Relative Frequencies

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Relative Frequencies · Grade 8 · Worksheet 2

1. Mere is a student council member analyzing whether there's an association between students' grade level (Grade 7 or Grade 8) and their preference for the school's new lunch menu options (Option A: Pizza or Option B: Wraps). She surveys 300 students and organizes the data into a two-way frequency table. Among the 120 Grade 7 students surveyed, 72 prefer Pizza and 48 prefer Wraps. Among the 180 Grade 8 students surveyed, 90 prefer Pizza and 90 prefer Wraps. Based on the relative frequencies, is there an association between grade level and lunch menu preference? Explain your reasoning. Answer: Yes, there is an association. Solution: Calculate the relative frequencies for Grade 7 students. Total Grade 7 students = 120. Pizza preference: 72/120 = 0.60 or 60%.
   Full step-by-step solution

   Step 1: Calculate the relative frequencies for Grade 7 students. Total Grade 7 students = 120. Pizza preference: 72/120 = 0.60 or 60%. Wrap preference: 48/120 = 0.40 or 40%. Step 2: Calculate the relative frequencies for Grade 8 students. Total Grade 8 students = 180. Pizza preference: 90/180 = 0.50 or 50%. Wrap preference: 90/180 = 0.50 or 50%. Step 3: Compare the relative frequencies. For Grade 7, 60% prefer Pizza and 40% prefer Wraps. For Grade 8, 50% prefer Pizza and 50% prefer Wraps. Step 4: Interpret the comparison. The preference distribution is different between the two grade levels. In Grade 7, there is a clear majority preferring Pizza (60%), while in Grade 8, the preferences are evenly split (50% each). This difference suggests that grade level is related to lunch menu preference. Conclusion: Yes, there is an association between grade level and lunch menu preference, because the relative frequencies of preferences differ between Grade 7 and Grade 8 students.

2. The two-way table below shows data from a survey of 120 Grade 8 students about whether they prefer reading fiction or non-fiction, and whether they are in the school book club. In Book Club Not in Book Club Total Prefers Fiction 24 36 60 Prefers Non-Fiction 16 44 60 Total 40 80 120 Calculate the relative frequencies (as percentages) of students who prefer fiction among those in the book club, and among those not in the book club. Do these relative frequencies suggest an association between being in the book club and preferring fiction? Explain. Answer: 60% of book club members prefer fiction; 45% of non-book club members prefer fiction. Yes, there is an association. Solution: Find the relative frequency of preferring fiction among those in the book club. Number in book club who prefer fiction = 24 Total in book club = 40 Relative frequency = (24 / 40) × 100 = 0.6 × 100 = 60% Find the relative frequency of preferring fiction among those not in the book club.
   Full step-by-step solution

   Step 1: Find the relative frequency of preferring fiction among those in the book club. Number in book club who prefer fiction = 24 Total in book club = 40 Relative frequency = (24 / 40) × 100 = 0.6 × 100 = 60% Step 2: Find the relative frequency of preferring fiction among those not in the book club. Number not in book club who prefer fiction = 36 Total not in book club = 80 Relative frequency = (36 / 80) × 100 = 0.45 × 100 = 45% Step 3: Compare the two percentages. 60% of book club members prefer fiction, while only 45% of non-book club members prefer fiction. The difference is 15 percentage points. Step 4: Describe the association. Since the percentages are different, there is an association between being in the book club and preferring fiction. Students in the book club are more likely to prefer fiction than those not in the book club. The answer is: 60% of book club members prefer fiction; 45% of non-book club members prefer fiction. Yes, there is an association.

3. Emma surveyed 200 Grade 8 students to investigate whether there is an association between participating in a sports team and completing daily homework on time. She recorded the results in the two-way frequency table below: | | Homework on Time | Homework Late | Total | |------------------------|------------------|---------------|-------| | On a Sports Team | 77 | 33 | 110 | | Not on a Sports Team | 45 | 45 | 90 | | Total | 122 | 78 | 200 | Calculate the relative frequencies (as percentages) of completing homework on time for students on a sports team and for students not on a sports team. Based on these relative frequencies, does there appear to be an association between being on a sports team and completing homework on time? Explain your reasoning. Answer: Yes, there is an association. Among students on a sports team, 70% complete homework on time (77/110 = 0.70 = 70%). Among students not on a sports team, 50% complete homework on time (45/90 = 0.50 = 50%). Since these percentages are different, there is an association: students on a sports team are more likely to complete homework on time. Solution: Find the relative frequency of completing homework on time for students on a sports team. Number of students on a sports team who complete homework on time: 77 Total number of students on a sports team: 110 Relative frequency = 77/110 = 0.70 = 70% Find the relative frequency of completing…
   Full step-by-step solution

   Step 1: Find the relative frequency of completing homework on time for students on a sports team. Number of students on a sports team who complete homework on time: 77 Total number of students on a sports team: 110 Relative frequency = 77/110 = 0.70 = 70% Step 2: Find the relative frequency of completing homework on time for students not on a sports team. Number of students not on a sports team who complete homework on time: 45 Total number of students not on a sports team: 90 Relative frequency = 45/90 = 0.50 = 50% Step 3: Compare the two relative frequencies. 70% of students on a sports team complete homework on time, while only 50% of students not on a sports team complete homework on time. Step 4: Draw a conclusion. Since the percentages are different (70% vs. 50%), there is an association between being on a sports team and completing homework on time. Students on a sports team are more likely to complete their homework on time than students not on a sports team.

4. Emma surveyed 150 students at her school to investigate whether there is an association between participation in a recycling program and awareness of environmental issues. She recorded the results in a two-way table. Among the 90 students who participate in the recycling program, 75 are environmentally aware and 15 are not. Among the 60 students who do not participate in the recycling program, 30 are environmentally aware and 30 are not. Based on the relative frequencies, is there an association between participation in the recycling program and environmental awareness? Explain your reasoning. Answer: Yes, there is an association because 83.3% of recycling participants are environmentally aware compared to only 50% of non-participants. Solution: Find the relative frequency of environmentally aware students among recycling program participants. There are 75 aware out of 90 participants. 75/90 = 0.8333...
   Full step-by-step solution

   Step 1: Find the relative frequency of environmentally aware students among recycling program participants. There are 75 aware out of 90 participants. 75/90 = 0.8333... = 83.3% (rounded to one decimal). Step 2: Find the relative frequency of environmentally aware students among non-participants. There are 30 aware out of 60 non-participants. 30/60 = 0.5 = 50%. Step 3: Compare the two percentages: 83.3% of participants are aware, but only 50% of non-participants are aware. This is a large difference (33.3 percentage points). Step 4: Since the relative frequencies are significantly different, there is an association between participating in the recycling program and being environmentally aware. The answer is yes, there is an association.