Construct and Interpret Two-Way Tables - 8th Grade Math

1. A group of 100 students was asked whether they prefer working in groups (Yes/No) and whether they like math (Yes/No). Results: Of the 60 students who like math, 36 prefer group work and 24 do not. Of the 40 students who do not like math, 12 prefer group work and 28 do not. What percent of students who do not like math prefer group work?
   Answer: $\frac{12}{40}=30\%$ This question tests calculating a row relative frequency in a two-way table for a conditional percentage. For don't like math (total 40), group work yes is 12/40=30%. Data: like math (60 total,36 yes group), don't (40,12 yes), focusing on conditional for don't like. Choice B is correct, using the proper row total. Errors: joint (A:12/100=12%), column (C:12/48=25%), marginal (D:40/100=40%). Divide cell by row total for percentage. Common issues: mixing conditional with overall or wrong denominators.
2. A teacher collected data from 80 students about grade level (6th or 8th) and whether they are in an after-school activity (Yes/No): - 6th grade: 25 in activities, 15 not in activities - 8th grade: 30 in activities, 10 not in activities How many students in the survey are not in an after-school activity?
   Answer: 25 This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Two-way table: rows for one variable (grade: 6th/8th), columns for other (activity: yes/no), cells show counts (25 students: 6th AND activity yes). To find total not in activity, add the 'no' cells: 15 (6th no) + 10 (8th no) = 25. Construction: identify variables, count combinations, organize in table with totals (e.g., row totals 40 each, column no total 25, grand 80). The correct number is 15+10=25, as in choice D, showing the addition of the relevant cells. A common error is confusing categories, like adding row totals in choice B (40) or grand total in choice C (80), or switching yes/no like in an inverted table. For association, compare row relatives (e.g., 25/40=62.5% vs 30/40=75%), but here focus on extracting totals from described data.
3. A student club surveyed 100 students about grade level and whether they participate in an after-school activity. - 6th grade: 25 participate, 15 do not - 8th grade: 30 participate, 10 do not Which statement best describes the association (if any) between grade level and participating in an after-school activity?
   Answer: There appears to be an association because $30/40=75\%$ of 8th graders participate, compared with $25/40=62.5\%$ of 6th graders. This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Row relatives indicate conditionals, such as 30/40 = 75% of 8th graders participate versus 25/40 = 62.5% of 6th graders, with the 12.5% difference suggesting association between grade and participation, while equal rates imply independence. For this 100-student survey (totals: 6th 40, 8th 40), choice A correctly identifies association by comparing row relatives (75% vs 62.5%) without causation. This matches the data's conditional rates. Errors: claiming no association from equal group sizes (B), using grand totals (25/100, 30/100 in C), or causation (D). For association: compare relatives across rows (differences suggest relation). Mistakes: ignoring conditionals or equating association with cause.
4. A school surveyed students about breakfast habits and academic performance. Of 180 students, 2/3 eat breakfast regularly. Among regular breakfast eaters, 75% perform well academically. Among irregular breakfast eaters, 40% perform well academically. What percentage of all students who perform well academically are regular breakfast eaters?
   Answer: 78.9% This is a conditional probability problem that requires you to work backwards from the given information. When you see questions asking "what percentage of Group A belongs to Category B," you're looking for a conditional probability. Start by finding the actual numbers. Of 180 students, $\frac{2}{3}$ eat breakfast regularly: $180 \times \frac{2}{3} = 120$ regular breakfast eaters and $180 - 120 = 60$ irregular breakfast eaters. Next, calculate how many students in each group perform well academically. Among regular breakfast eaters: $120 \times 0.75 = 90$ students perform well. Among irregular breakfast eaters: $60 \times 0.40 = 24$ students perform well. The total number of students who perform well academically is $90 + 24 = 114$ students. The question asks: of all students who perform well academically, what percentage are regular breakfast eaters? This means: $\frac{90}{114} = 0.789 = 78.9\%$ Choice A (75.0%) represents the percentage of regular breakfast eaters who perform well, but that's not what the question asks. Choice B (66.7%) is simply the fraction of all students who eat breakfast regularly ($\frac{2}{3}$), ignoring academic performance entirely. Choice D (83.3%) might result from calculation errors, possibly confusing the conditional relationships. Remember: conditional probability questions often flip the relationship. Pay careful attention to whether you're asked "percentage of A that are B" versus "percentage of B that are A" – these give very different answers.
5. A survey of 200 middle school students examined the relationship between having a smartphone and completing homework on time. The data shows: 140 students have smartphones, 110 students complete homework on time, and 85 students both have smartphones and complete homework on time. What is the relative frequency of completing homework on time among students who do NOT have smartphones?
   Answer: $\frac{25}{60} = 0.417$ Students without smartphones: 200 - 140 = 60. Students who complete homework on time but don't have smartphones: 110 - 85 = 25. Relative frequency = 25/60 = 0.417. Choice B uses total homework completers as denominator. Choice C uses smartphone owners who complete homework as numerator and total smartphone owners as denominator. Choice D incorrectly calculates students without smartphones who don't complete homework on time as numerator.
6. A neighborhood survey asked 100 families what type of home they live in and whether they have a pet. - Apartment: 15 have pets, 35 do not - House: 40 have pets, 10 do not Which two-way table correctly shows the frequencies (counts) and totals?
   Answer: | | Pet: Yes | Pet: No | Total | |--------------|----------|---------|-------| | Apartment | 15 | 40 | 55 | | House | 35 | 10 | 45 | | Total | 50 | 50 | 100 | This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Tables use rows for one variable (e.g., housing: apartment/house) and columns for another (pet: yes/no), with cells for counts like 15 apartments with pets, and totals like 50 apartments, 55 pets. Relatives might show 15/50 = 30% of apartments have pets vs 40/50 = 80% of houses, differing rates suggesting association, similar implying independence. For 100 families (apartment: 15 yes, 35 no, total 50; house: 40 yes, 10 no, total 50; totals yes 55, no 45), choice A correctly fills the table with accurate counts and totals. Errors: wrong totals (B), swapped cells (C), incorrect counts (D). Construction: (1) identify categories, (2) place counts, (3) sum rows/columns, (4) grand total. Mistakes: switching values or misadding.
7. At a club, 72 members were asked whether they play a sport (Yes/No) and whether they play a musical instrument (Yes/No). Data: Sport Yes & Instrument Yes = 18, Sport Yes & Instrument No = 30, Sport No & Instrument Yes = 6, Sport No & Instrument No = 18. What is the relative frequency out of the total for members who play a sport and an instrument?
   Answer: $\frac{18}{72}=25\%$ This question tests calculating joint relative frequencies relative to the grand total in two-way tables. Joint relatives are cell divided by grand total, like 18/72=25% for sport yes and instrument yes out of all members. Data: all cells given, sport yes total 48, no 24, grand 72, with target joint 18/72=25%. Choice C is correct, using the grand total denominator. Errors: row relative (A:18/48=37.5%), column (B:18/24=75%), or wrong total (D:18/36=50%). Compute by dividing joint count by sample size. Avoid confusing with conditional relatives or using incorrect denominators.
8. A school counselor surveyed 50 students about whether they have a curfew and whether they have regular chores at home. Results: - Curfew: Yes (30 students): 20 with chores, 10 without chores - Curfew: No (20 students): 5 with chores, 15 without chores What is the row relative frequency of students who have chores among students who have a curfew?
   Answer: $\frac{20}{30}\approx 0.67\approx 67\%$ This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Relative frequencies include row relatives, which show conditional probabilities, such as among students with curfew (30 total), 20/30 ≈ 67% have chores, versus 5/20 = 25% for those without curfew, with differing rates suggesting an association between curfew and chores, while similar rates would indicate independence. In this example with 50 students, the row relative frequency for chores among curfew students is 20/30 ≈ 67%, as calculated in choice B, using the row total as the denominator for the conditional percentage. This is the correct interpretation, as it focuses on the proportion within the curfew-yes row. A common error is using the wrong denominator, like the grand total (20/50 = 40% in A) or column total (20/25 = 80% in C), which computes a different relative frequency. To calculate row relatives: divide the cell count by its row total (e.g., 20/30 for curfew yes and chores yes); for association, compare these across rows. Mistakes include confusing row with column relatives or arithmetic errors, such as misdividing (e.g., treating 30/50 = 60% as the row relative in D).
9. A survey of 90 students recorded whether they prefer reading fiction (Yes/No) and whether they are in band (Yes/No). Results: Of 50 band students, 30 prefer fiction and 20 do not. Of 40 non-band students, 24 prefer fiction and 16 do not. Do the variables appear to be associated? Use row relative frequencies to decide.
   Answer: No, because $30/50=60\%$ and $24/40=60\%$ are the same. This question tests detecting associations in two-way tables by comparing row relative frequencies for independence. If conditional rates are similar, like fiction preference 30/50=60% for band and 24/40=60% for non-band, variables are independent with no association. Data: band (50 total, 30 yes fiction), non-band (40 total, 24 yes), equal 60% rates indicating no association. Choice C correctly identifies this using row relatives. Errors: comparing joints (A:30/90 vs 24/90), marginals (B), or causation (D). To assess: compute relatives across rows and compare—if equal, no association. Common pitfalls: ignoring conditionals or confusing correlation with causation.
10. A student council surveyed 60 students about whether they bring lunch from home (Yes/No) and whether they buy a snack after school (Yes/No). - Of the 24 students who bring lunch from home, 18 buy a snack and 6 do not. - Of the 36 students who do not bring lunch from home, 12 buy a snack and 24 do not. Which set of row relative frequencies is correct for comparing snack-buying between the two lunch groups?
    Answer: Bring lunch: $\frac{18}{24}=75\%$ buy a snack; No lunch from home: $\frac{12}{36}=33\frac{1}{3}\%$ buy a snack. This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Two-way table: rows for one variable (bring lunch: yes/no), columns for other (buy snack: yes/no), cells show counts (18: bring yes AND snack yes). Relative frequencies: row relatives show conditional—of bring lunch students, 18/24=75% buy snack vs 12/36≈33% of no-bring students. Different rates suggest association: bringing lunch relates to buying snacks (bringers more likely). Similar rates suggest independence. The correct set of row relatives is 18/24=75% and 12/36≈33%, as in choice A. A common error is using grand total like in choice B (18/60=30%, 12/60=20%) or inverting like in choice C (24/18=133%, 36/12=300%), or wrong cells like 6/24=25% in choice D (percent not buying).
11. A student club surveyed 100 students about grade level and whether they participate in an after-school activity. - 6th grade: 25 participate, 15 do not - 8th grade: 30 participate, 10 do not What percentage of 8th graders participate in an after-school activity?
    Answer: $\frac{30}{40}=75\%$ This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Row relatives are conditional, like for 8th graders (40 total), $30/40 = 75\%$ participate in activities versus $25/40 = 62.5\%$ for 6th graders, with differences suggesting association, while similarity means independence. Here, with 100 students (6th: 25 participate, 15 not, total 40; 8th: 30 participate, 10 not, total 40), the row relative for 8th graders participating is $30/40 = 75\%$, as in B. This uses the correct row total denominator for the conditional percentage. Common errors: using grand total ($30/100 = 30\%$ in A), inverting fraction ($40/30 \approx 133\%$ in C), or wrong row total ($30/55 \approx 55\%$ in D, perhaps miscounting). To calculate: divide cell by row total ($30/40$); compare for association. Mistakes include wrong denominators or arithmetic slips.
12. A teacher collected data from 80 students about grade level (6th or 8th) and whether they are in an after-school activity (Yes/No): - 6th grade: 25 in activities, 15 not - 8th grade: 30 in activities, 10 not Which two-way table correctly shows the frequencies (counts), including totals?
    Answer: | | Activity: Yes | Activity: No | Total | |-------------|--------------:|-------------:|------:| | 6th grade | 25 | 15 | 40 | | 8th grade | 30 | 10 | 40 | | Total | 55 | 25 | 80 | This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Two-way table: rows for one variable (grade: 6th/8th), columns for other (activity: yes/no), cells show counts (25 students: 6th AND activity yes, 15: 6th no, etc.). Construction: (1) identify variables and categories, (2) count each combination, (3) organize in table, (4) calculate totals (rows 40 each, columns 55 yes and 25 no, grand 80). The correct table is in choice A, matching the data with accurate cells and totals. A common error is switching cell values, like in choice D where yes and no are swapped, or incorrect totals like in choice B (wrong no counts leading to 35 and 45 rows). Relative frequencies: divide cell by row total for row relatives (e.g., 25/40=62.5% of 6th in activity). Mistakes: wrong denominator or not verifying totals add to 80.
13. A class surveyed 60 students about whether they prefer fiction or nonfiction and whether they read at least 20 minutes per day. - Fiction: 18 read 20+ minutes, 12 do not - Nonfiction: 10 read 20+ minutes, 20 do not How many students in the survey read at least 20 minutes per day?
    Answer: 28 This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Tables organize variables like preference (fiction/nonfiction) and reading time (20+ yes/no), with cells e.g., 18 fiction and 20+, totals enabling relatives like 18/30 = 60% of fiction preferers read 20+ vs 10/30 ≈ 33% nonfiction, difference suggesting association, similarity independence. For 60 students (fiction: 18 yes, 12 no, total 30; nonfiction: 10 yes, 20 no, total 30), total reading 20+ is column sum 18+10 = 28, as in B. This correctly adds the relevant cells. Errors: single cell (18 in A), wrong sum (30 in C), grand total (60 in D). Construction: fill cells, sum columns for totals like reading yes. Mistakes: adding incorrect cells or confusing variables.
14. A survey asked 150 teenagers about their social media usage and sleep quality. The data shows that 40% of teens use social media heavily, and among heavy users, 65% report poor sleep quality. Among light users, 30% report poor sleep quality. How many teens in the survey are light users with good sleep quality?
    Answer: 63 teens First, find the number of light users: 150 × (1 - 0.40) = 90 teens. Among light users, 30% have poor sleep quality, so 70% have good sleep quality. Therefore: 90 × 0.70 = 63 teens are light users with good sleep quality. Choice A incorrectly uses 70% of heavy users. Choice C gives total light users, not those with good sleep. Choice D gives light users with poor sleep quality (90 × 0.30).
15. A neighborhood survey asked 100 families what type of home they live in and whether they have a pet. - Apartment: 15 have pets, 35 do not - House: 40 have pets, 10 do not What is the row relative frequency of having a pet for families who live in a house?
    Answer: $\frac{40}{50}=80\%$ This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Row relatives are conditional, like for houses (50 total), 40/50 = 80% have pets versus 15/50 = 30% for apartments, differences suggesting association, similarity meaning independence. In this 100-family survey (house: 40 pets, 10 no), the row relative for pets in houses is 40/50 = 80%, as in B, using row total. This correctly conditions on housing type. Errors: grand total (40/100 = 40% in A), column total (40/55 ≈ 73% in C), inverted (50/40 = 125% in D). Calculate by dividing cell by row total; compare for association. Mistakes: wrong denominator or miscalculation.
16. A school surveyed 80 students about grade level (6th or 8th) and whether they participate in an extracurricular activity (Yes/No). Results: 6th grade: 25 Yes, 15 No. 8th grade: 30 Yes, 10 No. What percent of 8th graders participate in an activity?
    Answer: $\frac{30}{40}=75\%$ This question tests interpreting two-way table data to calculate a row relative frequency as a percentage within a specific category. For 8th graders (inferred total 40 from 30 yes + 10 no), the percentage participating is $30/40=75\%$, a conditional rate. Data: 6th grade (25 yes, 15 no, total 40), 8th (30 yes, 10 no, total 40), grand 80, focusing on 8th graders' participation. Choice B is correct, using the row total for 8th graders. Errors: overall proportion (A: $30/80=37.5\%$ ), wrong denominator like total yes (C: $30/55 \approx 54.5\%$ ), or marginal (D: $40/80=50\%$ ). Calculate by dividing the cell by its row total and converting to percent. Avoid using grand or column totals for row-specific questions.
17. A survey of 100 people recorded housing type (Apartment/House) and whether they own a pet (Yes/No). Results: In apartments, 15 people have pets and 35 do not. In houses, 40 people have pets and 10 do not. Based on row relative frequencies, which statement is best supported?
    Answer: There is no association because both housing types have 50 people total. This question tests identifying associations in two-way tables by comparing row relative frequencies to assess if variables are related. Row relatives show conditional rates, like percentage with pets within each housing type: apartments 15/50=30%, houses 40/50=80%, with differing rates suggesting association. Data: apartments (15 pets yes, 35 no, total 50), houses (40 yes, 10 no, total 50), so houses have higher pet ownership rate (80% vs 30%). Choice B correctly interprets this association using proper row relatives. Common errors: wrong denominators (A uses grand total), ignoring relatives (C focuses on equal totals), or claiming causation (D). To check association: compute row relatives for the outcome variable across categories and compare—if rates differ substantially, variables are associated. Mistakes include using joint frequencies without conditionals or inferring causation from correlation.
18. A school counselor surveyed 50 students about whether they have a curfew and whether they have regular chores at home. Results: - Curfew: Yes (30 students): 20 with chores, 10 without chores - Curfew: No (20 students): 5 with chores, 15 without chores Based on row relative frequencies, do curfew and chores appear to be associated?
    Answer: Yes. A higher percent of students with a curfew have chores ($20/30\approx 67\%$) than students without a curfew ($5/20=25\%$). This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Row relatives show conditionals like 20/30 ≈ 67% of curfew students have chores versus 5/20 = 25% without, with the difference (67% vs 25%) suggesting association between curfew and chores, as similar rates would indicate independence. In this 50-student example, comparing row relatives (67% vs 25%) shows a higher percentage with curfew have chores, indicating association without claiming causation, as in choice B. This is the correct interpretation based on conditional rates. Errors include ignoring conditionals by using grand totals (20/50 and 5/50 in A), misinterpreting balanced totals as independence (C), or claiming causation (D). For association: compare row relatives across rows (differing suggests related variables). Mistakes: not comparing conditionals or confusing association with causation (association shows correlation, not cause).
19. A neighborhood survey asked 100 families what type of home they live in and whether they have a pet. - Apartment: 15 have pets, 35 do not - House: 40 have pets, 10 do not Which statement best describes the association (if any) between housing type and pet ownership?
    Answer: Yes, housing type causes pet ownership because 80% is larger than 30%. This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Row relatives show conditionals like 15/50 = 30% of apartments have pets versus 40/50 = 80% of houses, the large difference suggesting association between housing and pets, while equal rates indicate independence. For 100 families (equal 50 each housing), choice B correctly states association by comparing row relatives (30% vs 80%) without causation. This fits the conditional analysis. Errors: no association from equal totals (A), irrelevant sums (C), causation claim (D). Association via comparing relatives across rows. Mistakes: overlooking conditionals or confusing with cause.
20. A neighborhood survey asked 100 families what type of home they live in (Apartment or House) and whether they have a pet (Yes/No). - Apartment: 15 have pets, 35 do not (50 total) - House: 40 have pets, 10 do not (50 total) Which conclusion is best supported by the data?
    Answer: Pet ownership appears associated with housing type because $\frac{15}{50}=30\%$ of apartment families have pets, but $\frac{40}{50}=80\%$ of house families have pets. This question tests constructing two-way tables for categorical data, calculating row/column relative frequencies (percentages), and identifying associations by comparing conditional rates. Two-way table: rows for one variable (housing: apartment/house), columns for other (pet: yes/no), cells show counts (15: apartment yes). Relative frequencies: row relatives show conditional—of apartment families, 15/50=30% have pets vs 40/50=80% of house families. Different rates (30% vs 80%) suggest association: housing type relates to pet ownership (houses more likely). Similar rates suggest independence. The correct conclusion is association due to differing rates, as in choice B, without claiming causation. A common error is claiming causation like in choice C, or misinterpreting equal row totals as no association in choice A.

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