Relative Frequencies

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Answer Key & Explanations

Relative Frequencies · Grade 8 · Worksheet 3

1. Kaia surveyed 150 students at her school about whether they walk to school or are driven. The results are shown in the two-way frequency table below. | | Walk to School | Driven to School | Total | |----------------|----------------|------------------|-------| | 6th Grade | 18 | 42 | 60 | | 7th Grade | 28 | 12 | 40 | | 8th Grade | 30 | 20 | 50 | | Total | 76 | 74 | 150 | Based on the relative frequencies, is there an association between grade level and how students get to school? Explain your reasoning by comparing the relative frequencies of walking to school for each grade level. Answer: Yes, there is an association. The relative frequency of walking to school decreases as grade level increases: 6th grade 30%, 7th grade 70%, 8th grade 60%. Wait, that's not a clear pattern. Let's recalculate: 6th grade 18/60 = 0.30 (30%), 7th grade 28/40 = 0.70 (70%), 8th grade 30/50 = 0.60 (60%). Since the percentages are not the same across grades, there is an association. Specifically, 7th graders have the highest percentage of walkers (70%), followed by 8th graders (60%), and 6th graders have the lowest (30%). Solution: Calculate the relative frequency of walking to school for each grade level by dividing the number who walk by the total in that grade. 6th Grade: 18 walk out of 60 total.
   Full step-by-step solution

   Step 1: Calculate the relative frequency of walking to school for each grade level by dividing the number who walk by the total in that grade. 6th Grade: 18 walk out of 60 total. 18/60 = 0.30 = 30% 7th Grade: 28 walk out of 40 total. 28/40 = 0.70 = 70% 8th Grade: 30 walk out of 50 total. 30/50 = 0.60 = 60% Step 2: Compare the relative frequencies. If there were no association between grade level and transportation method, we would expect the percentages to be roughly the same for all grades. However, we see: 6th grade = 30%, 7th grade = 70%, 8th grade = 60%. Step 3: Interpret the results. The percentages are clearly different. 7th graders have the highest rate of walking (70%), while 6th graders have the lowest (30%). This shows that the likelihood of walking to school depends on grade level. Conclusion: Yes, there is an association between grade level and how students get to school. The relative frequencies differ significantly across the grade levels.

2. A research team is studying the relationship between daily screen time and hours of sleep among middle school students. They surveyed 120 students and created this two-way relative frequency table: | Screen Time / Sleep | Less than 8 hours | 8 or more hours | Total | |---------------------|-------------------|-----------------|-------| | Less than 2 hours | 0.15 | 0.25 | 0.40 | | 2-4 hours | 0.20 | 0.20 | 0.40 | | More than 4 hours | 0.15 | 0.05 | 0.20 | | Total | 0.50 | 0.50 | 1.00 | Based on the relative frequencies, what association exists between daily screen time and hours of sleep? Explain your reasoning using the data. Answer: There is a negative association between screen time and sleep hours. Students with less screen time (under 2 hours) are more likely to get adequate sleep (25% get 8+ hours vs. 15% get less than 8), while students with more screen time (over 4 hours) are less likely to get adequate sleep (only 5% get 8+ hours vs. 15% get less than 8). Solution: In statistics, we use two-way relative frequency tables to identify associations between two categorical variables.
   Full step-by-step solution

   In statistics, we use two-way relative frequency tables to identify associations between two categorical variables. An association exists when the distribution of one variable changes across the categories of another variable. For example, if we were studying the relationship between exercise frequency and energy levels, we might notice that people who exercise more often tend to report higher energy levels. This would indicate a positive association. The strength of an association can be determined by how much the conditional distributions differ from the marginal distributions.

3. Mere, a school environmental officer, surveyed 240 students to investigate whether there is an association between grade level (Grade 7 or Grade 8) and whether they regularly recycle. The results are shown in the two-way frequency table below: | | Recycles | Does Not Recycle | Total | |---------------|----------|------------------|-------| | Grade 7 | 72 | 48 | 120 | | Grade 8 | 60 | 60 | 120 | | Total | 132 | 108 | 240 | Calculate the relative frequencies (as percentages) of students who recycle for each grade level. Based on these relative frequencies, is there an association between grade level and recycling behavior? Explain your reasoning. Answer: Grade 7: 60% recycle; Grade 8: 50% recycle. Yes, there is an association because the percentage of recyclers is higher in Grade 7 (60%) than in Grade 8 (50%). Solution: Calculate the relative frequency of recycling for Grade 7.
   Full step-by-step solution

   Step 1: Calculate the relative frequency of recycling for Grade 7. Number of Grade 7 students who recycle: 72 Total number of Grade 7 students: 120 Relative frequency = 72/120 = 0.60 = 60% Step 2: Calculate the relative frequency of recycling for Grade 8. Number of Grade 8 students who recycle: 60 Total number of Grade 8 students: 120 Relative frequency = 60/120 = 0.50 = 50% Step 3: Compare the relative frequencies. Grade 7: 60% recycle Grade 8: 50% recycle There is a 10 percentage point difference. Step 4: Draw a conclusion. Since the percentage of students who recycle differs between the two grade levels (60% vs 50%), there is an association between grade level and recycling behavior. Grade 7 students are more likely to recycle than Grade 8 students. Final answer: Grade 7: 60% recycle; Grade 8: 50% recycle. Yes, there is an association.

4. √(x² + 6x + 9) = 5, solve for x Answer: 2 Solution: Recognize that x² + 6x + 9 is a perfect square trinomial Factor it as (x + 3)² The equation becomes √((x + 3)²) = 5 Simplify to |x + 3| = 5 Solve the absolute value equation: x + 3 = 5 or x + 3 = -5 For x + 3 = 5: x = 5 - 3 = 2 For x + 3 = -5: x = -5 - 3 = -8 For x = 2: √(4 + 12 + 9) = √25 = 5 ✓…
   Full step-by-step solution

   Step 1: Recognize that x² + 6x + 9 is a perfect square trinomial Step 2: Factor it as (x + 3)² Step 3: The equation becomes √((x + 3)²) = 5 Step 4: Simplify to |x + 3| = 5 Step 5: Solve the absolute value equation: x + 3 = 5 or x + 3 = -5 Step 6: For x + 3 = 5: x = 5 - 3 = 2 Step 7: For x + 3 = -5: x = -5 - 3 = -8 Step 8: Check both solutions in the original equation Step 9: For x = 2: √(4 + 12 + 9) = √25 = 5 ✓ Step 10: For x = -8: √(64 - 48 + 9) = √25 = 5 ✓ Both solutions are valid. The answer is 2.

5. (2.4 × 10^5) × (3.0 × 10^-2) ÷ (6.0 × 10^2) = ? Answer: 12 Solution: First multiply the first two terms: (2.4 × 10^5) × (3.0 × 10^-2) Multiply coefficients: 2.4 × 3.0 = 7.2 Add exponents: 5 + (-2) = 3 Result is 7.2 × 10^3 Now divide by (6.0 × 10^2): (7.2 × 10^3) ÷ (6.0 × 10^2) Divide coefficients: 7.2 ÷ 6.0 = 1.2 Subtract exponents: 3 - 2 = 1 Result is 1.2 × 10^1…
   Full step-by-step solution

   Step 1: First multiply the first two terms: (2.4 × 10^5) × (3.0 × 10^-2) Step 2: Multiply coefficients: 2.4 × 3.0 = 7.2 Step 3: Add exponents: 5 + (-2) = 3 Step 4: Result is 7.2 × 10^3 Step 5: Now divide by (6.0 × 10^2): (7.2 × 10^3) ÷ (6.0 × 10^2) Step 6: Divide coefficients: 7.2 ÷ 6.0 = 1.2 Step 7: Subtract exponents: 3 - 2 = 1 Step 8: Result is 1.2 × 10^1 Step 9: Convert to standard form: 1.2 × 10 = 12 The answer is 12.