Relative Frequencies

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Relative Frequencies · Grade 8 · Worksheet 1

1. A survey of 240 students asked about their preferred sport (Soccer or Basketball) and whether they exercise daily. The results are shown in the two-way table below: | | Soccer | Basketball | Total | |--------------|--------|------------|-------| | Daily | 72 | 48 | 120 | | Not Daily | 48 | 72 | 120 | | Total | 120 | 120 | 240 | Calculate the relative frequency of students who prefer Soccer given that they exercise daily. Then, determine if the relative frequencies suggest an association between preferring Soccer and exercising daily. Answer: 0.6; No association Solution: Identify the condition: students who exercise daily. There are 120 daily exercisers total. Among daily exercisers, 72 prefer Soccer.
   Full step-by-step solution

   Step 1: Identify the condition: students who exercise daily. There are 120 daily exercisers total. Step 2: Among daily exercisers, 72 prefer Soccer. Step 3: Conditional relative frequency = 72 / 120 = 0.6. Step 4: Overall relative frequency of Soccer preference = 120 / 240 = 0.5. Step 5: Since 0.6 is close to 0.5 (difference of 0.1), there is no strong association between preferring Soccer and exercising daily. The answer is 0.6; No association.

2. Liam is analyzing data from a survey about students' favorite after-school activities and whether they play team sports. The two-way relative frequency table shows the following: Among students who play team sports, 60% prefer outdoor activities, 30% prefer creative activities, and 10% prefer academic clubs. Among students who do not play team sports, 20% prefer outdoor activities, 50% prefer creative activities, and 30% prefer academic clubs. Based on the relative frequencies, is there an association between playing team sports and activity preference? Explain what the data suggests. Answer: Yes, there is an association. Students who play team sports are much more likely to prefer outdoor activities (60%) compared to those who do not play sports (20%). Students who do not play sports show stronger preference for creative and academic activities. Solution: For each group, the table gives the percentage distribution of their favorite after-school activities: Outdoor, Creative, Academic. - Outdoor: 60% - Creative: 30% - Academic: 10% Total = 60 + 30 + 10 = 100% - Outdoor: 20% - Creative: 50% - Academic: 30% Total = 20 + 50 + 30 = 100% - Play sports:…
   Full step-by-step solution

   Step 1: Understand the data structure We have two groups: - Students who play team sports - Students who do not play team sports For each group, the table gives the percentage distribution of their favorite after-school activities: Outdoor, Creative, Academic. These are conditional relative frequencies: given a student is in one group (plays sports or not), what percentage prefers each activity. --- Step 2: Write down the given numbers For students who play team sports: - Outdoor: 60% - Creative: 30% - Academic: 10% Total = 60 + 30 + 10 = 100% For students who do not play team sports: - Outdoor: 20% - Creative: 50% - Academic: 30% Total = 20 + 50 + 30 = 100% --- Step 3: Compare percentages between the two groups Outdoor activities: - Play sports: 60% - Do not play sports: 20% Difference = 60 - 20 = 40 percentage points Creative activities: - Play sports: 30% - Do not play sports: 50% Difference = 50 - 30 = 20 percentage points Academic clubs: - Play sports: 10% - Do not play sports: 30% Difference = 30 - 10 = 20 percentage points --- Step 4: Interpret the differences If there were no association between playing sports and activity preference, the percentages for each activity would be roughly the same for both groups. Here, the percentages are very different: - Students who play sports are much more likely to prefer outdoor activities (60% vs 20%). - Students who do not play sports are more likely to prefer creative activities (50% vs 30%) and academic clubs (30% vs 10%). --- Step 5: Conclusion Yes, there is an association. The data suggests that playing team sports is associated with a higher preference for outdoor activities, while not playing sports is associated with higher preference for creative and academic activities.

3. The two-way table below shows the results of a survey of 300 students at a middle school. Students were asked if they prefer pizza or tacos for lunch, and if they play a sport or do not play a sport. | Prefer Pizza | Prefer Tacos | Total ------------------------------------------------- Plays a Sport | 84 | 36 | 120 ------------------------------------------------- Does Not Play | 96 | 84 | 180 ------------------------------------------------- Total | 180 | 120 | 300 Based on the relative frequencies in the table, is there an association between playing a sport and preferring pizza? Justify your answer by calculating the relative frequencies of preferring pizza for students who play a sport and for students who do not play a sport. Answer: Yes, there is an association. 70% of students who play a sport prefer pizza, while only about 53.3% of students who do not play a sport prefer pizza. Since these percentages are different, there is an association. Solution: Find the number of students who play a sport and prefer pizza. From the table, this is 84. Find the total number of students who play a sport.
   Full step-by-step solution

   Step 1: Find the number of students who play a sport and prefer pizza. From the table, this is 84. Step 2: Find the total number of students who play a sport. From the table, this is 120. Step 3: Calculate the relative frequency (percentage) of students who play a sport and prefer pizza: (84 / 120) x 100% = 0.7 x 100% = 70%. Step 4: Find the number of students who do not play a sport and prefer pizza. From the table, this is 96. Step 5: Find the total number of students who do not play a sport. From the table, this is 180. Step 6: Calculate the relative frequency of students who do not play a sport and prefer pizza: (96 / 180) x 100% = 0.5333... x 100% = about 53.3%. Step 7: Compare the two percentages. 70% is significantly different from 53.3%. This difference suggests that students who play a sport are more likely to prefer pizza than students who do not play a sport. Therefore, there is an association between playing a sport and preferring pizza. The answer is: Yes, there is an association. 70% of students who play a sport prefer pizza, while only about 53.3% of students who do not play a sport prefer pizza.

4. (2.4 × 10^5) × (3.0 × 10^-2) = ? Answer: 7200 Solution: Multiply the coefficients: 2.4 × 3.0 = 7.2 Add the exponents: 5 + (-2) = 3 Combine the results: 7.2 × 10^3 Convert to standard form: 7.2 × 1000 = 7200 The answer is 7200.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 2.4 × 3.0 = 7.2 Step 2: Add the exponents: 5 + (-2) = 3 Step 3: Combine the results: 7.2 × 10^3 Step 4: Convert to standard form: 7.2 × 1000 = 7200 The answer is 7200.

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.