- A teacher recorded whether students turned in homework on time and whether they attended a review session. Use the two-way table to answer the question.
What is the conditional relative frequency of turning in homework on time given that the student attended the review session? Give your answer as a percent.
| On Time | Not On Time | Total |
|---|
| Attended Review | 54 | 18 | 72 |
| Did Not Attend Review | 40 | 48 | 88 |
| Total | 94 | 66 | 160 |
Answer: 75%
The concept here is conditional relative frequency, which focuses on the proportion within a given category in two-way tables to explore potential associations. For conditional relative frequency, the denominator is the total of the specified condition, such as the review attendance subgroup. Locate the row for 'Attended Review' and the cell for 'On Time' which is 54, with the subgroup total of 72. Compute 54 divided by 72, equaling 0.75 or 75%. This shows that 75% of students who attended the review turned in homework on time. A common misconception is using counts directly, like assuming 54 is higher without proportion to the total. To transfer this, interpret 'given that the student attended' as restricting to the attended total of 72, then ratio the on-time cell to it.
- A teacher surveyed 160 students about whether they completed the optional practice set and whether they scored 80% or higher on the quiz. Based on the two-way table, what is the conditional relative frequency (as a percent) of Scored 80% or higher = Yes given that the student completed the practice set?
Answer: 75%
This conditional relative frequency asks for the proportion scoring 80% or higher among only those who completed the practice set. The phrase "given that the student completed the practice set" identifies our subgroup denominator. Find the row for students who completed the practice set and note the total in that row. Then find how many of these practice-set students scored 80% or higher. Calculate: (practice students scoring 80%+) ÷ (total practice students) × 100%. Don't use the overall 160 students or all high scorers—focus only on the practice-set subgroup. This conditional approach reveals whether completing practice correlates with quiz success.
- A school surveyed 130 students about whether they attended a tutoring session and whether they turned in the homework on time. Based on the two-way table, which statement best describes the association between attending tutoring and turning in homework on time?
Answer: There is an association because the percent who turned in homework on time is higher among students who attended tutoring than among those who did not.
Determining association between tutoring attendance and homework submission requires comparing conditional frequencies. Calculate the on-time homework rate for each group: (tutoring attendees submitting on time) ÷ (total tutoring attendees) versus (non-attendees submitting on time) ÷ (total non-attendees). If these percentages differ meaningfully, tutoring and homework timeliness are associated. The key is comparing proportions, not counts—more non-attendees might submit on time simply because that group is larger. Association exists when the likelihood of on-time submission changes based on tutoring attendance. Always use percentages to fairly compare groups of different sizes.
- A school surveyed 120 students about their preferred study location and whether they participate in an after-school club. Based on the two-way table, what is the conditional relative frequency (as a percent) of After-school club = Yes given that the student prefers studying in the Library?
Answer: 60%
This question asks for a conditional relative frequency, which means we need the proportion of students in the after-school club given they prefer the library. The key phrase "given that the student prefers studying in the Library" tells us to focus only on library students as our denominator. First, locate the library row in the table and find the total number of library students. Then find how many library students are in the after-school club. The conditional relative frequency equals (library students in club) ÷ (total library students) × 100%. A common mistake is using the grand total of 120 as the denominator instead of just the library subtotal. Remember: "given" always indicates which subgroup becomes your new denominator for the calculation.
- A school store recorded 160 purchases by whether the buyer used cash or a card, and whether the item purchased was a snack or a school supply. The two-way table shows the counts.
Which statement best describes the association between payment method and type of item purchased?
| Snack | School supply | Total |
|---|
| Cash | 50 | 30 | 80 |
| Card | 30 | 50 | 80 |
| Total | 80 | 80 | 160 |
Answer: There is an association: the percent buying snacks is higher among cash purchases than among card purchases.
This question investigates the association between payment method and item type by comparing conditional relative frequencies across methods. Calculate using row totals: for cash, snacks are 50 out of 80 or 62.5%; for card, 30 out of 80 or 37.5%. The higher snack percentage for cash implies an association. A common misconception is noting equal totals for cash and card without checking proportions within groups. To transfer this, find conditional percentages for the item within each payment type and determine if differences indicate an association.
- A school survey recorded whether students are in Band and whether they prefer Math or English. Use the two-way table to answer the question.
What is the conditional relative frequency of a student preferring Math given that the student is in Band? Give your answer as a percent.
| Prefer Math | Prefer English | Total |
|---|
| In Band | 36 | 24 | 60 |
| Not in Band | 30 | 60 | 90 |
| Total | 66 | 84 | 150 |
Answer: 60%
The concept here is conditional relative frequency, which examines the proportion of one category given another specific category, focusing on association in two-way tables. For conditional relative frequency, the denominator is the total of the given subgroup, not the grand total. Locate the row for 'In Band' and the cell for 'Prefer Math' which is 36, with the subgroup total of 60. Compute the frequency as 36 divided by 60, resulting in 0.6 or 60%. This indicates that among students in Band, 60% prefer Math. A common misconception is using the grand total of 150 as the denominator instead of the conditional subgroup total, which would incorrectly give 24%. To transfer this strategy, interpret 'given that the student is in Band' as using the Band total of 60, then form the ratio of the relevant cell to that total.
- Students were asked whether they have a part-time job and whether they participate in a school club. Use the two-way table to answer the question.
Which statement best describes the association between having a part-time job and participating in a school club?
| In a Club | Not in a Club | Total |
|---|
| Has a Job | 28 | 52 | 80 |
| No Job | 60 | 20 | 80 |
| Total | 88 | 72 | 160 |
Answer: There is an association: students with a job are less likely to be in a club than students with no job.
The concept here is association in two-way tables, determined by comparing conditional relative frequencies across groups to see if proportions differ meaningfully. For association, calculate conditional frequencies using subgroup totals as denominators, such as job status totals. Locate the cells for 'In a Club': 28 for has job (total 80) and 60 for no job (total 80). Compute 28/80 = 35% and 60/80 = 75%, showing students with a job are less likely to be in a club (35%) than those without (75%), implying a negative association. This difference in conditional frequencies suggests an association between having a job and club participation. A common misconception is comparing raw counts like 28 vs 60 without proportions, ignoring equal subgroup totals. To transfer this, translate 'association' into comparing conditionals like 'given job' vs 'given no job', and interpret differences as evidence of association.
- Students were asked whether they prefer working alone or in a group and whether they are in an AP class. Use the two-way table to answer the question.
What is the conditional relative frequency of preferring group work given that the student is in an AP class? Give your answer as a decimal.
| Prefer Alone | Prefer Group | Total |
|---|
| AP Class | 18 | 42 | 60 |
| Not AP | 54 | 36 | 90 |
| Total | 72 | 78 | 150 |
Answer: 0.70
The concept here is conditional relative frequency, which calculates the proportion of one attribute given a specific condition in two-way tables. For conditional relative frequency, the denominator is the total of the given group, such as AP class status. Locate the row for 'AP Class' and the cell for 'Prefer Group' which is 42, with the subgroup total of 60. Compute 42 divided by 60, equaling 0.70. This means 70% of AP students prefer group work. A common misconception is using the grand total of 150, yielding 42/150 = 0.28 incorrectly. To transfer this, translate 'given that the student is in an AP class' to using the AP total of 60, then form the ratio with the prefer group cell.
- A school tracked whether students usually eat breakfast and whether they arrive to first period on time. The results are shown in the two-way table.
What is the conditional relative frequency (as a percent) of students who arrive on time given that the student skips breakfast?
| On Time | Late | Total |
|---|
| Eats Breakfast | 70 | 30 | 100 |
| Skips Breakfast | 30 | 50 | 80 |
| Total | 100 | 80 | 180 |
Answer: 37.5%
Conditional relative frequency determines the proportion arriving on time given skipping breakfast. The denominator is the subgroup total for skips breakfast (80 students). Locate the cell for skips and on time (30 students) and the subgroup total (80). Compute the frequency as 30 divided by 80, which equals 0.375 or 37.5%. A common misconception is using the on time column total (100) instead, giving 30/100 = 30%. To transfer this strategy, interpret 'given that the student skips breakfast' as using the skips total (80) as denominator, then divide the relevant cell (30) by it for the ratio.
- A student council collected data on whether students prefer school events on weekdays or weekends and whether they are involved in student government. The results are shown in the two-way table.
What is the conditional relative frequency (as a decimal) of students who prefer weekends given that the student is involved in student government?
| Prefer Weekdays | Prefer Weekends | Total |
|---|
| Student Government: Yes | 18 | 42 | 60 |
| Student Government: No | 55 | 35 | 90 |
| Total | 73 | 77 | 150 |
Answer: 0.70
Conditional relative frequency finds the proportion preferring weekends given involvement in student government. The denominator is the subgroup total for student government yes (60 students). Locate the cell for yes and prefer weekends (42 students) and the subgroup total (60). Compute the frequency as 42 divided by 60, which equals 0.70. A common misconception is using the column total (77) instead, which would give 42/77 ≈ 0.545. To transfer this strategy, translate 'given that the student is involved in student government' to use the yes total (60) as denominator, then divide the relevant cell (42) by it for the ratio.
- A cafeteria tracked 200 student purchases by drink choice and whether the student bought a dessert. Based on the two-way table, what is the conditional relative frequency (as a percent) of Bought dessert = Yes given that the student chose Water?
Answer: 30%
To find the conditional relative frequency of buying dessert given water choice, focus only on water drinkers as your population. The phrase "given that the student chose Water" identifies this subgroup. Locate the water row and find both the total water drinkers and how many bought dessert. Calculate: (water drinkers buying dessert) ÷ (total water drinkers) × 100%. Don't use the full 200 students—conditional frequencies restrict to a specific subgroup. This reveals whether water drinkers have different dessert-purchasing habits compared to other drink choices. Remember to convert your decimal answer to a percentage.
- A school surveyed 140 students about their main mode of transportation and whether they arrive on time. Based on the two-way table, which statement best describes the association between transportation mode and arriving on time?
Answer: There is an association because the percent arriving on time is higher for Bus riders than for Car riders.
Association between transportation mode and arriving on time requires comparing conditional frequencies across different transportation types. Calculate the on-time percentage for each mode: (Bus riders on time) ÷ (total Bus riders) and (Car riders on time) ÷ (total Car riders). If these percentages differ substantially, an association exists between transportation and punctuality. The correct comparison uses proportions within each transportation group, not raw counts. A higher count for one group might simply reflect more students using that transportation mode. Focus on whether the on-time rate changes with transportation method, not absolute numbers.
- A cafeteria tracked whether students chose pizza or salad for lunch and whether they are in Grade 9 or Grade 10. The results are shown in the two-way table.
Which statement best describes the association between grade level and lunch choice?
| Pizza | Salad | Total |
|---|
| Grade 9 | 54 | 26 | 80 |
| Grade 10 | 36 | 54 | 90 |
| Total | 90 | 80 | 170 |
Answer: There is an association: Grade 10 students have a higher proportion choosing salad than Grade 9 students.
Association in two-way tables is determined by comparing conditional relative frequencies across groups to see if proportions differ meaningfully between grade levels for lunch choices. Calculate conditional frequencies using subgroup totals: for grade 9, salad is 26/80 = 0.325; for grade 10, salad is 54/90 = 0.60. Locate cells for salad in each grade (26 and 54) and divide by grade totals (80 and 90). The computation shows grade 10 has a higher proportion choosing salad (60%) than grade 9 (32.5%), implying an association where grade level relates to preference for salad over pizza. For association, compare these conditional frequencies across groups; a difference suggests the variables are related rather than independent. A common misconception is judging association by counts alone, like more pizza overall, instead of proportions within groups. To transfer, translate comparisons by computing ratios within each row total and noting if they differ substantially.
- At a high school, students were surveyed about whether they participate in an after-school club and whether they prefer studying in the morning or evening. The results are shown in the two-way table.
What is the conditional relative frequency (as a percent) of students who prefer studying in the morning given that the student participates in an after-school club?
| Prefer Morning | Prefer Evening | Total |
|---|
| Club: Yes | 28 | 52 | 80 |
| Club: No | 40 | 30 | 70 |
| Total | 68 | 82 | 150 |
Answer: 35.0%
Conditional relative frequency measures the proportion of one category within a specific subgroup, here the percentage of students preferring morning study given they participate in a club. The denominator is the subgroup total, which is the total for club participants (80 students). Locate the cell for club yes and prefer morning (28 students) and the subgroup total (80). Compute the frequency as 28 divided by 80, which equals 0.35 or 35%. A common misconception is using the overall total (150) instead of the subgroup, which would incorrectly give 28/150 ≈ 18.7%. To transfer this strategy, interpret 'given that the student participates in an after-school club' as using the club yes total (80) as the denominator, then divide the relevant cell (28) by it to find the ratio.
- A teacher recorded whether students turned in an assignment on time and whether they used a planner. The results are shown in the two-way table.
Which statement best describes the association between using a planner and turning in the assignment on time?
| On Time | Late | Total |
|---|
| Planner: Yes | 72 | 18 | 90 |
| Planner: No | 36 | 54 | 90 |
| Total | 108 | 72 | 180 |
Answer: There is an association: a higher proportion of planner users turn in the assignment on time than non-users.
Association is evaluated by comparing conditional relative frequencies across groups to check if turning in on time differs by planner use. Calculate conditional frequencies using subgroup totals: for planner yes, on time is 72/90 = 0.80; for no, it's 36/90 = 0.40. Locate cells for on time in each group (72 and 36) and divide by group totals (90 each). The computation shows planner users have a higher proportion on time (80%) than non-users (40%), implying an association between planner use and timeliness. For association, compare these conditional frequencies across groups; the difference suggests the variables are connected. A common misconception is using overall counts, like more on time total, instead of within-group proportions. To transfer, compute ratios within each row and compare to identify differences in behavior by group.
- Students in a school were asked whether they usually do homework at home or at the library and whether they have a part-time job. The results are shown in the two-way table.
Which value represents the joint relative frequency (as a decimal) of a student who has a part-time job and usually does homework at the library?
| Homework at Home | Homework at Library | Total |
|---|
| Part-time Job: Yes | 30 | 45 | 75 |
| Part-time Job: No | 60 | 15 | 75 |
| Total | 90 | 60 | 150 |
Answer: 0.30
Joint relative frequency represents the proportion of the total sample that falls into the intersection of two categories, such as having a part-time job and doing homework at the library. The denominator is the overall total (150 students). Locate the cell at the intersection of part-time job yes and homework at library (45 students) and the grand total (150). Compute the frequency as 45 divided by 150, which equals 0.30. A common misconception is confusing joint with conditional frequency by using a row or column total (like 75) instead, which would give 45/75 = 0.60. To transfer this strategy, identify the joint as the specific cell divided by the grand total to find the overall proportion.
- In a PE class, students reported whether they prefer team sports or individual sports and whether they participate in a school sport team. The results are shown in the two-way table.
Which statement best describes the association between participation on a school sport team and sports preference?
| Prefer Team Sports | Prefer Individual Sports | Total |
|---|
| On School Team: Yes | 48 | 12 | 60 |
| On School Team: No | 30 | 60 | 90 |
| Total | 78 | 72 | 150 |
Answer: There is an association: students on a school team have a higher proportion preferring team sports than students not on a team.
Association is assessed by comparing conditional relative frequencies across groups to determine if sports preference differs based on team participation. Calculate conditional frequencies using subgroup totals: for on team, prefer team sports is 48/60 = 0.80; for not on team, it's 30/90 ≈ 0.333. Locate cells for prefer team sports in each group (48 and 30) and divide by group totals (60 and 90). The computation shows students on a team have a higher proportion preferring team sports (80%) than those not on a team (33.3%), implying an association between participation and preference. For association, compare these conditional frequencies across groups; a substantial difference indicates the variables are related. A common misconception is basing association on total counts, like more not on team overall, rather than proportions within groups. To transfer, compute ratios within each row total and compare to see if preferences vary by group.
- A school recorded whether students prefer watching movies at home or in a theater and whether they have a streaming subscription. The results are shown in the two-way table.
What is the conditional relative frequency (as a percent) of students who prefer the theater given that the student does not have a streaming subscription?
| Prefer Home | Prefer Theater | Total |
|---|
| Streaming: Yes | 58 | 22 | 80 |
| Streaming: No | 24 | 56 | 80 |
| Total | 82 | 78 | 160 |
Answer: 70.0%
Conditional relative frequency calculates the proportion within a conditioned subgroup, here preferring theater given no streaming subscription. The denominator is the subgroup total for no streaming (80 students). Locate the cell for no streaming and prefer theater (56 students) and the subgroup total (80). Compute the frequency as 56 divided by 80, which equals 0.70 or 70%. A common misconception is using the column total (78) instead of the row, which would incorrectly give 56/78 ≈ 71.8%. To transfer this strategy, interpret 'given that the student does not have a streaming subscription' as using the no streaming total (80) as the denominator, then divide the relevant cell (56) by it to build the ratio.
- A school surveyed 180 students about whether they prefer studying alone or in a group, and whether they are in Grade 9 or Grade 12. The two-way table shows the counts.
What is the conditional relative frequency (as a percent) of students who prefer studying in a group given that the student is in Grade 12?
| Prefer alone | Prefer group | Total |
|---|
| Grade 9 | 54 | 36 | 90 |
| Grade 12 | 36 | 54 | 90 |
| Total | 90 | 90 | 180 |
Answer: 60.0%
This question seeks the conditional relative frequency of students who prefer studying in a group given that they are in Grade 12, emphasizing the proportion within that grade's subgroup. For conditional frequencies, use the subgroup total as the denominator, here 90 for Grade 12. Find the cell for Grade 12 students preferring a group, which is 54, and the row total of 90. Calculate 54 divided by 90, resulting in 0.6 or 60%. A common misconception is using counts like 54 without dividing by the subgroup, or mistakenly using the column total of 90 for groups overall. To transfer this, interpret 'given Grade 12' as restricting to that row's total, then form the ratio of the relevant cell to that total.
- A school collected data on whether students ride the bus or get a ride from family, and whether they arrive early or not. Use the two-way table to answer the question.
Which statement best describes the association between transportation method and arriving early?
| Arrives Early | Does Not Arrive Early | Total |
|---|
| Rides Bus | 35 | 65 | 100 |
| Family Ride | 45 | 15 | 60 |
| Total | 80 | 80 | 160 |
Answer: There is an association: students with a family ride are more likely to arrive early than students who ride the bus.
The concept here is association in two-way tables, assessed by comparing conditional relative frequencies between groups to detect differences in proportions. For association, use subgroup totals as denominators for each transportation method. Locate the 'Arrives Early' cells: 35 for bus (total 100) and 45 for family (total 60). Compute 35/100 = 35% and 45/60 = 75%, indicating students with family rides are more likely to arrive early (75%) than bus riders (35%), suggesting a positive association for family rides. This disparity implies an association between transportation and early arrival. A common misconception is comparing totals like early vs not early without conditioning on groups. To transfer this, compare conditionals like 'given bus' vs 'given family', with differences signaling association.
