Statistics Quiz: Summarizing Categorical Data Two Way Tables

What this quiz covers

This quiz focuses on Summarizing Categorical Data Two Way Tables, giving you a quick way to practice the rules, question types, and explanations that matter most for Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. 35.0%
2. 43.8%
3. 70.0% (correct answer)
4. 56.0%

Explanation: 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.

Question 2

1. 0.56
2. 0.28
3. 0.70 (correct answer)
4. 0.42

Explanation: 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.

Question 3

1. 16.7%
2. 55.6%
3. 37.5% (correct answer)
4. 30.0%

Explanation: 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.

Question 4

1. There is an association: Grade 10 students have a higher proportion choosing salad than Grade 9 students. (correct answer)
2. There is an association: Grade 9 students have a higher proportion choosing salad than Grade 10 students.
3. There is no association because the totals for Grade 9 and Grade 10 are different.
4. There is little to no association because the number choosing pizza is larger than the number choosing salad overall.

Explanation: 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.

Question 5

1. 18.0%
2. 29.0%
3. 30.0%
4. 11.25% (correct answer)

Explanation: Joint relative frequency is the proportion of the entire sample in the intersection of visited library and prefers nonfiction. The denominator is the grand total (160 students). Locate the cell for visited yes and prefer nonfiction (18 students) and the overall total (160). Compute the frequency as $18 \div 160$, which equals 0.1125 or 11.25%. A common misconception is using a marginal total like the nonfiction column (62) instead, giving $\frac{18}{62} \approx 29\%$. To transfer this strategy, find the joint by dividing the intersection cell by the grand total for the overall proportion.

Question 6

1. There is no association because the totals for planner users and non-users are the same.
2. There is an association: a higher proportion of non-users turn in the assignment on time than planner users.
3. There is an association: a higher proportion of planner users turn in the assignment on time than non-users. (correct answer)
4. There is little to no association because more students turned in the assignment on time than late overall.

Explanation: 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.

Question 7

1. 30.0%
2. 50.0%
3. 60.0% (correct answer)
4. 54.0%

Explanation: 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.

Question 8

1. There is no association because the number of fiction checkouts is greater than nonfiction checkouts overall.
2. There is no association because more students visited after school than at lunch.
3. There is an association: the percent checking out fiction is higher at lunch than after school.
4. There is an association: the percent checking out fiction is higher after school than at lunch. (correct answer)

Explanation: This question examines the association between visit time and book type by comparing conditional relative frequencies across the time groups. Use row totals as denominators for each subgroup: for lunch, fiction is 30 out of 50 or 60%; for after school, 70 out of 90 or about 77.8%. The higher percentage for after school implies an association favoring fiction then. A common misconception is relying on totals like more after-school visits without proportional comparisons. To transfer this, compute conditional percentages for the outcome within each category and assess if differences suggest an association.

Question 9

1. 60%
2. 57.1%
3. 80%
4. 48% (correct answer)

Explanation: This question asks for the joint relative frequency of students who ride the bus and arrive on time, representing that intersection's proportion of the entire sample. The denominator for joint frequencies is the grand total, here 100 students. Identify the cell for bus riders on time, which is 48, and divide by 100 to get 0.48 or 48%. A common misconception is confusing this with conditional by using the bus row total of 60, yielding 48/60 = 80%. To transfer this strategy, locate the joint cell at the categories' intersection and divide by the table's overall total for the joint proportion.

Question 10

1. 0.675
2. 0.438 (correct answer)
3. 0.700
4. 0.240

Explanation: This question requires the joint relative frequency of students who completed the packet and passed the quiz, which is the proportion of that specific combination relative to the overall total. For joint relative frequencies, the denominator is the grand total of all students, here 96. Locate the cell at the intersection of 'completed packet' and 'passed quiz,' which is 42. Compute the frequency as 42 divided by 96, equaling approximately 0.438. A common misconception is using a row or column total as the denominator instead of the grand total, such as 42/60 = 0.7, which would be conditional. To transfer this strategy, identify the joint cell and divide by the table's overall total to find the joint proportion.

Question 11

1. There is no association because more students chose pizza than salad in both school levels.
2. There is an association: the percent choosing salad is higher among high school students than among middle school students. (correct answer)
3. There is an association: the percent choosing salad is higher among middle school students than among high school students.
4. There is no association because the number of high school students is larger than the number of middle school students.

Explanation: This question explores the association between school level and lunch choice by comparing conditional relative frequencies across groups. For association, calculate percentages within each school level subgroup, using the row totals as denominators. For middle school, the salad choice is 15 out of 60, or 25%; for high school, it is 40 out of 90, or about 44.4%. The difference implies an association where salad preference is higher among high school students. A common misconception is comparing raw counts like 45 pizza versus 15 salad without proportions, ignoring group sizes. To transfer this, translate comparisons by computing conditional frequencies for each category and noting if they differ meaningfully across groups.

Question 12

1. There is no association because the totals for cash and card are the same.
2. There is an association: the percent buying snacks is higher among card purchases than among cash purchases.
3. There is an association: the percent buying snacks is higher among cash purchases than among card purchases. (correct answer)
4. There is no association because the total number of snacks equals the total number of school supplies.

Explanation: 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.

Question 13

1. There is little to no association because the percent who bring lunch from home is about the same in Grade 9 and Grade 10.
2. There is no association because the total number of students is 150.
3. There is an association because more Grade 10 students (in counts) bring lunch from home than Grade 9 students.
4. There is an association because a higher percent of Grade 9 students bring lunch from home than Grade 10 students. (correct answer)

Explanation: To determine association between grade level and bringing lunch from home, we must compare conditional relative frequencies across the two grade levels. Association exists when the percent bringing lunch differs meaningfully between Grade 9 and Grade 10 students. Calculate the conditional frequency for each grade: (Grade 9 students bringing lunch) ÷ (total Grade 9 students) versus (Grade 10 students bringing lunch) ÷ (total Grade 10 students). If these percentages are notably different, there's an association. A common error is comparing raw counts instead of percentages—Grade 10 might have more students bringing lunch simply because there are more Grade 10 students total. Always use proportions, not counts, when assessing association between categorical variables.

Question 14

1. There is no association because each transportation mode has a different row total.
2. There is an association because the percent arriving on time is higher for Bus riders than for Car riders. (correct answer)
3. There is no association because the number arriving on time is higher for Car riders than for Bus riders.
4. There is an association because most students in the survey arrive on time overall.

Explanation: 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.

Question 15

1. 75% (correct answer)
2. 65%
3. 50%
4. 35%

Explanation: 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.

Question 16

1. 60% (correct answer)
2. 30%
3. 40%
4. 50%

Explanation: 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.

Question 17

1. There is an association because the number who turned in homework on time is higher for students who did not attend tutoring.
2. There is no association because the grand total is 130.
3. There is no association because more students did not attend tutoring than attended tutoring.
4. 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. (correct answer)

Explanation: 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.

Question 18

1. 20%
2. 40%
3. 60%
4. 30% (correct answer)

Explanation: 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.

Question 19

1. There is no association because more students prefer afternoon practice overall.
2. There is an association: students who play a fall sport are more likely to prefer morning practice than students who do not play. (correct answer)
3. There is an association: students who do not play a fall sport are more likely to prefer morning practice than students who play.
4. There is no association because the totals for plays vs does not play are different.

Explanation: The concept here is association in two-way tables, evaluated by contrasting conditional relative frequencies across categories to identify proportional differences. For association, denominators are the totals for each sport participation group. Locate the 'Prefer Morning' cells: 44 for plays (total 80) and 30 for does not play (total 100). Compute 44/80 = 55% and 30/100 = 30%, showing fall sport players are more likely to prefer morning (55%) than non-players (30%), indicating a positive association. This difference suggests an association between playing a fall sport and morning preference. A common misconception is using raw counts like 44 vs 30 without adjusting for different group sizes. To transfer this, compare conditionals such as 'given plays' vs 'given does not', interpreting gaps as association evidence.

Question 20

1. 24%
2. 40%
3. 55%
4. 60% (correct answer)

Explanation: 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.