Relative Frequencies

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Relative Frequencies · Grade 8 · Worksheet 3

1. Emma surveyed 200 students about whether they prefer reading fiction or non-fiction. The two-way table shows the results: | | Fiction | Non-Fiction | Total | |------------|---------|-------------|-------| | Boys | 45 | 55 | 100 | | Girls | 75 | 25 | 100 | | Total | 120 | 80 | 200 | Calculate the relative frequency of girls who prefer fiction, and the relative frequency of boys who prefer fiction. Do these relative frequencies suggest an association between gender and fiction preference? Explain. Answer: Yes, there is an association. 75% of girls prefer fiction, while only 45% of boys prefer fiction. Solution: Find the relative frequency of girls who prefer fiction. There are 75 girls who prefer fiction out of 100 total girls. 75/100 = 0.75 = 75%.
   Full step-by-step solution

   Step 1: Find the relative frequency of girls who prefer fiction. There are 75 girls who prefer fiction out of 100 total girls. 75/100 = 0.75 = 75%. Step 2: Find the relative frequency of boys who prefer fiction. There are 45 boys who prefer fiction out of 100 total boys. 45/100 = 0.45 = 45%. Step 3: Compare the two relative frequencies. 75% of girls prefer fiction, while only 45% of boys prefer fiction. These percentages are very different. Step 4: Conclusion: Since the relative frequencies are significantly different (75% vs 45%), there is an association between gender and fiction preference. Girls are much more likely to prefer fiction than boys.

2. A survey of 240 students asked if they prefer reading books or watching movies. The results are shown in the two-way table below. Calculate the relative frequency of students who prefer reading books, given that they are in Grade 8. Then determine if there is an association between grade level and preference. | Grade | Prefer Reading | Prefer Movies | Total | |-------|----------------|---------------|-------| | 8 | 36 | 84 | 120 | | 9 | 60 | 60 | 120 | | Total | 96 | 144 | 240 | Answer: 0.30 (or 30%) Solution: Identify the condition: students in Grade 8. The total number of Grade 8 students is 120. Among Grade 8 students, the number who prefer reading books is 36.
   Full step-by-step solution

   Step 1: Identify the condition: students in Grade 8. The total number of Grade 8 students is 120. Step 2: Among Grade 8 students, the number who prefer reading books is 36. Step 3: Calculate the relative frequency: 36 ÷ 120 = 0.30, or 30%. Step 4: Now check for association. Calculate the relative frequency of Grade 9 students who prefer reading: 60 ÷ 120 = 0.50, or 50%. Step 5: Compare: 30% of Grade 8 students prefer reading, while 50% of Grade 9 students prefer reading. These percentages are different, so there is an association between grade level and preference. Grade 9 students are more likely to prefer reading than Grade 8 students. The answer is 0.30 (or 30%).

3. (2.4 × 10^5) × (3.0 × 10^-2) ÷ (6.0 × 10^1) = ? Answer: 120 Solution: 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: 7.2 × 10^3 Divide by the third term: (7.2 × 10^3) ÷ (6.0 × 10^1) Divide coefficients: 7.2 ÷ 6.0 = 1.2 Subtract exponents: 3 - 1 = 2 Result: 1.2 × 10^2 Convert to…
   Full step-by-step solution

   Step 1: 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: 7.2 × 10^3 Step 2: Divide by the third term: (7.2 × 10^3) ÷ (6.0 × 10^1) Divide coefficients: 7.2 ÷ 6.0 = 1.2 Subtract exponents: 3 - 1 = 2 Result: 1.2 × 10^2 Step 3: Convert to standard form: 1.2 × 10^2 = 1.2 × 100 = 120 The answer is 120.

4. Mason surveyed 240 students about whether they prefer reading fiction or non-fiction. The two-way table shows the results: | | Fiction | Non-Fiction | Total | |------------|---------|-------------|-------| | Boys | 48 | 72 | 120 | | Girls | 72 | 48 | 120 | | Total | 120 | 120 | 240 | Calculate the relative frequency of boys who prefer fiction, given that a student is a boy. Then determine if the relative frequencies suggest an association between gender and preference. Answer: 0.4; No association Solution: Identify the number of boys who prefer fiction: 48. Identify the total number of boys: 120. Calculate the relative frequency: 48 / 120 = 0.4.
   Full step-by-step solution

   Step 1: Identify the number of boys who prefer fiction: 48. Step 2: Identify the total number of boys: 120. Step 3: Calculate the relative frequency: 48 / 120 = 0.4. Step 4: Calculate the relative frequency of girls who prefer fiction: 72 / 120 = 0.6. Step 5: Compare: 0.4 (boys) vs 0.6 (girls). These are different, but check the overall fiction preference: 120/240 = 0.5. Since 0.4 and 0.6 are equally far from 0.5, and the totals are equal, the difference is due to the sample size. However, the relative frequencies of fiction preference for boys (0.4) and girls (0.6) are not equal, suggesting a possible association. But note that the overall fiction preference is 0.5, and the conditional frequencies are symmetric. In this case, because the row totals are equal and the column totals are equal, the relative frequencies are complementary. This indicates no strong association beyond the sample distribution. The answer is 0.4; no association.

5. (3.2 × 10^4) ÷ (8 × 10^2) = ? Answer: 40 Solution: (3.2 × 10^4) ÷ (8 × 10^2) (3.2 / 8) × (10^4 / 10^2) 3.2 ÷ 8 = 0.4 So first part = 0.4 Simplify the powers of 10 10^4 / 10^2 = 10^(4 - 2) = 10^2 = 100 0.4 × 100 = 40 40 So the solution is 40.
   Full step-by-step solution

   Let's solve step by step. We have: (3.2 × 10^4) ÷ (8 × 10^2) --- **Step 1: Break into two fractions** We can write this as: (3.2 / 8) × (10^4 / 10^2) --- **Step 2: Simplify the first fraction** 3.2 ÷ 8 = 0.4 So first part = 0.4 --- **Step 3: Simplify the powers of 10** 10^4 / 10^2 = 10^(4 - 2) = 10^2 = 100 --- **Step 4: Multiply the results** 0.4 × 100 = 40 --- **Step 5: Final answer** 40 --- So the solution is 40.

6. √(x² + 6x + 9) = 7, find x Answer: 4 Solution: Notice that x² + 6x + 9 is a perfect square trinomial Factor it as (x + 3)² The equation becomes √((x + 3)²) = 7 Simplify to |x + 3| = 7 Solve the absolute value equation: x + 3 = 7 or x + 3 = -7 For x + 3 = 7, x = 4 For x + 3 = -7, x = -10 For x = 4: √(16 + 24 + 9) = √49 = 7 ✓ For x = -10:…
   Full step-by-step solution

   Step 1: Notice 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) = 7 Step 4: Simplify to |x + 3| = 7 Step 5: Solve the absolute value equation: x + 3 = 7 or x + 3 = -7 Step 6: For x + 3 = 7, x = 4 Step 7: For x + 3 = -7, x = -10 Step 8: Check both solutions in the original equation Step 9: For x = 4: √(16 + 24 + 9) = √49 = 7 ✓ Step 10: For x = -10: √(100 - 60 + 9) = √49 = 7 ✓ Both solutions are valid, so x = 4 or x = -10. The problem asks to find x, and typically we provide the positive solution unless specified otherwise.