariate Data

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Bivariate Data · Grade 11 · Worksheet 3

1. Sophia, a high school science student, is investigating whether there is a relationship between the number of hours students spend on social media per week and their average exam score (out of 100). She surveys 30 students and records the following bivariate data. The data is summarized in the two-way table below, where hours on social media per week is categorized as Low (0–7 hours), Medium (8–14 hours), or High (15+ hours), and exam score is categorized as Low (below 70), Medium (70–84), or High (85–100). | Exam Score → | Low | Medium | High | Total | |--------------|-----|--------|------|-------| | Low Social Media | 2 | 7 | 5 | 14 | | Medium Social Media | 4 | 6 | 2 | 12 | | High Social Media | 3 | 1 | 0 | 4 | | Total | 9 | 14 | 7 | 30 | Based on this table, what percentage of students who spend High hours on social media scored in the High exam score category? Express your answer as a percentage (rounded to one decimal place, if necessary). Answer: 0.0% Solution: Identify the row for High Social Media. The row shows: Low exam = 3, Medium exam = 1, High exam = 0, Total = 4. The number of High Social Media students who scored High is 0.
   Full step-by-step solution

   Step 1: Identify the row for High Social Media. The row shows: Low exam = 3, Medium exam = 1, High exam = 0, Total = 4. Step 2: The number of High Social Media students who scored High is 0. Step 3: The total number of High Social Media students is 4. Step 4: Calculate the percentage: (0 / 4) * 100% = 0%. Step 5: Rounded to one decimal place, the answer is 0.0%. The answer is 0.0%.

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. Construct a two-way table for Kaia's survey of 75 students: 33 prefer online learning, 17 prefer in-person learning, and 25 are undecided. Among those who prefer online, 21 are female; among those who prefer in-person, 9 are male; among the undecided, 13 are female. Determine the number of male students who are undecided. Answer: 12 Solution: Set up a two-way table with rows Male and Female, columns Online, In-person, Undecided, and a Total row/column. Total students = 75. Online total = 33, In-person total = 17, Undecided total = 25.
   Full step-by-step solution

   Step 1: Set up a two-way table with rows Male and Female, columns Online, In-person, Undecided, and a Total row/column. Step 2: Total students = 75. Online total = 33, In-person total = 17, Undecided total = 25. Step 3: Online: 21 female, so male online = 33 - 21 = 12. Step 4: In-person: 9 male, so female in-person = 17 - 9 = 8. Step 5: Undecided: 13 female, so male undecided = 25 - 13 = 12. Step 6: Check totals: Male total = 12 (online) + 9 (in-person) + 12 (undecided) = 33. Female total = 21 + 8 + 13 = 42. Total = 33 + 42 = 75. Correct. The number of male students who are undecided is 12.

4. 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 = 12.4 - 0.35x, where x represents water temperature and ŷ represents predicted coral growth rate. The standard error of the slope is 0.08. She wants to test if there is a statistically significant negative relationship at α = 0.01. Calculate the test statistic for this hypothesis test about the population slope β₁. Answer: B. t = -4.38 Solution: Identify the estimated slope from the regression equation From ŷ = 12.4 - 0.35x, the slope b₁ = -0.35 Standard error of slope = 0.08 Recall the formula for the t-test statistic for regression slope t = (b₁ - β₁) / SE(b₁) Under the null hypothesis, β₁ = 0 (no relationship) So t = (-0.35 - 0) /…
   Full step-by-step solution

   Step 1: Identify the estimated slope from the regression equation From ŷ = 12.4 - 0.35x, the slope b₁ = -0.35 Step 2: Identify the standard error of the slope Standard error of slope = 0.08 Step 3: Recall the formula for the t-test statistic for regression slope t = (b₁ - β₁) / SE(b₁) Step 4: Under the null hypothesis, β₁ = 0 (no relationship) So t = (-0.35 - 0) / 0.08 Step 5: Calculate the test statistic t = -0.35 / 0.08 = -4.375 Step 6: Round to two decimal places t = -4.38 The correct answer is t = -4.38.